A176121 Triangle, read by rows, T(n,k) = binomial(n+k+1, n+1) * Sum_{j=0..k} j!*binomial(n,j)*binomial(k, j).

1, 1, 6, 1, 12, 70, 1, 20, 195, 1190, 1, 30, 441, 4088, 26334, 1, 42, 868, 11424, 105210, 714252, 1, 56, 1548, 27480, 344850, 3208392, 22869132, 1, 72, 2565, 59070, 970695, 11938212, 113011899, 842483070, 1, 90, 4015, 116380, 2425995, 38200162, 466911445, 4511398320, 35048431990

Offset

0,3

Comments

Row sums are: {1, 7, 83, 1406, 30894, 831797, 26451459, 968465584, 40067488398, 1846559924606, 93734743085246, ...}.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n,k) = binomial(n+k+1, n+1) * Sum_{j=0..k} j!*binomial(n,j)*binomial(k, j).

Example

Triangle begins as:
  1;
  1,  6;
  1, 12,   70;
  1, 20,  195,  1190;
  1, 30,  441,  4088,  26334;
  1, 42,  868, 11424, 105210,   714252;
  1, 56, 1548, 27480, 344850,  3208392,  22869132;
  1, 72, 2565, 59070, 970695, 11938212, 113011899, 842483070;
  ...

Maple

b:=binomial; T(n, k):=b(n+k+1, n+1)*add(j!*b(n, j)*b(k, j), j=0..k); seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 27 2019

Mathematica

T[n_, k_] = Binomial[n+k+1, n+1]*Sum[Binomial[n, j]*Binomial[k, j]*j!, {j, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

Prog

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

Crossrefs

Sequence in context: A013613 A122508 A171006 * A062190 A080211 A146997

Adjacent sequences: A176118 A176119 A176120 * A176122 A176123 A176124

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 09 2010

Status

approved