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A181145 G.f.: exp( Sum_{n>=1} [Sum_{k=0..2n} C(2n,k)^2*y^k]*x^n/n ) = Sum_{n>=0,k=0..2n} T(n,k)*x^n*y^k, as a triangle of coefficients T(n,k) read by rows. 1
1, 1, 4, 1, 1, 12, 27, 12, 1, 1, 24, 134, 236, 134, 24, 1, 1, 40, 410, 1540, 2380, 1540, 410, 40, 1, 1, 60, 975, 6260, 18386, 26216, 18386, 6260, 975, 60, 1, 1, 84, 1981, 19320, 91441, 227052, 306495, 227052, 91441, 19320, 1981, 84, 1, 1, 112, 3612, 49672 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare g.f. to that of the triangle A034870:

* exp( Sum_{n>=1} [Sum_{k=0..2n} C(2n,k)*y^k]*x^n/n )

which consists of the even numbered rows of Pascal's triangle.

LINKS

Table of n, a(n) for n=0..52.

FORMULA

Row sums form A066357 (with offset), the number of ordered trees on 2n nodes with every subtree at the root having an even number of edges.

EXAMPLE

G.f.: A(x,y) = 1 + (1+ 4*y+ y^2)*x + (1 + 12*y+ 27*y^2+ 12*y^3+ y^4)*x^2 + (1+ 24*y+ 134*y^2+ 236*y^3+ 134*y^4+ 24*y^5+ y^6)*x^3 +...

The logarithm of the g.f. begins:

log(A(x,y)) = (1 + 2^2*y + y^2)*x

+ (1 + 4^2*y + 6^2*y^2 + 4^2*y^3 + y^4)*x^2/2

+ (1 + 6^2*y + 15^2*y^2 + 20^2*y^3 + 15^2*y^4 + 6^2*y^5 + y^6)*x^3/3

+ (1 + 8^2*y + 28^2*y^2 + 56^2*y^3 + 70^2*y^4 + 56^2*y^5 + 28^2*y^6 + 8^2*y^7 + y^8)*x^4/4 +...

Triangle begins:

1;

1, 4, 1;

1, 12, 27, 12, 1;

1, 24, 134, 236, 134, 24, 1;

1, 40, 410, 1540, 2380, 1540, 410, 40, 1;

1, 60, 975, 6260, 18386, 26216, 18386, 6260, 975, 60, 1;

1, 84, 1981, 19320, 91441, 227052, 306495, 227052, 91441, 19320, 1981, 84, 1;

1, 112, 3612, 49672, 344260, 1312080, 2883562, 3740572, 2883562, 1312080, 344260, 49672, 3612, 112, 1; ...

PROG

(PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, sum(j=0, 2*m, binomial(2*m, j)^2*y^j)*x^m/m)+O(x^(n+1))), n, x), k, y)}

CROSSREFS

Cf. A066357 (row sums), A181146 (main diagonal).

Cf. variants: A181143, A181144, A001263, A034870.

Sequence in context: A146990 A051433 A163366 * A227203 A140070 A158815

Adjacent sequences:  A181142 A181143 A181144 * A181146 A181147 A181148

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Oct 16 2010

EXTENSIONS

Comment and example corrected by Paul D. Hanna, Oct 16 2010

STATUS

approved

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Last modified September 20 17:16 EDT 2020. Contains 337265 sequences. (Running on oeis4.)