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%I #30 Jan 23 2024 02:08:38
%S 1,1,3,5,9,14,24,35,55,81,120,171,248,345,486,669,920,1246,1690,2256,
%T 3014,3984,5253,6870,8970,11618,15022,19306,24745,31557,40154,50845,
%U 64244,80850,101501,126982,158514,197218,244865,303143,374497,461435
%N The number phi_2(n) of Frobenius partitions that allow up to 2 repetitions of an integer in a row.
%C Sum of products of multiplicities of odd parts in all partitions of n (if there are no odd parts in a partition then product of multiplicities is considered to be 1). - _Vladeta Jovovic_, Feb 16 2005
%C The sequence A077285 is the same but with multiplicities of all parts.
%D George E. Andrews, Generalized Frobenius partitions, Memoirs of the American Mathematical Society, Number 301, May 1984.
%H Alois P. Heinz, <a href="/A053993/b053993.txt">Table of n, a(n) for n = 0..1000</a>
%H Brian Drake, <a href="http://dx.doi.org/10.1016/j.disc.2008.11.020">Limits of areas under lattice paths</a>, Discrete Math. 309 (2009), no. 12, 3936-3953. See formula (18) on page 3944.
%F Expansion of q^(1/12) * eta(q^4) * eta(q^6)^2 / (eta(q) * eta(q^2) * eta(q^3) * eta(q^12)) in powers of q. - _Michael Somos_, Mar 09 2011
%F Euler transform of period 12 sequence [ 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, ...]. - _Michael Somos_, Mar 09 2011
%F G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(12*k - 10)) * (1 - x^(12*k - 9)) * (1 - x^(12*k - 3)) * (1 - x^(12*k - 2)))^(-1). [Andrews, p. 10, equation (5.9)]
%F a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (6*sqrt(2)*n). - _Vaclav Kotesovec_, Nov 28 2015
%e 1 + x + 3*x^2 + 5*x^3 + 9*x^4 + 14*x^5 + 24*x^6 + 35*x^7 + 55*x^8 + ...
%e q^-1 + q^11 + 3*q^23 + 5*q^35 + 9*q^47 + 14*q^59 + 24*q^71 + 35*q^83 + ...
%e a(6) = 24 since the 5 partitions 6 = 5+1 = 4+2 = 3+2+1 = 2+2+2 each contribute 1, the 3 partitions 4+1+1 = 3+3 = 2+2+1+1 each contribute 2, the partition 3+1+1+1 contributes 3, the partition 2+1+1+1+1 contributes 4, and the partition 1+1+1+1+1+1 contributes 6 to give total 24 = 5*1 + 3*2 + 1*3 + 1*4 + 1*6. - _Michael Somos_, Mar 09 2011
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)
%p +add(b(n-i*j, i-1)*`if`(irem(i, 2)=1, j, 1), j=1..n/i)))
%p end:
%p a:= n-> b(n, n):
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Jul 16 2013
%t b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-1] * If[Mod[i, 2] == 1, j, 1], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jul 15 2015, after _Alois P. Heinz_ *)
%t QP = QPochhammer; s = QP[q^4] * (QP[q^6]^2 / (QP[q] * QP[q^2] * QP[q^3] * QP[q^12])) + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 09 2015, adapted from PARI *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x + A) * eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))} /* _Michael Somos_, Mar 09 2011 */
%Y Cf. A077285, A052992, A097242, A247661, A247662.
%K easy,nonn
%O 0,3
%A _James A. Sellers_, Apr 04 2000
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