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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). 12
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
From Paul D. Hanna, Dec 12 2006: (Start)
Consider the row reversal, which is A124496 with an additional left column (A000110 = Bell numbers). The matrix inverse of this triangle is very simple:
1;
-1, 1;
-1, -1, 1;
-1, -2, -1, 1;
-1, -3, -3, -1, 1;
-1, -4, -6, -4, -1, 1;
-1, -5, -10, -10, -5, -1, 1;
-1, -6, -15, -20, -15, -6, -1, 1;
-1, -7, -21, -35, -35, -21, -7, -1, 1;
-1, -8, -28, -56, -70, -56, -28, -8, -1, 1; ...
This gives the recurrence and explains why the Bell numbers appear. (End)
Triangle A160185 = reversal then deletes right border of 1's. - Gary W. Adamson, May 03 2009
LINKS
Robert Dougherty-Bliss, Gosper's algorithm and Bell numbers, arXiv:2210.13520 [cs.SC], 2022.
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.
EXAMPLE
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;
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();
# 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 *)
PROG
(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
(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
CROSSREFS
Diagonals, reading from the right, are A000110, A040027, A045501, A045499, A045500.
A124496 is a very similar triangle, obtained by reversing the rows and appending a rightmost diagonal which is A000110, the Bell numbers. See also A046936, A298804, A186020, A160185.
T(2n,n) gives A297924.
Sequence in context: A135722 A258306 A049513 * A097084 A143327 A094954
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, based on email from R. J. Mathar, Dec 11 2006
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)