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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 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.)