A176080 Triangle, read by rows, T(n, k) = Sum_{j=0..k} (n+k)!/((n-j)!*(k-j)!*j!).

1, 1, 4, 1, 9, 42, 1, 16, 130, 680, 1, 25, 315, 2555, 14630, 1, 36, 651, 7616, 63126, 389592, 1, 49, 1204, 19236, 219450, 1871562, 12314148, 1, 64, 2052, 42960, 647130, 7346592, 64578228, 449324304, 1, 81, 3285, 87285, 1679535, 24557247, 280146867, 2537661555, 18555052230

Offset

0,3

Comments

Row sums are: {1, 5, 52, 827, 17526, 461022, 14425650, 521941331, 21399188086, 979196554118, 49420150452256, ...}.

Links

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

Formula

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

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

Example

Triangle begins as:
1;
1,  4;
1,  9,   42;
1, 16,  130,   680;
1, 25,  315,  2555,  14630;
1, 36,  651,  7616,  63126,  389592;
1, 49, 1204, 19236, 219450, 1871562, 12314148;
1, 64, 2052, 42960, 647130, 7346592, 64578228, 449324304;

Maple

b:=binomial; T(n, k):=b(n+k, n)*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_]:= Sum[(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, n]*HypergeometricPFQ[{-n, -k}, {}, 1], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 27 2019 *)

Prog

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

Crossrefs

Sequence in context: A028941 A364109 A348180 * A328302 A346748 A348184

Adjacent sequences: A176077 A176078 A176079 * A176081 A176082 A176083

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 08 2010

Status

approved