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A203776 Number of partitions of n into distinct parts 5k+1 or 5k+4. 15

%I

%S 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,

%T 8,12,15,15,13,12,16,22,22,18,18,24,30,31,27,26,33,42,43,37,37,47,57,

%U 58,53,52,63,78,80,71,72,88,103,106,99,98,116,139,142

%N Number of partitions of n into distinct parts 5k+1 or 5k+4.

%C Convolution of A281243 and A280454. - _Vaclav Kotesovec_, Jan 18 2017

%H Seiichi Manyama, <a href="/A203776/b203776.txt">Table of n, a(n) for n = 0..1000</a> (first 251 terms from Reinhard Zumkeller)

%H M. Somos, <a href="http://cis.csuohio.edu/~somos/multiq.pdf">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F 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

%F 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

%F Euler transform of period 10 sequence [ 1, -1, 0, 1, 0, 1, 0, -1, 1, 0, ...]. - _Michael Somos_, Mar 23 2013

%F G.f.: Product_{k>0} (1 + x^(5*k - 1)) * (1 + x^(5*k - 4)). - _Michael Somos_, Mar 23 2013

%F 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

%e a(10) = #{9+1, 6+4} = 2;

%e a(20) = #{19+1, 16+4, 14+6, 11+9, 9+6+4+1} = 5.

%e 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 + ...

%e q + q^61 + q^241 + q^301 + q^361 + q^421 + q^541 + 2*q^601 + 2*q^661 + q^721 + ...

%t 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 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^5] QPochhammer[ -x^4, x^5], {x, 0, n}] (* _Michael Somos_, Mar 23 2013 *)

%o (Haskell)

%o a203776 = p a047209_list where

%o p _ 0 = 1

%o p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

%o (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 */

%Y Cf. A047209, A003114.

%K nonn

%O 0,11

%A _Reinhard Zumkeller_, Jan 05 2012

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Last modified October 13 16:44 EDT 2019. Contains 327968 sequences. (Running on oeis4.)