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A092885 Number of partitions of n in which no parts are multiples of 25. 4

%I #28 Oct 18 2018 03:07:30

%S 1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,

%T 1002,1255,1575,1957,2435,3008,3715,4560,5597,6831,8334,10121,12280,

%U 14841,17921,21560,25914,31050,37162,44352,52877,62876,74685,88507

%N Number of partitions of n in which no parts are multiples of 25.

%H Seiichi Manyama, <a href="/A092885/b092885.txt">Table of n, a(n) for n = 0..10000</a>

%H Kevin Acres, David Broadhurst, <a href="https://arxiv.org/abs/1810.07478">Eta quotients and Rademacher sums</a>, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.

%H T. Horie and N. Kanou, <a href="http://dx.doi.org/10.1007/BF02941667">Certain modular functions similar to the Dedekind eta function</a>, Abh. Math. Sem. Univ. Hamburg 72 (2002), 89-117. MR1941549 (2003j:11043).

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.

%F Expansion of q^(-1) * eta(q^25) / eta(q) in powers of q.

%F Euler transform of period 25 sequence [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, ...].

%F Given g.f. A(x), then B(x) = x * A(x) satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^3 + v^3 - 5*(u*v)^2 - 2*u*v *(u+v) - u*v.

%F G.f.: Product_{k>0} (1 - x^(25*k)) / (1 - x^k).

%F a(n) ~ exp(4*Pi*sqrt(n)/5) / (5*sqrt(10)*n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015

%F a(n) = (1/n)*Sum_{k=1..n} A227131(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Jun 16 2017

%e G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + ...

%e G.f. = q + q^2 + 2*q^3 + 3*q^4 + 5*q^5 + 7*q^6 + 11*q^7 + 15*q^8 + 22*q^9 + 30*q^10 + ...

%t a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 25, n, 25}] / Product[ 1 - x^k, {k, n}], {x, 0, n}];

%t a[ n_] := SeriesCoefficient[(QPochhammer[ x^25] / QPochhammer[ x]), {x, 0, n}]; (* _Michael Somos_, May 13 2014 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^25 + A) / eta(x + A), n))};

%o (PARI) {a(n) = local(A, m); if( n<0, 0, n++; m=5; A = x + O(x^6); while( m<n, m*=5; A = x * subst((A * (1 - 2*A + 4*A^2 - 3*A^3 + A^4 ) / (1 + 3*A + 4*A^2 + 2*A^3 + A^4) / x)^(1/5), x, x^5)); polcoeff( 1 / (1/A - A -1), n))};

%Y Cf. A000041, A096562, A227131.

%K nonn

%O 0,3

%A _Michael Somos_, Mar 10 2004

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