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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified June 24 20:41 EDT 2021. Contains 345425 sequences. (Running on oeis4.)