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A137634 Square array where T(n,k) = Sum_{j=0..k} C(n+2*j,j)*C(n+2*j,k-j), read by antidiagonals.

%I A137634 #2 Mar 30 2012 18:37:09
%S A137634 1,1,2,1,4,10,1,6,19,46,1,8,32,94,226,1,10,49,170,474,1136,1,12,70,
%T A137634 282,899,2431,5810,1,14,95,438,1577,4764,12609,30080,1,16,124,646,
%U A137634 2600,8701,25318,65972,157162,1,18,157,914,4076,15000,47682,134964,347524,826992
%N A137634 Square array where T(n,k) = Sum_{j=0..k} C(n+2*j,j)*C(n+2*j,k-j), read by antidiagonals.
%F A137634 G.f.: A(x,y) = R(y)/(1 - x*G(y)), so that the g.f. of row n = R(y)*G(y)^n, where R(y) = 1/sqrt(1-4*y*(1+y)^2) and G(y) = (1-sqrt(1-4*y*(1+y)^2))/(2*y*(1+y)) is the g.f. of A073157.
%e A137634 Square array begins:
%e A137634 1, 2, 10, 46, 226, 1136, 5810, 30080, 157162, ...;
%e A137634 1, 4, 19, 94, 474, 2431, 12609, 65972, 347524, ...;
%e A137634 1, 6, 32, 170, 899, 4764, 25318, 134964, 721562, ...;
%e A137634 1, 8, 49, 282, 1577, 8701, 47682, 260384, 1419436, ...;
%e A137634 1, 10, 70, 438, 2600, 15000, 85102, 477808, 2664539, ...;
%e A137634 1, 12, 95, 646, 4076, 24643, 145099, 839620, 4800849, ...;
%e A137634 1, 14, 124, 914, 6129, 38868, 237842, 1420660, 8342297, ...;
%e A137634 1, 16, 157, 1250, 8899, 59201, 376740, 2325088, 14036647, ...; ...
%o A137634 (PARI) {T(n,k)=sum(j=0,k,binomial(n+2*j,j)*binomial(n+2*j,k-j))} /* Using the g.f.: */ {T(n,k)=local(Oy=y*O(y^(n+k))); polcoeff(polcoeff(1/sqrt(1-4*y*(1+y)^2+Oy)* 1/(1-x*((1-sqrt(1-4*y*(1+y)^2+Oy))/(2*y*(1 + y+Oy))+x*O(x^n))),n,x),k,y)}
%Y A137634 Cf. A137635, A137636, A137637, A137638, A073157.
%K A137634 nonn,tabl
%O A137634 0,3
%A A137634 _Paul D. Hanna_, Jan 31 2008