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.

1, 1, 2, 1, 4, 10, 1, 6, 19, 46, 1, 8, 32, 94, 226, 1, 10, 49, 170, 474, 1136, 1, 12, 70, 282, 899, 2431, 5810, 1, 14, 95, 438, 1577, 4764, 12609, 30080, 1, 16, 124, 646, 2600, 8701, 25318, 65972, 157162, 1, 18, 157, 914, 4076, 15000, 47682, 134964, 347524, 826992

Offset

0,3

Links

Table of n, a(n) for n=0..54.

Formula

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.

Example

Square array begins:
1, 2, 10, 46, 226, 1136, 5810, 30080, 157162, ...;
1, 4, 19, 94, 474, 2431, 12609, 65972, 347524, ...;
1, 6, 32, 170, 899, 4764, 25318, 134964, 721562, ...;
1, 8, 49, 282, 1577, 8701, 47682, 260384, 1419436, ...;
1, 10, 70, 438, 2600, 15000, 85102, 477808, 2664539, ...;
1, 12, 95, 646, 4076, 24643, 145099, 839620, 4800849, ...;
1, 14, 124, 914, 6129, 38868, 237842, 1420660, 8342297, ...;
1, 16, 157, 1250, 8899, 59201, 376740, 2325088, 14036647, ...; ...

Prog

(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)}

Crossrefs

Cf. A137635, A137636, A137637, A137638, A073157.

Sequence in context: A343909 A117338 A279927 * A100229 A071949 A297506

Adjacent sequences: A137631 A137632 A137633 * A137635 A137636 A137637

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 31 2008

Status

approved