A291589 - OEIS

%I #9 Aug 27 2017 06:57:10

%S 0,0,0,2,3,8,13,26,40,69,104,165,241,363,517,750,1046,1473,2018,2779,

%T 3746,5063,6733,8959,11769,15454,20082,26068,33549,43108,54997,70037,

%U 88645,111979,140714,176462,220280,274418,340480,421593,520154,640481,786104,962976

%N Column 4 of A060244.

%H Vaclav Kotesovec, <a href="/A291589/b291589.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: x^4 * (2 + x + x^2 - x^3 - x^4 - x^5 - x^6 + x^7) / ((1 - x)^3 * (1 + x)^2 * (1 + x^2) * (1 + x + x^2)) * Product_{k>=1} 1/(1 - x^k).

%F a(n) ~ exp(Pi*sqrt(2*n/3)) * sqrt(n) / (8*sqrt(2)*Pi^3).

%F a(n) ~ sqrt(3) * n^(3/2) * A000041(n) / (2^(3/2) * Pi^3).

%t nmax = 50; col = 4; Flatten[{0, 0, 0, CoefficientList[Coefficient[Normal[Series[Product[Product[1/(1 - x^(i - j)*y^j), {j, 0, i}], {i, 2, nmax + col}], {x, 0, col}, {y, 0, nmax}]], x^col], y]}]

%t Rest[CoefficientList[Series[x^4*(2 + x + x^2 - x^3 - x^4 - x^5 - x^6 + x^7) / ((1 - x)^3 * (1 + x)^2 * (1 + x^2) * (1 + x + x^2)) / QPochhammer[x], {x, 0, 100}], x]]

%t Table[Sum[(7*k^2/48 + 47*(k-4)/48 + Floor[(k-3)/4]/2 - (2*k + 19)*Floor[(k-3)/2]/8 + Floor[(k-2)/3]/3 - Floor[k/3]/3) * PartitionsP[n-k], {k, 4, n}], {n, 1, 50}]

%Y Cf. A060244.

%K nonn

%O 1,4

%A _Vaclav Kotesovec_, Aug 27 2017