A176331 Triangle read by rows: T(n, k) = Sum_{j=0..n} C(j, n-k) * C(j, k) * (-1)^(n - j).
1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 13, 28, 13, 1, 1, 21, 79, 79, 21, 1, 1, 31, 181, 315, 181, 31, 1, 1, 43, 361, 971, 971, 361, 43, 1, 1, 57, 652, 2511, 3876, 2511, 652, 57, 1, 1, 73, 1093, 5713, 12606, 12606, 5713, 1093, 73, 1, 1, 91, 1729, 11789, 35246, 50358, 35246, 11789, 1729, 91, 1
Offset: 0
Examples
Triangle begins 1; 1, 1; 1, 3, 1; 1, 7, 7, 1; 1, 13, 28, 13, 1; 1, 21, 79, 79, 21, 1; 1, 31, 181, 315, 181, 31, 1; 1, 43, 361, 971, 971, 361, 43, 1; 1, 57, 652, 2511, 3876, 2511, 652, 57, 1; 1, 73, 1093, 5713, 12606, 12606, 5713, 1093, 73, 1; 1, 91, 1729, 11789, 35246, 50358, 35246, 11789, 1729, 91, 1;
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Crossrefs
Programs
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GAP
T:= function(n,k) return Sum([0..n], j-> (-1)^(n-j)*Binomial(j,k)*Binomial(j,n-k) ); end; Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Dec 07 2019 -
Magma
T:= func< n,k | &+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]] >; [T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 07 2019
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Maple
T:= proc(n, k) option remember; add( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k), j=0..n); end: seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Dec 07 2019 T := (n, k) -> binomial(n, k)^2*hypergeom([1, -k, -n + k], [-n, -n], -1): seq(seq(simplify(T(n, k)), k = 0..n), n = 0..9); # Peter Luschny, May 13 2024
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Mathematica
T[n_, k_]:= Sum[(-1)^(n-j)*Binomial[j, k]*Binomial[j, n-k], {j,0,n}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 07 2019 *) -
PARI
T(n,k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k)); \\ G. C. Greubel, Dec 07 2019
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Sage
@CachedFunction def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n)) [[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 07 2019
Formula
T(n, k) = T(n, n-k).
T(n, k) = binomial(n, k)^2*hypergeom([1, -k, -n + k], [-n, -n], -1). - Peter Luschny, May 13 2024