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A369549
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Expansion of g.f. A(x) satisfying A(x) = A( x^2*(1+x)^6 ) / (x*(1+x)^5).
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7
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1, 1, 1, 7, 22, 48, 120, 440, 1941, 8621, 35496, 135200, 491951, 1742491, 6016240, 20329172, 68107458, 229526914, 786489953, 2761171223, 9982883073, 37215348447, 142623131714, 558706789122, 2222790569335, 8927649606731, 36019589692877, 145421164021659, 585841066191547
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OFFSET
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1,4
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COMMENTS
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The radius of convergence r of g.f. A(x) solves r*(1+r)^6 = 1 where r = 0.2554228710768465432050...
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A(x) = A( x^2*(1+x)^6 ) / (x*(1+x)^5).
(2) R(x*(1+x)^5*A(x)) = x^2*(1+x)^6, where R(A(x)) = x.
(3) A(x) = x * Product_{n>=1} F(n), where F(1) = 1+x, and F(n+1) = 1 + (F(n) - 1)^2 * F(n)^6 for n >= 1.
(4) A(x)^6 = x^5*B(x) where B(x) is the g.f. of A369556.
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EXAMPLE
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G.f.: A(x) = x + x^2 + x^3 + 7*x^4 + 22*x^5 + 48*x^6 + 120*x^7 + 440*x^8 + 1941*x^9 + 8621*x^10 + 35496*x^11 + 135200*x^12 + ...
RELATED SERIES.
A(x)^6/x^5 = x + 6*x^2 + 21*x^3 + 92*x^4 + 432*x^5 + 1704*x^6 + 6276*x^7 + 24096*x^8 + 95628*x^9 + ... + A369556(n)*x^n + ...
Let R(x) be the series reversion of A(x),
R(x) = x - x^2 + x^3 - 7*x^4 + 16*x^5 - 27*x^6 + 150*x^7 - 539*x^8 + 903*x^9 - 4826*x^10 + 21651*x^11 - 40859*x^12 + ...
then R(x) and g.f. A(x) satisfy:
(1) R(A(x)) = x,
(2) R(x*(1+x)^5*A(x)) = x^2*(1 + x)^6.
GENERATING METHOD.
Define F(n), a polynomial in x of order 8^(n-1), by the following recurrence:
F(1) = (1 + x),
F(2) = (1 + x^2 * (1+x)^6),
F(3) = (1 + x^4 * (1+x)^12 * F(2)^6),
F(4) = (1 + x^8 * (1+x)^24 * F(2)^12 * F(3)^6),
F(5) = (1 + x^16 * (1+x)^48 * F(2)^24 * F(3)^12 * F(4)^6),
...
F(n+1) = 1 + (F(n) - 1)^2 * F(n)^6
...
Then the g.f. A(x) equals the infinite product:
A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...
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PROG
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(PARI) {a(n) = my(A=[1], F); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = polcoeff( subst(F, x, x^2*(1 + x)^6 ) - x*(1 + x)^5*F , #A+1) ); A[n]}
for(n=1, 35, 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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