A176093 Triangle, T(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!), read by rows.

2, 1, 1, -5, -6, -5, 81, 2, 2, 81, -1049, 20, 150, 20, -1049, 14113, -2418, -553, -553, -2418, 14113, -170939, 69727, 8302, -7896, 8302, 69727, -170939, 700129, -1872204, -25884, 61770, 61770, -25884, -1872204, 700129, 79909831, 50456772, -2661816, -1523148, 634590, -1523148, -2661816, 50456772, 79909831

Offset

0,1

Comments

Row sums are: {2, 2, -16, 166, -1908, 22284, -193716, -2272378, 252997868, -13311131684, 619536514192, ...}.

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!) + Sum_{j=0..n-k} (-1)^j*(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, -(n-k); -; -1), where 2F0 is a hypergeometric function. - G. C. Greubel, Nov 27 2019

Example

Triangle begins as:
        2;
        1,        1;
       -5,       -6,     -5;
       81,        2,      2,    81;
    -1049,       20,    150,    20, -1049;
    14113,    -2418,   -553,  -553, -2418,  14113;
  -170939,    69727,   8302, -7896,  8302,  69727,  -170939;
   700129, -1872204, -25884, 61770, 61770, -25884, -1872204, 700129;

Maple

b:=binomial; T:= proc(n, k) option remember; 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-k, j)*b(n, j), j=0..n-k); end;
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 27 2019

Mathematica

T[n_, m_] = Sum[(-1)^k*(n+m)!/((n-k)!*(m-k)!*k!), {k, 0, m}] + Sum[(-1)^k*(2*n- m)!/((n-k)!*((n-m-k)!*k!), {k, 0, (n-m)}]; Table[T[n, m], {n, 0, 10}, {m, 0, n}]//Flatten
T[n_, k_]:= Binomial[n+k, k]*HypergeometricPFQ[{-n, -k}, {}, -1] + Binomial[2*n-k, n]*HypergeometricPFQ[{-n, k-n}, {}, -1]; Table[T[n, k], {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, (-1)^j*j!*b(n, j)*b(k, j)) + b(2*n-k, n)*sum(j=0, n-k, (-1)^j*j!*b(n-k, j)*b(n, j)); \\ G. C. Greubel, Nov 27 2019
(Magma) B:=Binomial; T:= func< n, k | 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]]) >;
[T(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 27 2019
(Sage) b=binomial;
def T(n, k): 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))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 27 2019
(GAP) B:=Binomial;; F:=Factorial;;
T:= function(n, k) return B(n+k, n)*Sum([0..k], j-> (-1)^j*F(j)*B(n, j) *B(k, j)) + B(2*n-k, n)*Sum([0..n-k], j-> (-1)^j*F(j)*B(n, j)*B(n-k, j)); end;
Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 27 2019

Crossrefs

Sequence in context: A266572 A266681 A210664 * A092437 A064814 A051012

Adjacent sequences: A176090 A176091 A176092 * A176094 A176095 A176096

Keyword

sign,tabl

Author

Roger L. Bagula, Apr 08 2010

Status

approved