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A220420 Express sum_{n>=0} p(n)*x^n where p(n) is the partition function as a product prod_{k>=1} (1+a(k)*x^k). 5
1, 2, 1, 4, 1, 0, 1, 14, 1, -4, 1, -8, 1, -16, 1, 196, 1, -54, 1, -92, 1, -184, 1, 144, 1, -628, 1, -1040, 1, -2160, 1, 41102, 1, -7708, 1, -12932, 1, -27592, 1, 54020, 1, -98496, 1, -173720, 1, -364720, 1, 853624, 1, -1341970, 1, -2383916, 1, -4918536, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This is the PPE (power product expansion) of A000041.

When n is odd, a(n) = 1.

When n is even, a(n) = 2, 4, 0, 14, -4, -8, -16, 196, -54, -92, -184, 144, -628, -1040, -2160, 41102, ...

LINKS

Table of n, a(n) for n=1..55.

G. Alkauskas, One curious proof of Fermat's little theorem, arXiv:0801.0805 [math.NT], 2008; Amer. Math. Monthly, 116 (4), (April 2009), 362-364.

Giedrius Alkauskas, Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems, arXiv:1609.09842 [math.NT], 2016.

MATHEMATICA

terms = 55;

sol[0] = {}; sol[m_] := sol[m] = Join[sol[m-1], If[OddQ[m], {a[m] -> 1}, Solve[Thread[Table[PartitionsP[n], {n, 0, m}] == CoefficientList[Product[ 1+a[n]*x^n, {n, 1, m}] /. sol[m-1] + O[x]^(m+1), x]]][[1]]]];

A220420 = Array[a, terms] /. sol[terms] (* Jean-Fran├žois Alcover, Dec 06 2018 *)

PROG

(PARI) a(m) = {default(seriesprecision, m+1); ak = vector(m); pol = 1 / eta(x + x * O(x^m)); ak[1] = polcoeff(pol, 1); for (k=2, m, pol = taylor(pol / (1+ak[k-1]*x^(k-1)), x); ak[k] = polcoeff(pol, k, x); ); for (k=1, m, print1(ak[k], ", "); ); }

CROSSREFS

Sequence in context: A140505 A295881 A117971 * A190616 A238018 A131642

Adjacent sequences:  A220417 A220418 A220419 * A220421 A220422 A220423

KEYWORD

sign,changed

AUTHOR

Michel Marcus, Dec 14 2012

STATUS

approved

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Last modified December 17 06:45 EST 2018. Contains 318192 sequences. (Running on oeis4.)