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A109085
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G.f. A(x) satisfies: A(x) = P(x*A(x)) where P(x) = A(x/P(x)) is the g.f. of the partition numbers A000041.
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22
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1, 1, 3, 10, 38, 153, 646, 2816, 12585, 57343, 265401, 1244256, 5896512, 28200365, 135935424, 659754072, 3221354296, 15812501100, 77985955410, 386254209762, 1920391362054, 9580985321554, 47951223856445, 240680464689600
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OFFSET
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0,3
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COMMENTS
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a(n) = Sum[Product(1 + n/h(v)^2)]/(n+1), where the product is over all boxes v in the Ferrers diagram of a partition L of n, h(v) is the hook length of v and the summation is over all partitions L of n. Example: a(3)=10 because for the partitions L=(3), (2,1), (1,1,1) of n=3 the hook length multi-sets are {3,2,1}, {3,1,1},{3,2,1}, respectively, the products are (1+3/9)(1+3/4)(1+3/1)=28/3, (1+3/9)(1+3/1)(1+3/1)=64/3, (1+3/9)(1+3/4)(1+3/1)=28/3 and now a(3)=(1/4)(28+64+28)/3=10. - Emeric Deutsch, May 15 2008
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion(x*eta(x)), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.
(2) A(x) = 1/G(x) where G(x) is g.f. of A109084.
(3) A(x) = 1 / Product_{n>=1} (1 - x^n*A(x)^n).
(4) A(x) = Sum_{n>=0} x^n*A(x)^n / Product_{k=1..n} (1-x^k*A(x)^k).
(5) A(x) = Sum_{n>=0} (x*A(x))^(n^2) / Product_{k=1..n} (1-x^k*A(x)^k)^2.
(6) A(x) = exp( Sum_{n>=1} (x^n/n) * A(x)^n/(1 - x^n*A(x)^n) ). - Paul D. Hanna, Jun 01 2011
Logarithmic derivative yields A008485, where A008485(n) is the number of partitions of n into parts of n kinds. - Paul D. Hanna, Feb 06 2012
a(n) = ([x^n] 1/((x; x)_inf)^(n+1))/(n+1), where (a; q)_inf is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 20 2016
a(n) ~ c * d^n / n^(3/2), where d = A270915 = 5.3527013334866426877724... and c = A366022 = 0.489635226684303373081541660578468619322416625... . - Vaclav Kotesovec, Nov 21 2016
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 153*x^5 + 646*x^6 + ...
G.f. satisfies: P(x*A(x)) = A(x) where P(x) is the partition function:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
The g.f. A = A(x) also satisfies the identities:
(1) A(x) = 1/((1-x*A) * (1-x^2*A^2) * (1-x^3*A^3) * (1-x^4*A^4) * ...).
(2) A(x) = 1 + x*A/(1-x*A) + x^2*A^2/((1-x*A)*(1-x^2*A^2)) + x^3*A^3/((1-x*A)*(1-x^2*A^2)*(1-x^3*A^3)) + ...
(3) A(x) = 1 + x*A/(1-x*A)^2 + x^4*A^4/((1-x*A)*(1-x^2*A^2))^2 + x^9*A^9/((1-x*A)*(1-x^2*A^2)*(1-x^3*A^3))^2 + ...
The logarithm of the g.f. is given by:
log(A(x)) = x*A(x)/(1-x*A(x)) + x^2*A(x)^2/(2*(1-x^2*A(x)^2)) + x^3*A(x)^3/(3*(1-x^3*A(x)^3)) + x^4*A(x)^4/(4*(1-x^4*A(x)^4)) + x^5*A(x)^5/(5*(1-x^5*A(x)^5)) + ...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 105*x^4/4 + 506*x^5/5 + 2492*x^6/6 + 12405*x^7/7 + 62337*x^8/8 + ... + A008485(n)*x^n/n + ...
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MATHEMATICA
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A109085list[n_] := Module[{m = 1, A = 1 + x}, For[i = 1, i <= n, i++, A = 1/Product[(1 - x^k*(A + x*O[x]^n)^k), {k, 1, n}]]; CoefficientList[A, x][[1 ;; n]]]; A109085list[24] (* Jean-François Alcover, Apr 21 2016, adapted from PARI *)
Table[SeriesCoefficient[1/QPochhammer[x, x]^(n+1), {x, 0, n}]/(n+1), {n, 0, 24}] (* Vladimir Reshetnikov, Nov 20 2016 *)
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PROG
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(PARI) {a(n)=polcoeff(1/x*serreverse(x*eta(x+x*O(x^n))), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-x^k*(A+x*O(x^n))^k))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*A^m/prod(k=1, m, (1-x^k*(A+x*O(x^n))^k)))); polcoeff(A, n)} \\ Paul D. Hanna, Nov 24 2012
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, sqrtint(n+1), (x*A)^(m^2)/prod(k=1, m, (1-x^k*(A+x*O(x^n))^k)^2))); polcoeff(A, n)} \\ Paul D. Hanna, Nov 24 2012
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (x^m*A^m/m)/(1-x^m*A^m+x*O(x^n)) ))); polcoeff(A, n)} \\ Paul D. Hanna, Jun 01 2011
(PARI) {A008485(n)=polcoeff(prod(k=1, n, 1/(1-x^k +x*O(x^n))^n), 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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