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A214130 Partitions of n into parts congruent to +-2, +-3 (mod 13). 3
1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 7, 9, 9, 11, 12, 14, 15, 18, 19, 23, 24, 28, 30, 35, 37, 43, 46, 52, 56, 64, 68, 77, 84, 93, 101, 113, 121, 135, 146, 161, 174, 193, 207, 229, 247, 272, 292, 322, 346, 379, 408, 446, 479, 524, 562, 613, 659 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of f(-x^13)^2 / (f(-x^2, -x^11) * f(-x^3, -x^10)) in powers of x where f() is Ramanujan's two-variable theta function.

Euler transform of period 13 sequence [ 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, ...].

G.f.: 1 / (Product_{k>0} (1 - x^(13*k - 2)) * (1 - x^(13*k - 3)) * (1 - x^(13*k - 10)) * (1 - x^(13*k - 11))).

A214129(n) = a(n) + A214131(n-1).

EXAMPLE

1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + ...

q^-1 + q^11 + q^17 + q^23 + q^29 + 2*q^35 + q^41 + 2*q^47 + 2*q^53 + ...

MAPLE

with (numtheory):

a:= proc(n) option remember; `if`(n=0, 1, add(add(

      `if`(irem(d, 13) in [2, 3, 10, 11], d, 0),

          d=divisors(j)) *a(n-j), j=1..n)/n)

    end:

seq(a(n), n=0..100);  # Alois P. Heinz, Oct 23 2013

MATHEMATICA

a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ q^2, q^13] QPochhammer[ q^3, q^13] QPochhammer[ q^10, q^13] QPochhammer[ q^11, q^13]), {q, 0, n}]

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[If[MemberQ[{2, 3, 10, 11}, Mod[d, 13]], d, 0], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 100}] (* Jean-Fran├žois Alcover, Nov 11 2015, after Alois P. Heinz *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - [ 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0][k%13 + 1] * x^k, 1 + x * O(x^n)), n))}

CROSSREFS

Cf. A214129, A214131.

Sequence in context: A029168 A035454 A161228 * A029172 A240864 A241322

Adjacent sequences:  A214127 A214128 A214129 * A214131 A214132 A214133

KEYWORD

nonn

AUTHOR

Michael Somos, Jul 04 2012

STATUS

approved

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Last modified July 22 03:22 EDT 2017. Contains 289648 sequences.