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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. 20
 1, 1, 3, 10, 38, 153, 646, 2816, 12585, 57343, 265401, 1244256, 5896512, 28200365, 135935424, 659754072, 3221354296, 15812501100, 77985955410, 386254209762, 1920391362054, 9580985321554, 47951223856445, 240680464689600 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO]; see p.5 and p.32 Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications,arXiv:0805.1398v1 [math.CO], see p.5 Eric Weisstein's World of Mathematics, q-Pochhammer Symbol FORMULA 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 = 0.489635226684303373081541660578468619322416625... . - Vaclav Kotesovec, Nov 21 2016 EXAMPLE 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 + ... MATHEMATICA 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 *) InverseSeries[x QPochhammer[x] + O[x]^25][[3]] (* Vladimir Reshetnikov, Nov 17 2016 *) Table[SeriesCoefficient[1/QPochhammer[x, x]^(n+1), {x, 0, n}]/(n+1), {n, 0, 24}] (* Vladimir Reshetnikov, Nov 20 2016 *) PROG (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)} {a(n)=polcoeff(exp(sum(m=1, n, A008485(m)*x^m/m)+x*O(x^n)), n)} \\ Paul D. Hanna, Feb 06 2012 CROSSREFS Cf. A109084, A184362, A008485, A000041; related: A106336. Sequence in context: A151059 A151060 A151061 * A259690 A001002 A151062 Adjacent sequences:  A109082 A109083 A109084 * A109086 A109087 A109088 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 18 2005 STATUS approved

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Last modified August 6 09:38 EDT 2020. Contains 336245 sequences. (Running on oeis4.)