A110541 A number triangle of sums of binomial products.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 5, 7, 4, 1, 1, 1, 8, 19, 13, 5, 1, 1, 1, 13, 51, 46, 21, 6, 1, 1, 1, 21, 141, 166, 89, 31, 7, 1, 1, 1, 34, 393, 610, 393, 151, 43, 8, 1, 1, 1, 55, 1107, 2269, 1761, 776, 235, 57, 9, 1, 1, 1, 89, 3139, 8518, 7985, 4056, 1363, 344, 73, 10, 1, 1

Offset: 0

Author

Paul Barry, Jul 25 2005

Keywords

• easy
• nonn
• tabl

Comments

Columns include A000045, A002426, A026641. Rows include A000012, A000027, A002061(n+1). Row sums are A110542.

Examples

			Rows begin
  1;
  1,   1;
  1,   1,   1;
  1,   2,   1,   1;
  1,   3,   3,   1,   1;
  1,   5,   7,   4,   1,   1;
  1,   8,  19,  13,   5,   1,   1;
  1,  13,  51,  46,  21,   6,   1,   1;
  1,  21, 141, 166,  89,  31,   7,   1,   1;
As a number square read by antidiagonals, rows begin
  1,   1,   1,   1,   1,   1, ...
  1,   1,   1,   1,   1,   1, ...
  1,   2,   3,   4,   5,   6, ...
  1,   3,   7,  13,  21,  31, ...
  1,   5,  19,  46,  89, 151, ...
  1,   8,  51, 166, 393, 776, ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Programs

GAP

Flat(List([0..12], n-> List([0..n], k-> Sum([0..n-k], j-> Binomial((k-1)*(n-k)-(k-2)*j, j)*Binomial(j, n-k-j) )))); # G. C. Greubel, Feb 19 2019

Magma

[[(&+[Binomial((k-1)*(n-k)-(k-2)*j, j)*Binomial(j, n-k-j): j in [0..n-k]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Feb 19 2019

Mathematica

T[n_, k_] := Sum[Binomial[(k-1)*(n-k) - (k-2)*j, j]*Binomial[j, n-k-j], {j, 0, n-k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 31 2017 *)

PARI

for(n=0,20, for(k=0,n, print1(sum(j=0,n-k, binomial((k-1)*(n-k) -(k-2)*j, j)*binomial(j, n-k-j)), ", "))) \\ G. C. Greubel, Aug 31 2017

Sage

[[sum(binomial((k-1)*(n-k) -(k-2)*j, j)*binomial(j, n-k-j) for j in (0..n-k)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Feb 19 2019

Formula

T(n,k) = Sum_{j=0..n-k} C((k-1)*(n-k)-(k-2)*j, j)*C(j, n-k-j).