A121207 Triangle read by rows. The definition is by diagonals. The r-th diagonal from the right, for r >= 0, is given by b(0) = b(1) = 1; b(n+1) = Sum_{k=0..n} binomial(n+2,k+r)*a(k).
1, 1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 9, 15, 1, 1, 5, 14, 31, 52, 1, 1, 6, 20, 54, 121, 203, 1, 1, 7, 27, 85, 233, 523, 877, 1, 1, 8, 35, 125, 400, 1101, 2469, 4140, 1, 1, 9, 44, 175, 635, 2046, 5625, 12611, 21147, 1, 1, 10, 54, 236, 952, 3488, 11226, 30846, 69161, 115975
Offset: 0
Examples
Triangle begins (compare also table 9.2 in the Gould-Quaintance reference): 1; 1, 1; 1, 1, 2; 1, 1, 3, 5; 1, 1, 4, 9, 15; 1, 1, 5, 14, 31, 52; 1, 1, 6, 20, 54, 121, 203; 1, 1, 7, 27, 85, 233, 523, 877; 1, 1, 8, 35, 125, 400,1101, 2469, 4140; 1, 1, 9, 44, 175, 635,2046, 5625, 12611, 21147; 1, 1, 10, 54, 236, 952,3488,11226, 30846, 69161, 115975; 1, 1, 11, 65, 309,1366,5579,20425, 65676,180474, 404663, 678570; 1, 1, 12, 77, 395,1893,8494,34685,126817,407787,1120666,2512769,4213597;
Links
- Alois P. Heinz, Rows n = 0..140, flattened
- Robert Dougherty-Bliss, Gosper's algorithm and Bell numbers, arXiv:2210.13520 [cs.SC], 2022.
- Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See pp. 69-70.
- H. W. Gould and Jocelyn Quaintance, A linear binomial recurrence and the Bell numbers and polynomials, Applicable Analysis and Discrete Mathematics, 1 (2007), 371-385.
Crossrefs
Programs
-
Julia
function Gould_diag(diag, size) size < 1 && return [] size == 1 && return [1] L = [1, 1] accu = ones(BigInt, diag) for _ in 1:size-2 accu = cumsum(vcat(accu[end], accu)) L = vcat(L, accu[end]) end L end # Peter Luschny, Mar 30 2022 -
Maple
# This is the Jovovic formula with general index 'd' # where A040027, A045499, etc. use one explicit integer # Index n+1 is shifted to n from the original formula. Gould := proc(n, d) local k; if n <= 1 then return 1 else return add(binomial(n-1+d, k+d)*Gould(k, d), k=0..n-1); fi end: # row and col refer to the extrapolated super-table: # working up to row, not row-1, shows also the Bell numbers # at the end of each row. for row from 0 to 13 do for col from 0 to row do # 'diag' is constant for one of A040027, A045499 etc. diag := row - col; printf("%4d, ", Gould(col, diag)); od; print(); od; # R. J. Mathar # second Maple program: T:= proc(n, k) option remember; `if`(k=0, 1, add(T(n-j, k-j)*binomial(n-1, j-1), j=1..k)) end: seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Jan 08 2018
-
Mathematica
g[n_ /; n <= 1, ] := 1; g[n, d_] := g[n, d] = Sum[ Binomial[n-1+d, k+d]*g[k, d], {k, 0, n-1}]; Flatten[ Table[ diag = row-col; g[col, diag], {row, 0, 13}, {col, 0, row}]] (* Jean-François Alcover, Nov 25 2011, after R. J. Mathar *) T[n_, k_] := T[n, k] = If[k == 0, 1, Sum[T[n-j, k-j] Binomial[n-1, j-1], {j, 1, k}]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 26 2018, after Alois P. Heinz *)
-
Python
# Computes the n-th diagonal of the triangle reading from the right. from itertools import accumulate def Gould_diag(diag, size): if size < 1: return [] if size == 1: return [1] L, accu = [1,1], [1]*diag for _ in range(size-2): accu = list(accumulate([accu[-1]] + accu)) L.append(accu[-1]) return L # Peter Luschny, Apr 24 2016
Comments