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A181145
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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.
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1
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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; ...
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PROG
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(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)}
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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