A176124 Triangle, read by rows, T(n, k) = Sum_{j=0..k} (-1)^j*(k-j+1)^n*binomial(n+1, j) *binomial(n+2, j)/(j+1).

1, 1, -1, 1, -2, -9, 1, -2, -33, -56, 1, 1, -109, -209, 50, 1, 11, -324, -894, 1641, 10485, 1, 36, -867, -4262, 12951, 85926, 211435, 1, 92, -2085, -20516, 74369, 625164, 1435939, 1005536, 1, 211, -4419, -93989, 344617, 4306671, 9337441, -7280909, -94801266

Offset

0,5

Comments

Row sums are: {1, 0, -10, -90, -266, 10920, 305220, 3118500, -88191642, -5436127620, -116211623892, ...}.

Links

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

Formula

T(n, k) = Sum_{j=0..k} (-1)^j*(k-j+1)^n*binomial(n+1,j)*binomial(n+2,j)/(j+1).

Example

Triangle begins as:
  1;
  1, -1;   1, -2,    -9;
  1, -2,   -33,    -56;
  1,  1,  -109,   -209,    50;
  1, 11,  -324,   -894,  1641,  10485;
  1, 36,  -867,  -4262, 12951,  85926,  211435;
  1, 92, -2085, -20516, 74369, 625164, 1435939, 1005536;

Maple

b:=binomial; T(n, k):=add((-1)^j*(k-j+1)^n*b(n+1, j)*b(n+2, j)/(j+1), j=0..k); seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 27 2019

Mathematica

T[n_, K_]: = Sum[(-1)^j*(k-j+1)^n*Binomial[n+1, j]*Binomial[n+2, j]/(j+1), {j, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(PARI) b=binomial; T(n, k) = sum(j=0, k, (-1)^j*(k-j+1)^n*b(n, j)*b(k, j)/(j+1)); \\ G. C. Greubel, Nov 27 2019
(Magma) B:=Binomial; [(&+[(-1)^j*(k-j+1)^n*B(n+1, j)*B(n+2, j)/(j+1): j in [0..k]]) : k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 27 2019
(Sage) b=binomial; [[sum((-1)^j*(k-j+1)^n*b(n+1, j)*b(n+2, j)/(j+1) 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-> Sum([0..k], j-> (-1)^j*(k-j+1)^n*B(n+1, j)*B(n+2, j)/(j+1)) ))); # G. C. Greubel, Nov 27 2019

Crossrefs

Sequence in context: A073009 A011064 A086773 * A190411 A190142 A173995

Adjacent sequences: A176121 A176122 A176123 * A176125 A176126 A176127

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 09 2010

Status

approved