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A109085 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.

%I A109085 #63 Feb 16 2025 08:32:58
%S A109085 1,1,3,10,38,153,646,2816,12585,57343,265401,1244256,5896512,28200365,
%T A109085 135935424,659754072,3221354296,15812501100,77985955410,386254209762,
%U A109085 1920391362054,9580985321554,47951223856445,240680464689600
%N A109085 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.
%C A109085 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
%H A109085 Vaclav Kotesovec, <a href="/A109085/b109085.txt">Table of n, a(n) for n = 0..1000</a>
%H A109085 Guo-Niu Han, <a href="http://arxiv.org/abs/0804.1849">An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths</a>, arXiv:0804.1849 [math.CO]; see p.5 and p.32
%H A109085 Guo-Niu Han, <a href="http://arxiv.org/abs/0805.1398">The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications</a>,arXiv:0805.1398v1 [math.CO], see p.5
%H A109085 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>
%F A109085 G.f. A(x) satisfies:
%F A109085 (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.
%F A109085 (2) A(x) = 1/G(x) where G(x) is g.f. of A109084.
%F A109085 (3) A(x) = 1 / Product_{n>=1} (1 - x^n*A(x)^n).
%F A109085 (4) A(x) = Sum_{n>=0} x^n*A(x)^n / Product_{k=1..n} (1-x^k*A(x)^k).
%F A109085 (5) A(x) = Sum_{n>=0} (x*A(x))^(n^2) / Product_{k=1..n} (1-x^k*A(x)^k)^2.
%F A109085 (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
%F A109085 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
%F A109085 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
%F A109085 a(n) ~ c * d^n / n^(3/2), where d = A270915 = 5.3527013334866426877724... and c = A366022 = 0.489635226684303373081541660578468619322416625... . - _Vaclav Kotesovec_, Nov 21 2016
%e A109085 G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 153*x^5 + 646*x^6 + ...
%e A109085 G.f. satisfies: P(x*A(x)) = A(x) where P(x) is the partition function:
%e A109085 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 + ...
%e A109085 The g.f. A = A(x) also satisfies the identities:
%e A109085 (1) A(x) = 1/((1-x*A) * (1-x^2*A^2) * (1-x^3*A^3) * (1-x^4*A^4) * ...).
%e A109085 (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)) + ...
%e A109085 (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 + ...
%e A109085 The logarithm of the g.f. is given by:
%e A109085 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)) + ...
%e A109085 Explicitly,
%e A109085 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 + ...
%t A109085 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 *)
%t A109085 InverseSeries[x QPochhammer[x] + O[x]^25][[3]] (* _Vladimir Reshetnikov_, Nov 17 2016 *)
%t A109085 Table[SeriesCoefficient[1/QPochhammer[x, x]^(n+1), {x, 0, n}]/(n+1), {n, 0, 24}] (* _Vladimir Reshetnikov_, Nov 20 2016 *)
%o A109085 (PARI) {a(n)=polcoeff(1/x*serreverse(x*eta(x+x*O(x^n))),n)}
%o A109085 for(n=0,30,print1(a(n),", "))
%o A109085 (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)}
%o A109085 (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
%o A109085 for(n=0,30,print1(a(n),", "))
%o A109085 (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
%o A109085 for(n=0,30,print1(a(n),", "))
%o A109085 (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
%o A109085 (PARI) {A008485(n)=polcoeff(prod(k=1,n,1/(1-x^k +x*O(x^n))^n),n)}
%o A109085 {a(n)=polcoeff(exp(sum(m=1,n,A008485(m)*x^m/m)+x*O(x^n)),n)} \\ _Paul D. Hanna_, Feb 06 2012
%Y A109085 Cf. A109084, A184362, A008485, A000041; related: A106336.
%K A109085 nonn
%O A109085 0,3
%A A109085 _Paul D. Hanna_, Jun 18 2005