A176092 Triangle, read by rows, T(n, k) = Sum_{j=0..k} (-1)^j*(n+k)!/((n-j)!*(k-j)!*j!).

1, 1, 0, 1, -3, -6, 1, -8, 10, 80, 1, -15, 75, 35, -1050, 1, -24, 231, -784, -2394, 14112, 1, -35, 532, -3948, 7770, 69762, -170940, 1, -48, 1044, -12480, 74250, -26928, -1872156, 700128, 1, -63, 1845, -31515, 317295, -1491633, -2663661, 50456835, 79909830

Offset

0,5

Comments

Row sums are: {1, 1, -8, 83, -954, 11142, -96858, -1136189, 126498934, -6655565842, 309768257096, ...}.

Links

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

Formula

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

T(n, k) = binomial(n+k,k)*2F0(-n, -k; -; -1), where 2F0 is a hypergeometric function. - G. C. Greubel, Nov 27 2019

Example

Triangle begins as:
  1;
  1,   0;
  1,  -3,   -6;
  1,  -8,   10,     80;
  1, -15,   75,     35, -1050;
  1, -24,  231,   -784, -2394,  14112;
  1, -35,  532,  -3948,  7770,  69762,  -170940;
  1, -48, 1044, -12480, 74250, -26928, -1872156, 700128;

Maple

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

Mathematica

T[n_, k_] = Sum[(-1)^j*(n+k)!/((n-j)!*(k-j)!*j!), {j, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten
Table[Binomial[n+k, k]*HypergeometricPFQ[{-n, -k}, {}, -1], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 27 2019 *)

Prog

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

Crossrefs

Sequence in context: A290481 A259501 A118948 * A118012 A022836 A090911

Adjacent sequences: A176089 A176090 A176091 * A176093 A176094 A176095

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 08 2010

Status

approved