A176122 Triangle, read by rows, T(n,k) = Sum_{j=1..k} binomial(n-1, j-1)*binomial(k, j - 1)*(j-1)!.

1, 1, 3, 1, 5, 13, 1, 7, 28, 73, 1, 9, 49, 185, 501, 1, 11, 76, 381, 1426, 4051, 1, 13, 109, 685, 3331, 12607, 37633, 1, 15, 148, 1121, 6756, 32593, 125882, 394353, 1, 17, 193, 1713, 12361, 73129, 354033, 1401409, 4596553, 1, 19, 244, 2485, 20926, 147295, 865936, 4233673, 17209234, 58941091

Offset

1,3

Comments

Row sums are: {1, 4, 19, 109, 745, 5946, 54379, 560869, 6439409, 81420904, ...}.

Links

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Formula

T(n,k) = Sum_{j=1..k} binomial(n-1, j-1)*binomial(k, j - 1)*(j-1)!.

Example

Triangle begins as:
  1;
  1,  3;
  1,  5,  13;
  1,  7,  28,   73;
  1,  9,  49,  185,   501;
  1, 11,  76,  381,  1426,   4051;
  1, 13, 109,  685,  3331,  12607,  37633;
  1, 15, 148, 1121,  6756,  32593, 125882,  394353;
  1, 17, 193, 1713, 12361,  73129, 354033, 1401409,  4596553;
  1, 19, 244, 2485, 20926, 147295, 865936, 4233673, 17209234, 58941091;

Maple

b:=binomial; T(n, k):=add((j-1)!*b(n-1, j-1)*b(k, j-1), j=1..k); seq(seq(T(n, k), k=1..n), n=1..10); # G. C. Greubel, Nov 27 2019

Mathematica

T[n_, k_]:= Sum[Binomial[n-1, j-1]*Binomial[k, j-1]*(j-1)!, {j, k}]; Table[T[n, k], {n, 10}, {k, n}]//Flatten

Prog

(PARI) b=binomial; T(n, k) = sum(j=1, k, (j-1)!*b(n-1, j-1)*b(k, j-1)); \\ G. C. Greubel, Nov 27 2019
(Magma) B:=Binomial; [(&+[Factorial(j-1)*B(n-1, j-1)*B(k, j-1): j in [1..k]]) : k in [1..n], n in [1..10]]; // G. C. Greubel, Nov 27 2019
(Sage) b=binomial; [[sum(factorial(j-1)*b(n-1, j-1)*b(k, j-1) for j in (1..k)) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Nov 27 2019
(GAP) B:=Binomial;; Flat(List([1..10], n-> List([1..n], k-> Sum([0..k], j-> Factorial(j-1)*B(n-1, j-1)*B(k, j-1)) ))); # G. C. Greubel, Nov 27 2019

Crossrefs

Sequence in context: A266033 A105064 A073496 * A370380 A091623 A215474

Adjacent sequences: A176119 A176120 A176121 * A176123 A176124 A176125

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 09 2010

Status

approved