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 A155495 Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m). 5
 1, 1, 1, 1, 12, 1, 1, 45, 45, 1, 1, 112, 420, 112, 1, 1, 225, 2100, 2100, 225, 1, 1, 396, 7425, 18480, 7425, 396, 1, 1, 637, 21021, 105105, 105105, 21021, 637, 1, 1, 960, 50960, 448448, 900900, 448448, 50960, 960, 1, 1, 1377, 110160, 1559376, 5513508, 5513508, 1559376, 110160, 1377, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS T(n,k) equals (-1)^k times the coefficient of x^k in 3F2(-n,-n,-n+1/2;1,1/2;x); see Mathematica code below. - John M. Campbell, Oct 23 2011 LINKS Robert Israel, Table of n, a(n) for n = 0..10010(rows 0 to 140, flattened) FORMULA T(n, k) = binomial(n, k)*binomial(2*n, 2*k). Sum_{k=0..n} T(n, k) = A288470(n). EXAMPLE Table starts: 1; 1, 1; 1, 12, 1; 1, 45, 45, 1; 1, 112, 420, 112, 1; 1, 225, 2100, 2100, 225, 1; 1, 396, 7425, 18480, 7425, 396, 1; 1, 637, 21021, 105105, 105105, 21021, 637, 1; 1, 960, 50960, 448448, 900900, 448448, 50960, 960, 1; 1, 1377, 110160, 1559376, 5513508, 5513508, 1559376, 110160, 1377, 1; 1, 1900, 218025, 4651200, 26453700, 46558512, 26453700, 4651200, 218025, 1900, 1; MAPLE seq(seq(binomial(2*n, 2*m)*binomial(n, m), m=0..n), n=0..10); # Robert Israel, Jun 12 2017 MATHEMATICA T[n_, k_]:= Binomial[2*n, 2*k]*Binomial[n, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten Abs[Flatten[Table[CoefficientList[HypergeometricPFQ[{-n, -n, -n+1/2}, {1, 1/2}, x], x], {n, 1, 20}]]] (* or *) T[n_, k_]:= (-1)^k*Coefficient[HypergeometricPFQ[{-n, -n, -n+1/2}, {1, 1/2}, x], x^k] (* John M. Campbell, Oct 23 2011 *) PROG (Magma) [Binomial(n, k)*Binomial(2*n, 2*k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 29 2021 (Sage) flatten([[binomial(n, k)*binomial(2*n, 2*k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 29 2021 CROSSREFS Cf. A155497, A155516, A288470 (row sums). Sequence in context: A176489 A174039 A174148 * A157273 A350729 A168518 Adjacent sequences: A155492 A155493 A155494 * A155496 A155497 A155498 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Jan 23 2009 STATUS approved

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Last modified June 15 10:19 EDT 2024. Contains 373407 sequences. (Running on oeis4.)