A203776 Number of partitions of n into distinct parts 5k+1 or 5k+4.
1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 2, 3, 3, 2, 2, 3, 5, 5, 3, 3, 5, 7, 7, 6, 5, 7, 11, 11, 8, 8, 12, 15, 15, 13, 12, 16, 22, 22, 18, 18, 24, 30, 31, 27, 26, 33, 42, 43, 37, 37, 47, 57, 58, 53, 52, 63, 78, 80, 71, 72, 88, 103, 106, 99, 98, 116, 139, 142
Offset: 0
Keywords
Examples
a(10) = #{9+1, 6+4} = 2;
a(20) = #{19+1, 16+4, 14+6, 11+9, 9+6+4+1} = 5.
1 + x + x^4 + x^5 + x^6 + x^7 + x^9 + 2*x^10 + 2*x^11 + x^12 + x^13 + 2*x^14 + ...
q + q^61 + q^241 + q^301 + q^361 + q^421 + q^541 + 2*q^601 + 2*q^661 + q^721 + ...
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000 (first 251 terms from Reinhard Zumkeller)
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Haskell
a203776 = p a047209_list where p _ 0 = 1 p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
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Mathematica
a[ n_] := SeriesCoefficient[ Product[ (1 + x^(5 k - 1)) (1 + x^(5 k - 4)), {k, Ceiling[ n / 5]}], {x, 0, n}] (* Michael Somos, Mar 23 2013 *) a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^5] QPochhammer[ -x^4, x^5], {x, 0, n}] (* Michael Somos, Mar 23 2013 *) -
PARI
{a(n) = polcoeff( prod( k=1, ceil(n / 5), (1 + x^(5*k - 1)) * (1 + x^(5*k - 4)), 1 + x * O(x^n)), n)} /* Michael Somos, Mar 23 2013 */
Formula
Expansion of f( x, x^4) / f(-x^5, -x^10) in powers of x where f() is the Ramanujan two-variable theta function. - Michael Somos, Mar 23 2013
Expansion of (-x; x^5)oo (-x^4; x^5)_oo in powers of x where (x; q)_oo is the q-Pochhammer symbol. - _Michael Somos, Mar 23 2013
Euler transform of period 10 sequence [ 1, -1, 0, 1, 0, 1, 0, -1, 1, 0, ...]. - Michael Somos, Mar 23 2013
G.f.: Product_{k>0} (1 + x^(5*k - 1)) * (1 + x^(5*k - 4)). - Michael Somos, Mar 23 2013
a(n) ~ exp(sqrt(2*n/15)*Pi) / (2*30^(1/4)*n^(3/4)) * (1 + (Pi/(60*sqrt(30)) - 3*sqrt(15/2)/(8*Pi)) / sqrt(n)). - Vaclav Kotesovec, Jan 18 2017, extended Jan 24 2017
Comments