A035386 Number of partitions of n into parts congruent to 2 mod 3.
1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 4, 4, 6, 5, 7, 7, 9, 9, 12, 11, 15, 15, 18, 19, 23, 23, 29, 29, 35, 37, 43, 45, 53, 55, 64, 68, 78, 82, 95, 99, 114, 121, 136, 145, 164, 173, 196, 208, 232, 248, 276, 294, 328, 349, 386, 413, 456, 486, 537, 572, 629, 673, 737, 787
Offset: 0
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
- James Mc Laughlin, Andrew V. Sills, Peter Zimmer, Rogers-Ramanujan-Slater Type Identities , arXiv:1901.00946 [math.NT]
Programs
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Maple
g:=add(x^(n*(3*n-1))/mul((1-x^(3*k))*(1-x^(3*k-1)), k = 1..n), n = 0..6): gser:=series(g,x,101): seq(coeff(gser,x,n), n = 0..100); # Peter Bala, Feb 02 2021
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Mathematica
nmax=100; CoefficientList[Series[Product[1/(1-x^(3*k+2)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 26 2015 *) nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 3] == 2, Do[poly[[j + 1]] -= poly[[j - k + 1]], {j, nmax, k, -1}];], {k, 2, nmax}]; poly2 = Take[poly, {2, nmax + 1}]; poly3 = 1 + Sum[poly2[[n]]*x^n, {n, 1, Length[poly2]}]; CoefficientList[Series[1/poly3, {x, 0, Length[poly2]}], x] (* Vaclav Kotesovec, Jan 13 2017 *) nmax = 50; s = Range[0, nmax/3]*3 + 2; Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *) -
PARI
{a(n)= if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - (k%3==2) * x^k, 1 + x * O(x^n)), n))} /* Michael Somos, Jul 24 2007 */
Formula
a(n) = 1/n*Sum_{k=1..n} A078182(k)*a(n-k), a(0) = 1. - Vladeta Jovovic, Nov 21 2002
Euler transform of period 3 sequence [ 0, 1, 0, ...]. - Michael Somos, Jul 24 2007
a(n) ~ Gamma(2/3) * exp(sqrt(2*n)*Pi/3) / (2^(11/6) * sqrt(3) * Pi^(1/3) * n^(5/6)) * (1 + (Pi/72 - 5/(3*Pi)) / sqrt(2*n)). - Vaclav Kotesovec, Feb 26 2015, extended Jan 24 2017
G.f.: A(x) = Sum_{n >= 0} x^(n*(3*n-1))/Product_{k = 1..n} ((1 - x^(3*k)) *(1 - x^(3*k-1))). (Set z = x^2 and q = x^3 in Mc Laughlin et al., Section 1.3, Entry 7.) - Peter Bala, Feb 02 2021
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