A291596 Column 6 of A060244.

0, 0, 0, 0, 0, 4, 7, 19, 35, 69, 116, 204, 323, 523, 799, 1225, 1809, 2675, 3843, 5515, 7756, 10869, 14998, 20621, 27996, 37865, 50701, 67612, 89419, 117806, 154101, 200838, 260168, 335824, 431202, 551824, 702890, 892503, 1128577, 1422846, 1787183, 2238554

Offset: 1

Author

Vaclav Kotesovec, Aug 27 2017

Keywords

• nonn

Comments

Conjecture: column k of A060244 is asymptotic to (6*n)^((k-1)/2) * A000041(n) / (Pi^(k-1) * k!) ~ 2^((k-5)/2) * 3^((k-2)/2) * n^((k-3)/2) * exp(Pi*sqrt(2*n/3)) / (Pi^(k-1) * k!).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A060244.

Programs

Mathematica

nmax = 30; col = 6; Flatten[{0, 0, 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]}]
Rest[CoefficientList[Series[x^6 * (4 + 3*x + 4*x^2 + 2*x^3 - 4*x^5 - 6*x^6 - 6*x^7 - 5*x^8 - x^9 + x^10 + 5*x^11 + 3*x^12 + 3*x^13 + x^14 - x^15 - x^16 - x^17) / ((1 - x)^5 * (1 + x)^3 * (1 + x^2) * (1 - x + x^2) * (1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)) / QPochhammer[x], {x, 0, 100}], x]]
Table[Sum[(k^4/17280 + 101*k^3/8640 + 1661*k^2/4320 - 23017*k/17280 - 2563/576 + Floor[(k-5)/4]/8 - 7*Floor[(k-5)/3]/18 - (19/192 + 7*k/12 + k^2/96) * Floor[(k-5)/2] + Floor[(k-4)/6]/6 - Floor[(k-4)/4]/8 - (4/3 + k/18) * Floor[(k-4)/3] - Floor[(k-3)/5]/5 + Floor[(k-2)/5]/5) * PartitionsP[n-k], {k, 6, n}], {n, 1, 100}]

Formula

G.f.: x^6 * (4 + 3*x + 4*x^2 + 2*x^3 - 4*x^5 - 6*x^6 - 6*x^7 - 5*x^8 - x^9 + x^10 + 5*x^11 + 3*x^12 + 3*x^13 + x^14 - x^15 - x^16 - x^17) / ((1 - x)^5 * (1 + x)^3 * (1 + x^2) * (1 - x + x^2) * (1 + x + x^2)^2*(1 + x + x^2 + x^3 + x^4)) * Product_{k>=1} 1/(1 - x^k).

a(n) ~ exp(Pi*sqrt(2*n/3)) * n^(3/2) / (40*sqrt(2)*Pi^5).

a(n) ~ sqrt(3/2) * n^(5/2) * A000041(n) / (10*Pi^5).