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A103261 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. 4
1, 20, 200, 1360, 7200, 32024, 125280, 443680, 1450240, 4435940, 12827888, 35346800, 93377920, 237675640, 585229760, 1398704736, 3253934080, 7386124520, 16392493800, 35634450320, 75992326592, 159199081600, 328027789600 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

LINKS

Table of n, a(n) for n=0..22.

FORMULA

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

a(n) ~ exp(Pi*sqrt(5*n)) * 5^(1/4) / (64 * sqrt(2) * n^(3/4)). - Vaclav Kotesovec, Aug 28 2015

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

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

Sequence in context: A008420 A045758 A035474 * A120796 A120787 A223753

Adjacent sequences:  A103258 A103259 A103260 * A103262 A103263 A103264

KEYWORD

nonn

AUTHOR

Noureddine Chair, Feb 16 2005

EXTENSIONS

Example corrected by Vaclav Kotesovec, Sep 01 2015

STATUS

approved

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Last modified March 28 14:04 EDT 2020. Contains 333089 sequences. (Running on oeis4.)