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
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Apr 08 2010
STATUS
approved