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

0,9

Comments

a(n) = A116376(3*n). - _Reinhard Zumkeller_, Feb 15 2006

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]

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

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

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 *)

Prog

(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 */

Crossrefs

Cf. A035382, A035451, A262928.

Sequence in context: A051274 A267806 A025797 * A244327 A319318 A029164

Adjacent sequences: A035383 A035384 A035385 * A035387 A035388 A035389

Keyword

nonn,easy

Author

_Olivier Gérard_

Status

approved