A036033 - OEIS

A036033

Number of partitions of n into parts not of form 4k+2, 24k, 24k+9 or 24k-9. Also number of partitions in which no odd part is repeated, with at most 4 parts of size less than or equal to 2 and where differences between parts at distance 5 are greater than 1 when the smallest part is odd and greater than 2 when the smallest part is even.

1, 1, 1, 2, 3, 4, 5, 7, 10, 12, 15, 20, 26, 32, 39, 49, 62, 75, 90, 111, 136, 163, 194, 234, 282, 334, 394, 469, 557, 654, 765, 900, 1058, 1232, 1431, 1669, 1943, 2248, 2595, 3002, 3470, 3990, 4580, 5265, 6045, 6915, 7897, 9026, 10307, 11733, 13338, 15170

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OFFSET

0,4

COMMENTS

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Case k=6, i=5 of Gordon/Goellnitz/Andrews Theorem.

REFERENCES

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 114.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..5000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of f(-q^9,-q^15)/psi(-q) in powers of q where psi(),f() are Ramanujan theta functions. - Michael Somos, Oct 28 2006

Euler transform of period 24 sequence [ 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, ...]. - Michael Somos, Oct 28 2006

a(n) ~ 5^(1/4) * sqrt(2 + sqrt(2)) * exp(sqrt(5*n/3)*Pi/2) / (8*3^(3/4)*n^(3/4)). - Vaclav Kotesovec, May 09 2018

MATHEMATICA

f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; CoefficientList[Series[f[-q^9, -q^15]/f[-q, -q^3], {q, 0, 50}], q] (* G. C. Greubel, Apr 15 2018 *)

nmax = 50; CoefficientList[Series[Product[(1 - x^(4*k - 2))*(1 - x^(24*k))*(1 - x^(24*k + 9 - 24))*(1 - x^(24*k - 9))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 09 2018 *)

PROG

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)/(eta(x+A)*eta(x^4+A))*eta(x^24+A)* prod(k=1, ceil(n/24), (1-x^(24*k-9))*(1-x^(24*k-15)), 1+A), n))} /* Michael Somos, Oct 28 2006 */

CROSSREFS

Sequence in context: A089597 A022957 A036028 * A124243 A132975 A213267

Adjacent sequences: A036030 A036031 A036032 * A036034 A036035 A036036

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

EXTENSIONS

Name corrected by Vaclav Kotesovec, May 09 2018

STATUS

approved