A176081 Triangle, read by rows, T(n, k) = Sum_{j=0..k} (n+k)!/((n-j)!*(k-j)!*j!) + Sum_{j=0..n-k} (2*n-k)!/((n-j)!*(n-k-j)!*j!).

2, 5, 5, 43, 18, 43, 681, 146, 146, 681, 14631, 2580, 630, 2580, 14631, 389593, 63162, 8267, 8267, 63162, 389593, 12314149, 1871611, 220654, 38472, 220654, 1871611, 12314149, 449324305, 64578292, 7348644, 690090, 690090, 7348644, 64578292, 449324305

Offset

0,1

Comments

Row sums are: {2, 10, 104, 1654, 35052, 922044, 28851300, 1043882662, 42798376172, 1958393108236, 98840300904512, ...}.

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!) + Sum_{j=0..n-k} (2*n-k)!/((n - j)!*(n-k-j)!*j!).

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

Example

Triangle begins as:
         2;
         5,       5;
        43,      18,     43;
       681,     146,    146,   681;
     14631,    2580,    630,  2580,  14631;
    389593,   63162,   8267,  8267,  63162,  389593;
  12314149, 1871611, 220654, 38472, 220654, 1871611, 12314149;

Maple

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

Mathematica

T[n_, m_]= Sum[(n+m)!/((n-k)!*(m-k)!*k!), {k, 0, m}] + Sum[(n+n-m)!/((n-k)!*(n-m- k)!*k!), {k, 0, n-m}]; Table[T[n, m], {n, 0, 10}, {k, 0, n}]//Flatten
Table[Binomial[n+k, n]*HypergeometricPFQ[{-n, -k}, {}, 1] + Binomial[2*n-k, n]* HypergeometricPFQ[{-n, k-n}, {}, 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)) + b(2*n-k, n)* sum(j=0, n-k, j!*b(n, j)*b(n-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]]) + B(2*n-k, n)*(&+[Factorial(j)*B(n, j)*B(n-k, j): j in [0..n-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)) + b(2*n-k, n)*sum(factorial(j)*b(n, j)*b(n-k, j) for j in (0..n-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)) +B(2*n-k, n)*Sum([0..n-k], j-> Factorial(j)*B(n, j)*B(n-k, j)) ))); # G. C. Greubel, Nov 27 2019

Crossrefs

Sequence in context: A154918 A176862 A352392 * A271222 A073339 A180092

Adjacent sequences: A176078 A176079 A176080 * A176082 A176083 A176084

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 08 2010

Status

approved