A176094 Triangle, read by rows, T(n,k) = f(n,k) - f(n,0) + 1, where f(n, k) = Sum_{j=0..k} (-1)^j*(n+k)!/((n-j)!*(k-j)!*j!) + Sum_{j=0..n-k} (-1)^j*(2*n - k)!/((n-j)!*(n-k-j)!*j!).

1, 1, 1, 1, 0, 1, 1, -78, -78, 1, 1, 1070, 1200, 1070, 1, 1, -16530, -14665, -14665, -16530, 1, 1, 240667, 179242, 163044, 179242, 240667, 1, 1, -2572332, -726012, -638358, -638358, -726012, -2572332, 1, 1, -29453058, -82571646, -81432978, -79275240, -81432978, -82571646, -29453058, 1

Offset

0,8

Comments

Row sums are: {1, 2, 2, -154, 3342, -62388, 1002864, -7873402, -466190602, 41337748316, -2470134563444, ...}.

Links

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

Formula

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

f(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 28 2019

Example

Triangle begins as:
  1;
  1,      1;
  1,      0,      1;
  1,    -78,    -78,      1;
  1,   1070,   1200,   1070,      1;
  1, -16530, -14665, -14665, -16530,      1;
  1, 240667, 179242, 163044, 179242, 240667, 1;

Maple

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

Mathematica

(* First program *)
f[n_, k_]:= Sum[(-1)^j*(n+k)!/((n-j)!*(k-j)!*j!), {j, 0, k}] + Sum[(-1)^j*(2*n-k)! /((n-j)!*(n-k-j)!*j!), {j, 0, (n-m)}]; Table[f[n, m] -f[n, 0] +1, {n, 0, 10}, {k, 0, n}]//Flatten
(* Second program *)
f[n_, k_]:= Binomial[n+k, n]*HypergeometricPFQ[{-n, -k}, {}, -1] + Binomial[2*n-k, n]*HypergeometricPFQ[{-n, k - n}, {}, -1]; Table[f[n, k] - f[n, 0] + 1, {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 27 2019 *)

Prog

(PARI) b=binomial; f(n, k) = b(n+k, n)*sum(j=0, k, (-1)^j*j!*b(n, j)*b(k, j)) + b(2*n-k, n)* sum(j=0, n-k, (-1)^j*j!*b(n, j)*b(n-k, j));
T(n, k) = f(n, k) - f(n, 0) + 1; \\ G. C. Greubel, Nov 27 2019
(Magma)
function f(n, k)
  B:=Binomial;
  return B(n+k, n)*(&+[(-1)^j*Factorial(j)*B(n, j)*B(k, j): j in [0..k]]) + B(2*n-k, n)* (&+[(-1)^j*Factorial(j)*B(n, j)*B(n-k, j): j in [0..n-k]]); end function;
[f(n, k) -f(n, 0) +1: k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 27 2019
(Sage)
def f(n, k):
    b=binomial;
    return b(n+k, n)*sum((-1)^j*factorial(j)*b(n, j)*b(k, j) for j in (0..k)) + b(2*n-k, n)*sum((-1)^j*factorial(j)*b(n, j)*b(n-k, j) for j in (0..n-k))
[[f(n, k) -f(n, 0) +1 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 27 2019
(GAP)
B:=Binomial;;
f:= function(n, k) return B(n+k, n)*Sum([0..k], j-> (-1)^j*Factorial(j)*B(n, j)* B(k, j)) + B(2*n-k, n)*Sum([0..n-k], j-> (-1)^j*Factorial(j)*B(n, j)*B(n-k, j ));  end;
Flat(List([0..10], n-> List([0..n], k-> f(n, k)-f(n, 0)+1 ))); # G. C. Greubel, Nov 27 2019

Crossrefs

Sequence in context: A117330 A033398 A204376 * A124289 A181467 A344812

Adjacent sequences: A176091 A176092 A176093 * A176095 A176096 A176097

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 08 2010

Status

approved