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A155495 Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m). 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are: {1, 2, 14, 92, 646, 4652, 34124, 253528, 1901638, 14368844, 109208164,  ...}

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,m) = Binomial[2*n, 2*m]*Binomial[n, m].

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

Clear[t, n, m]; t[n_, m_] := Binomial[2*n, 2*m]*Binomial[n, m]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

Abs[Flatten[Table[CoefficientList[HypergeometricPFQ[{-n, -n, -n+1/2}, {1, 1/2}, x], x], {n, 1, 20}]]] (* also *)

T[n_, k_] := (-1)^k*Coefficient[HypergeometricPFQ[{-n, -n, -n+1/2}, {1, 1/2}, x], x^k] (* John M. Campbell, Oct 23 2011 *)

CROSSREFS

Cf. A288470 (row sums).

Sequence in context: A176489 A174039 A174148 * A157273 A168518 A174667

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 September 24 01:21 EDT 2020. Contains 337315 sequences. (Running on oeis4.)