A214129 -id:A214129 - cp's OEIS frontend

A214131 Partitions of n into parts congruent to +-4, +-6 (mod 13).

1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 10, 12, 12, 15, 15, 18, 19, 22, 23, 27, 28, 32, 35, 39, 41, 47, 50, 56, 60, 67, 71, 80, 85, 94, 101, 113, 119, 132, 141, 156, 166, 183, 195, 215, 229, 250, 268, 293, 313, 341

Offset: 0

Michael Somos, Jul 04 2012

nonn

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

			1 + x^4 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + 2*x^12 + 2*x^13 + 2*x^14 + ...
q^5 + q^29 + q^41 + q^47 + q^53 + q^59 + q^65 + q^71 + 2*q^77 + 2*q^83 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A214129, A214130.

a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ q^4, q^13] QPochhammer[ q^6, q^13] QPochhammer[ q^7, q^13] QPochhammer[ q^9, q^13]), {q, 0, n}]

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

Expansion of f(-x^13)^2 / (f(-x^4, -x^9) * f(-x^6, -x^7)) in powers of x where f() is Ramanujan's two-variable theta function.
Euler transform of period 13 sequence [ 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, ...].
G.f.: 1 / (Product_{k>0} (1 - x^(13*k - 4)) * (1 - x^(13*k - 6)) * (1 - x^(13*k - 7)) * (1 - x^(13*k - 9))).
A214129(n) = A214130(n) + a(n-1).

A214130 Partitions of n into parts congruent to +-2, +-3 (mod 13).

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jul 04 2012

nonn

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

			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 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A214129, A214131.

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

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-François Alcover, Nov 11 2015, after Alois P. Heinz *)

{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))}

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).

cp's OEIS Frontend

A214131 Partitions of n into parts congruent to +-4, +-6 (mod 13).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula