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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-Fran├žois 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.)