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A253136 The number of overpartitions of n into parts congruent to 2, 4, or 5 modulo 6. 1
1, 0, 2, 0, 4, 2, 6, 4, 10, 8, 18, 14, 28, 24, 44, 42, 68, 66, 102, 104, 154, 160, 226, 238, 330, 354, 476, 516, 676, 742, 958, 1056, 1342, 1486, 1862, 2076, 2568, 2872, 3516, 3940, 4782, 5370, 6464, 7268, 8686, 9774, 11606, 13070, 15428, 17380, 20408, 22986 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

J. Lovejoy, A theorem on seven-colored overpartitions and its applications, Int. J. Number Theory 1 (2005), 215-224.

FORMULA

a(n) ~ Pi^(5/6) * exp(Pi*sqrt(n/2)) / (2^(7/4) * 3^(1/6) * Gamma(1/6) * n^(11/12)). - Vaclav Kotesovec, Jan 14 2021

MAPLE

series(mul((1+x^(6*k+2))*(1+x^(6*k+4))*(1+x^(6*k+5))/((1-x^(6*k+2))*(1-x^(6*k+4))*(1-x^(6*k+5))), k=0..100), x=0, 100);

# second Maple program:

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+

     `if`(irem(i, 6) in [2, 4, 5], add(2*b(n-i*j, i-1), j=1..n/i), 0)))

    end:

a:= n-> b(n$2):

seq(a(n), n=0..60);  # Alois P. Heinz, Jan 04 2019

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[ MemberQ[{2, 4, 5}, Mod[i, 6]], Sum[2b[n - i j, i-1], {j, 1, n/i}], 0]]];

a[n_] := b[n, n];

a /@ Range[0, 60] (* Jean-Fran├žois Alcover, Nov 23 2020, after Alois P. Heinz *)

nmax = 60; CoefficientList[Series[Product[(1 + x^(6*k+2)) * (1 + x^(6*k+4)) * (1 + x^(6*k+5)) / ((1 - x^(6*k+2)) * (1 - x^(6*k+4)) * (1 - x^(6*k+5))), {k, 0, nmax/6}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 14 2021 *)

CROSSREFS

Cf. A056970.

Sequence in context: A307704 A139716 A168232 * A216960 A285348 A163123

Adjacent sequences:  A253133 A253134 A253135 * A253137 A253138 A253139

KEYWORD

nonn

AUTHOR

Jeremy Lovejoy, Mar 23 2015

STATUS

approved

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Last modified October 16 00:29 EDT 2021. Contains 348034 sequences. (Running on oeis4.)