A103261 - OEIS

%I #14 Sep 01 2015 04:36:52

%S 1,20,200,1360,7200,32024,125280,443680,1450240,4435940,12827888,

%T 35346800,93377920,237675640,585229760,1398704736,3253934080,

%U 7386124520,16392493800,35634450320,75992326592,159199081600,328027789600

%N Number of partitions of 2n into parts with 10 types c^1 c^2...C^10 of each part. The even parts appear with multiplicity 1 for each type . The odd parts occur with multiplicity 2 for each part.

%C This is also Sequence(A080054)^(10) or sequence(A007096)^(5).

%C In general, if j > 0 and g.f. = Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^j, then a(n) ~ exp(Pi*sqrt(j*n/2)) * j^(1/4) / (2^(j/2 + 7/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 28 2015

%F G.f.:(theta_4(0, x^2)/theta_4(0, x))^10= (theta_3(0, x)/theta_4(0, x))^5.

%F a(n) ~ exp(Pi*sqrt(5*n)) * 5^(1/4) / (64 * sqrt(2) * n^(3/4)). - _Vaclav Kotesovec_, Aug 28 2015

%e a(2)=200 because we have 10 types of 4, 45 ways of writing 4 in terms of ten of 2's only or ten of 11's only and 100 ways of writing 2's combined with 11's so the total number of ways of writing 4 is 200.

%p series(product(((1+x^k)*(1-x^(2*k)))^(10)/((1-x^k)*(1+x^(2*k)))^(10),k=1..100),x=0,100);

%t nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^10,{k,0,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Aug 28 2015 *)

%Y Cf. A080054 (j=1), A007096 (j=2), A261647 (j=3), A014969 (j=4), A261648 (j=5), A014970 (j=6), A014972 (j=8).

%K nonn

%O 0,2

%A _Noureddine Chair_, Feb 16 2005

%E Example corrected by _Vaclav Kotesovec_, Sep 01 2015