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 A176331 Triangle read by rows: T(n, k) = Sum_{j=0..n} C(j, n-k) * C(j, k) * (-1)^(n - j). 4
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS G. C. Greubel, Rows n = 0..100 of triangle, flattened 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 EXAMPLE 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; 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 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 *) PROG (PARI) T(n, k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, 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 (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 (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 CROSSREFS Row sums are A176332. Diagonal sums are A176334. Central coefficients T(2*n, n) are A176335. Sequence in context: A347971 A082039 A367505 * A157836 A205497 A063394 Adjacent sequences: A176328 A176329 A176330 * A176332 A176333 A176334 KEYWORD easy,nonn,tabl AUTHOR Paul Barry, Apr 15 2010 STATUS approved

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Last modified September 18 20:23 EDT 2024. Contains 376002 sequences. (Running on oeis4.)