A176078 - OEIS

%I #10 Sep 08 2022 08:45:52

%S 1,1,1,1,19,1,1,161,161,1,1,1051,2451,1051,1,1,6049,24949,24949,6049,

%T 1,1,32341,206977,368677,206977,32341,1,1,164737,1510081,4200769,

%U 4200769,1510081,164737,1,1,810811,10077211,40347451,63050131,40347451,10077211,810811,1

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

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

%H G. C. Greubel, <a href="/A176078/b176078.txt">Rows n = 0..100 of triangle, flattened</a>

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

%F T(n, k) = binomial(2*n,n)*( binomial(n,k)^2 - 1) + 1. - _G. C. Greubel_, Nov 27 2019

%e Triangle begins as:

%e   1;

%e   1,      1;

%e   1,     19,       1;

%e   1,    161,     161,       1;

%e   1,   1051,    2451,    1051,       1;

%e   1,   6049,   24949,   24949,    6049,       1;

%e   1,  32341,  206977,  368677,  206977,   32341,      1;

%e   1, 164737, 1510081, 4200769, 4200769, 1510081, 164737, 1;

%p 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

%t 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

%o (PARI) b=binomial; T(n,k) = b(2*n,n)*(b(n,k)^2 -1)+1; \\ _G. C. Greubel_, Nov 27 2019

%o (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

%o (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

%o (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

%Y Cf. A141902, A000984

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Apr 08 2010