A046935 -id:A046935 - cp's OEIS frontend

A046934 Same rule as Aitken triangle (A011971) except a(0,0)=1, a(1,0)=0.

1, 0, 1, 1, 1, 2, 2, 3, 4, 6, 6, 8, 11, 15, 21, 21, 27, 35, 46, 61, 82, 82, 103, 130, 165, 211, 272, 354, 354, 436, 539, 669, 834, 1045, 1317, 1671, 1671, 2025, 2461, 3000, 3669, 4503, 5548, 6865, 8536, 8536, 10207, 12232, 14693, 17693, 21362, 25865, 31413, 38278, 46814

Offset: 0

N. J. A. Sloane, R. K. Guy

First differences of A046935 = this triangle seen as flattened list without the initial term. - Reinhard Zumkeller, Nov 10 2013

			[0] [   1]
[1] [   0,     1]
[2] [   1,     1,     2]
[3] [   2,     3,     4,     6]
[4] [   6,     8,    11,    15,    21]
[5] [  21,    27,    35,    46,    61,    82]
[6] [  82,   103,   130,   165,   211,   272,   354]
[7] [ 354,   436,   539,   669,   834,  1045,  1317,  1671]
[8] [1671,  2025,  2461,  3000,  3669,  4503,  5548,  6865,  8536]
[9] [8536, 10207, 12232, 14693, 17693, 21362, 25865, 31413, 38278, 46814]

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Borders give A032347 and A032346. Cf. A046935.

a046934 n k = a046934_tabl !! n !! k
a046934_row n = a046934_tabl !! n
a046934_tabl = [1] : iterate (\row -> scanl (+) (last row) row) [0,1]
a046934_list = concat a046934_tabl
-- Reinhard Zumkeller, Nov 10 2013

alias(PS = ListTools:-PartialSums):
A046934Triangle := proc(len) local a, k, P, T; a := 0; P := [1]; T := [];
for k from 1 to len  do T := [op(T), P]; P := PS([a, op(P)]); a := P[-1] od;
ListTools:-Flatten(T) end: A046934Triangle(10);  # Peter Luschny, Mar 29 2022

a[0, 0] = 1; a[1, 0] = 0; a[n_, 0] := a[n, 0] = a[n-1, n-1]; a[n_, k_] := a[n, k] = a[n, k-1] + a[n-1, k-1]; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 14 2013 *)

A275871 Row sums and second diagonal of A046934.

Original entry on oeis.org

1, 1, 4, 15, 61, 272, 1317, 6865, 38278, 227093, 1426921, 9457918, 65898275, 481177881, 3672102116, 29218285875, 241873478425, 2079079678176, 18524191138689, 170803860905237, 1627465240969382

Offset: 0

Olivier Gérard, Aug 11 2016

The offset corresponds to the definition of A046934.

Differences of A032346 and of A032347.

Cf. A046934, A046935, A032346, A032347.

Cf. A005493 (first differences of Bell numbers, second diagonal and row sum of A011971).

Clear[d]; d[0] = 1; d[1] = 0; d[n_] := d[n] =
  1 + Sum[Binomial[n - 1, j]*d[j], {j, 2, n - 1}]; Table[
d[n + 2] - d[n + 1], {n, 0, 22}] (* From the code by  J.-F. Alcover and Jon Perry in A032347 *)

cp's OEIS Frontend

A046934 Same rule as Aitken triangle (A011971) except a(0,0)=1, a(1,0)=0.

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