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A109085
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G.f. A(x) satisfies: A(x) = G000041(x*A(x)) where G000041(x) is the g.f. of the partition numbers A000041.
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8
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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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 (deutsch(AT)duke.poly.edu), May 15 2008
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REFERENCES
| Guo-Niu Han, An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths, arXiv:0804.1849v3 [math.CO] 9 May 2008 (p. 5).
Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, arXiv:0805.1398v1 [math.CO] 9 May 2008 (p. 5).
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FORMULA
| G.f.: A(x) = (1/x)*series_reversion(x*eta(x)).
G.f.: A(x) = 1/G109084(x) where G109084(x) is g.f. of A109084.
G.f. satisfies: A(x) = exp( Sum_{n>=1} (x^n*A(x)^n/(1 - x^n*A(x)^n))/n ). [From Paul D. Hanna, Jun 1 2011]
Logarithmic derivative yields A008485, where A008485(n) is the number of partitions of n into parts of n kinds. [From Paul D. Hanna, Feb 06 2012]
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EXAMPLE
| G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 153*x^5 + 646*x^6 +...
log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 105*x^4/4 + 506*x^5/5 + 2492*x^6/6 +...
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PROG
| (PARI) a(n)=polcoeff(1/x*serreverse(x*eta(x+x*O(x^n))), n)
(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=exp(sum(m=1, n, x^m*A^m/(1-x^m*A^m+x*O(x^n))/m))); polcoeff(A, n)} [From Paul D. Hanna, Jun 1 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 */
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CROSSREFS
| Cf. A109084, A008485, A000041; related: A106336.
Sequence in context: A151059 A151060 A151061 * A001002 A151062 A000902
Adjacent sequences: A109082 A109083 A109084 * A109086 A109087 A109088
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KEYWORD
| nonn,changed
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Jun 18 2005
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