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
Examples
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;
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Programs
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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
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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
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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
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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 -
PARI
b=binomial; T(n,k) = b(2*n,n)*(b(n,k)^2 -1)+1; \\ G. C. Greubel, Nov 27 2019
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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
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
Comments