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A036008 Number of partitions of n into parts not of the form 25k, 25k+9 or 25k-9. Also number of partitions with at most 8 parts of size 1 and differences between parts at distance 11 are greater than 1. 1
1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 54, 74, 96, 128, 165, 215, 274, 353, 445, 566, 708, 890, 1105, 1377, 1697, 2097, 2569, 3151, 3837, 4677, 5664, 6863, 8271, 9967, 11957, 14340, 17127, 20448, 24326, 28921, 34276, 40595, 47942, 56575, 66594 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Case k=12,i=9 of Gordon Theorem.

REFERENCES

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

Expansion of f(-q^9, -q^16) / f(-q) in powers of q where f(,) is the Ramanujan two-variable theta function. - Michael Somos, Apr 29 2008

G.f.: Product_{k>0} (1 - x^(25*k-9)) * (1 - x^(25*k-16)) * (1 - x^(25*k)) / (1 - x^k). - corrected by Vaclav Kotesovec, May 09 2018

a(n) ~ exp(2*Pi*sqrt(11*n/3)/5) * 11^(1/4) * cos(7*Pi/50) / (3^(1/4) * 5^(3/2) * n^(3/4)). - Vaclav Kotesovec, May 09 2018

EXAMPLE

1 + q + 2*q^2 + 3*q^3 + 5*q^4 + 7*q^5 + 11*q^6 + 15*q^7 + 22*q^8 + 29*q^9 + ...

MATHEMATICA

f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; a:= CoefficientList[Series[f[-q^9, -q^16]/f[-q, -q^2], {q, 0, 100}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Apr 17 2018 *)

nmax = 50; CoefficientList[Series[Product[(1 - x^(25*k - 9))*(1 - x^(25*k - 16))*(1 - x^(25*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 09 2018 *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n\25+1, (1 - x^(25*k-9)) * (1 - x^(25*k-16)) * (1 - x^(25*k)), 1 + x * O(x^n)) / eta(x + x * O(x^n)), n))} /* Michael Somos, Apr 29 2008 */

CROSSREFS

Sequence in context: A023029 A035987 A035997 * A104502 A027343 A184644

Adjacent sequences:  A036005 A036006 A036007 * A036009 A036010 A036011

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified September 19 23:51 EDT 2020. Contains 337199 sequences. (Running on oeis4.)