A176078 Triangle, read by rows, T(n, k) = (2*n)!/((n-k)! * k!)^2 - (2*n)!/(n!)^2 + 1.

1, 1, 1, 1, 19, 1, 1, 161, 161, 1, 1, 1051, 2451, 1051, 1, 1, 6049, 24949, 24949, 6049, 1, 1, 32341, 206977, 368677, 206977, 32341, 1, 1, 164737, 1510081, 4200769, 4200769, 1510081, 164737, 1, 1, 810811, 10077211, 40347451, 63050131, 40347451, 10077211, 810811, 1

Offset

0,5

Comments

Row sums are: {1, 2, 21, 324, 4555, 61998, 847315, 11751176, 165521079, 2363418210, 34132747231, ...}.

Links

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

Formula

T(n, k) = (2*n)!/((n-k)! * k!)^2 - (2*n)!/(n!)^2 + 1.

T(n, k) = binomial(2*n,n)*( binomial(n,k)^2 - 1) + 1. - G. C. Greubel, Nov 27 2019

Example

Triangle begins as:
  1;
  1,      1;
  1,     19,       1;
  1,    161,     161,       1;
  1,   1051,    2451,    1051,       1;
  1,   6049,   24949,   24949,    6049,       1;
  1,  32341,  206977,  368677,  206977,   32341,      1;
  1, 164737, 1510081, 4200769, 4200769, 1510081, 164737, 1;

Maple

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

Mathematica

T[n_, k_] = (2*n)!/((n-k)!*k!)^2 - (2*n)!/(n!)^2 + 1; Table[T[n, k], {n, 0, 10}, (k, 0, n)]//Flatten

Prog

(PARI) b=binomial; T(n, k) = b(2*n, n)*(b(n, k)^2 -1)+1; \\ G. C. Greubel, Nov 27 2019
(Magma) B:=Binomial; [B(2*n, n)*(B(n, k)^2 -1)+1: k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 27 2019
(Sage) b=binomial; [[b(2*n, n)*(b(n, k)^2 -1)+1 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(2*n, n)*(B(n, k)^2 -1)+1 ))); # G. C. Greubel, Nov 27 2019

Crossrefs

Cf. A141902, A000984

Sequence in context: A040361 A174097 A174040 * A022182 A015145 A040369

Adjacent sequences: A176075 A176076 A176077 * A176079 A176080 A176081

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 08 2010

Status

approved