20:
132:
677:
402:
186:
666:
397:{\displaystyle {\begin{array}{cc|cccccc}&k&0&1&2&3&4&5\\n&&\\\hline 0&&1\\1&&1&2\\2&&1&3&4\\3&&1&4&7&8\\4&&1&5&11&15&16\\5&&1&6&16&26&31&32\end{array}}}
458:
411:, each component of Bernoulli's triangle is the sum of two components of the previous row, except for the last number of each row, which is double the last number of the previous row. For example, if
191:
126:
442:
661:{\displaystyle B_{n,k}={\begin{cases}1&{\mbox{if }}n=0\\B_{n-1,k}+B_{n-1,k-1}&{\mbox{if }}k<n\\2B_{n-1,k-1}=2^{n}&{\mbox{if }}k=n\end{cases}}}
943:
681:
688:
As in Pascal's triangle and other similarly constructed triangles, sums of components along diagonal paths in
Bernoulli's triangle result in the
963:
968:
735:
786:
152:
136:
700:
68:
39:
408:
148:
486:
754:
831:(4) (1968) 221–234; Hoggatt, Jr, V. E., Convolution triangles for generalized Fibonacci numbers,
19:
848:
131:
414:
689:
676:
23:
Derivation of
Bernoulli's triangle (blue bold text) from Pascal's triangle (pink italics)
957:
35:
849:
Links
Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers
716:
774:
180:
th-order binomial coefficients. The first rows of
Bernoulli's triangle are:
813:
675:
130:
31:
18:
942:
The sequence of numbers formed by
Bernoulli's triangle on the
947:
925:
911:
897:
883:
869:
734:= 4) gives the maximum number of regions in the problem of
654:
824:
Hoggatt, Jr, V. E., A new angle on Pascal's triangle,
636:
570:
495:
753:+ 1)th column gives the maximum number of regions in
715:= 3) is the three-dimensional analogue, known as the
461:
417:
189:
71:
699:= 2) is a triangular number plus one, it forms the
660:
436:
396:
120:
16:Array of partial sums of the binomial coefficients
109:
96:
695:As the third column of Bernoulli's triangle (
672:Sequences derived from the Bernoulli triangle
121:{\displaystyle \sum _{p=0}^{k}{n \choose p},}
8:
169:ordered partitions form Bernoulli's triangle
944:On-Line Encyclopedia of Integer Sequences
814:On-Line Encyclopedia of Integer Sequences
682:On-Line Encyclopedia of Integer Sequences
635:
627:
596:
569:
543:
518:
494:
481:
466:
460:
422:
416:
190:
188:
108:
95:
93:
87:
76:
70:
806:
7:
147: +1 ordered partitions form
100:
14:
42:. For any non-negative integer
785:. It also gives the number of
711:≥ 2. The fourth column (
137:compositions
1:
964:Factorial and binomial topics
444:denotes the component in row
853:Journal of Integer Sequences
847:Neiter, D. & Proag, A.,
736:dividing a circle into areas
173:i.e., the sum of the first
985:
948:https://oeis.org/A008949
789:(ordered partitions) of
701:lazy caterer's sequence
684:in Bernoulli's triangle
437:{\displaystyle B_{n,k}}
54:, the component in row
50:included between 0 and
772:− 1)-dimensional
685:
662:
438:
398:
170:
122:
92:
24:
679:
663:
439:
399:
134:
123:
72:
40:binomial coefficients
22:
969:Triangles of numbers
797:+ 1 or fewer parts.
459:
415:
187:
69:
46:and for any integer
28:Bernoulli's triangle
833:Fibonacci Quarterly
826:Fibonacci Quarterly
680:Sequences from the
838:(2) (1970) 158–171
758:-dimensional space
742:+ 1 points, where
730:The fifth column (
686:
658:
653:
640:
574:
499:
434:
394:
392:
171:
135:As the numbers of
118:
25:
749:In general, the (
690:Fibonacci numbers
639:
573:
498:
409:Pascal's triangle
151:, the numbers of
149:Pascal's triangle
107:
976:
930:
929:
926:"A008861 - Oeis"
922:
916:
915:
912:"A006261 - Oeis"
908:
902:
901:
898:"A000127 - Oeis"
894:
888:
887:
884:"A000125 - Oeis"
880:
874:
873:
870:"A000124 - Oeis"
866:
860:
845:
839:
822:
816:
811:
773:
766:
667:
665:
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632:
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127:
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159: +1 into
143: +1 into
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67:
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17:
12:
11:
5:
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952:
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937:External links
935:
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859:(2016) 16.8.3.
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478:
473:
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455:
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451:
447:
429:
426:
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419:
410:
407:Similarly to
387:
382:
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367:
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356:
349:
344:
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334:
329:
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316:
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306:
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115:
104:
101:
88:
83:
80:
77:
73:
65:
64:
63:
62:is given by:
61:
57:
53:
49:
45:
41:
37:
33:
29:
21:
920:
906:
892:
878:
864:
856:
852:
843:
835:
832:
828:
825:
820:
809:
794:
790:
787:compositions
782:
778:
769:
762:
755:
750:
748:
743:
739:
731:
729:
724:
723:cuts, where
720:
717:cake numbers
712:
708:
707:cuts, where
704:
696:
694:
687:
449:
445:
406:
177:
174:
172:
165:
164:
160:
156:
153:compositions
144:
140:
59:
55:
51:
47:
43:
36:partial sums
27:
26:
775:hyperplanes
746:≥ 4.
727:≥ 3.
448:and column
58:and column
958:Categories
801:References
760:formed by
163: +1
793:+ 1 into
765:− 1
613:−
601:−
560:−
548:−
523:−
74:∑
781:≥
638:if
572:if
497:if
452:, then:
166:or fewer
38:of the
777:, for
719:, for
30:is an
32:array
738:for
703:for
580:<
155:of
139:of
34:of
960::
946::
857:19
855:,
851:,
692:.
388:32
383:31
378:26
373:16
350:16
345:15
340:11
950:.
928:.
914:.
900:.
886:.
872:.
836:8
829:6
795:k
791:n
783:k
779:n
770:k
768:(
763:n
756:k
751:k
744:n
740:n
732:k
725:n
721:n
713:k
709:n
705:n
697:k
649:n
646:=
643:k
629:n
625:2
621:=
616:1
610:k
607:,
604:1
598:n
594:B
590:2
583:n
577:k
563:1
557:k
554:,
551:1
545:n
541:B
537:+
532:k
529:,
526:1
520:n
516:B
508:0
505:=
502:n
490:1
484:{
479:=
474:k
471:,
468:n
464:B
450:k
446:n
430:k
427:,
424:n
420:B
368:6
363:1
357:5
335:5
330:1
324:4
317:8
312:7
307:4
302:1
296:3
289:4
284:3
279:1
273:2
266:2
261:1
255:1
248:1
242:0
233:n
226:5
221:4
216:3
211:2
206:1
201:0
196:k
178:n
175:k
161:k
157:n
145:k
141:n
116:,
110:)
105:p
102:n
97:(
89:k
84:0
81:=
78:p
60:k
56:n
52:n
48:k
44:n
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