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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
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.)