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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
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.
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Oct 16 2010
EXTENSIONS
Comment and example corrected by Paul D. Hanna, Oct 16 2010
STATUS
approved