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A218298
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G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..2*n} A084606(n,k)^2 * x^k * A(x)^(2*k) ), where A084606(n,k) = [x^k] (1 + 2*x + 2*x^2)^n.
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2
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1, 1, 6, 37, 274, 2154, 17896, 153981, 1361702, 12297022, 112935652, 1051549970, 9903781784, 94183796404, 903135799468, 8722680673357, 84776578857670, 828531289070582, 8137311780855076, 80272417524869462, 795011346686319212, 7902010696389037900
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. satisfies:
(1) A(x) = (1 + x*A(x)^2)*(1 + 4*x^3*A(x)^6)*(1 + 4*x^4*A(x)^8)/(1 - 2*x^2*A(x)^4)^2.
(2) A(x) = sqrt( (1/x)*Series_Reversion( x*(1 - 2*x^2)^4 / ((1 + x)*(1 + 4*x^3)*(1 + 4*x^4))^2 ) ).
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EXAMPLE
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G.f.: A(x) = 1 + x + 6*x^2 + 37*x^3 + 274*x^4 + 2154*x^5 + 17896*x^6 +...
Let A = g.f. A(x), then the logarithm of the g.f. equals the series:
log(A(x)) = (1 + 2^2*x*A^2 + 2^2*x^2*A^4)*x*A +
(1 + 4^2*x*A^2 + 8^2*x^2*A^4 + 8^2*x^3*A^6 + 4^2*x^4*A^8)*x^2*A^2/2 +
(1 + 6^2*x*A^2 + 18^2*x^2*A^4 + 32^2*x^3*A^6 + 36^2*x^4*A^8 + 24^2*x^5*A^10 + 8^2*x^6*A^12)*x^3*A^3/3 +
(1 + 8^2*x*A^2 + 32^2*x^2*A^4 + 80^2*x^3*A^6 + 136^2*x^4*A^8 + 160^2*x^5*A^10 + 128^2*x^6*A^12 + 64^2*x^7*A^14 + 16^2*x^8*A^16)*x^4*A^4/4 +...
which involves the squares of the trinomial coefficients A084606(n,k):
1;
1, 2, 2;
1, 4, 8, 8, 4;
1, 6, 18, 32, 36, 24, 8;
1, 8, 32, 80, 136, 160, 128, 64, 16;
1, 10, 50, 160, 360, 592, 720, 640, 400, 160, 32; ...
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PROG
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(PARI) /* G.f. A(x) using the squares of the trinomial coefficients A084606: */
{A084606(n, k)=polcoeff((1 + 2*x + 2*x^2)^n, k)}
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, A084606(m, k)^2*x^k*(A+x*O(x^n))^(2*k))*x^m*A^m/m))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=sqrt(serreverse( x*(1 - 2*x^2)^4 / ((1 + x)*(1 + 4*x^3)*(1 + 4*x^4 +x*O(x^n)))^2)/x)); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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