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A155495 Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).

%I A155495 #20 Sep 08 2022 08:45:40
%S A155495 1,1,1,1,12,1,1,45,45,1,1,112,420,112,1,1,225,2100,2100,225,1,1,396,
%T A155495 7425,18480,7425,396,1,1,637,21021,105105,105105,21021,637,1,1,960,
%U A155495 50960,448448,900900,448448,50960,960,1,1,1377,110160,1559376,5513508,5513508,1559376,110160,1377,1
%N A155495 Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).
%C A155495 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
%H A155495 Robert Israel, <a href="/A155495/b155495.txt">Table of n, a(n) for n = 0..10010</a>(rows 0 to 140, flattened)
%F A155495 T(n, k) = binomial(n, k)*binomial(2*n, 2*k).
%F A155495 Sum_{k=0..n} T(n, k) = A288470(n).
%e A155495 Table starts:
%e A155495   1;
%e A155495   1,    1;
%e A155495   1,   12,      1;
%e A155495   1,   45,     45,       1;
%e A155495   1,  112,    420,     112,        1;
%e A155495   1,  225,   2100,    2100,      225,        1;
%e A155495   1,  396,   7425,   18480,     7425,      396,        1;
%e A155495   1,  637,  21021,  105105,   105105,    21021,      637,       1;
%e A155495   1,  960,  50960,  448448,   900900,   448448,    50960,     960,      1;
%e A155495   1, 1377, 110160, 1559376,  5513508,  5513508,  1559376,  110160,   1377,    1;
%e A155495   1, 1900, 218025, 4651200, 26453700, 46558512, 26453700, 4651200, 218025, 1900, 1;
%p A155495 seq(seq(binomial(2*n,2*m)*binomial(n,m), m=0..n),n=0..10); # _Robert Israel_, Jun 12 2017
%t A155495 T[n_, k_]:= Binomial[2*n,2*k]*Binomial[n,k]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
%t A155495 Abs[Flatten[Table[CoefficientList[HypergeometricPFQ[{-n,-n,-n+1/2}, {1,1/2}, x], x], {n, 1, 20}]]] (* or *)
%t A155495 T[n_,k_]:= (-1)^k*Coefficient[HypergeometricPFQ[{-n,-n,-n+1/2}, {1,1/2}, x], x^k] (* _John M. Campbell_, Oct 23 2011 *)
%o A155495 (Magma) [Binomial(n, k)*Binomial(2*n, 2*k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 29 2021
%o A155495 (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
%Y A155495 Cf. A155497, A155516, A288470 (row sums).
%K A155495 nonn,tabl
%O A155495 0,5
%A A155495 _Roger L. Bagula_, Jan 23 2009