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

0,8

Comments

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

Links

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

Formula

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

Example

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, ...

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 *)

Prog

(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
(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
(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
(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

Crossrefs

Sequence in context: A323718 A130580 A220708 * A331461 A238016 A185812

Adjacent sequences: A110538 A110539 A110540 * A110542 A110543 A110544

Keyword

easy,nonn,tabl

Author

Paul Barry, Jul 25 2005

Status

approved