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 A117957 Number of partitions of n into parts larger than 1 and congruent to 1 mod 4. 3

%I

%S 1,0,0,0,0,1,0,0,0,1,1,0,0,1,1,1,0,1,2,1,1,1,2,2,1,2,3,3,2,2,4,4,3,3,

%T 5,6,5,4,6,8,7,6,8,10,10,9,10,13,13,12,14,17,18,16,18,22,23,22,23,28,

%U 31,29,30,36,39,39,39,45,51,50,51,57,64,65,65,73,81,83,84,91,102,106,106

%N Number of partitions of n into parts larger than 1 and congruent to 1 mod 4.

%C Also number of partitions of n such that 2k and 2k+1 occur with the same multiplicities. Example: a(26)=3 because we have [11,10,3,2], [9,8,5,4] and [7,7,6,6]. It is easy to find a bijection between these partitions and those described in the definition.

%F G.f.: 1/product(1-x^(4i+1), i=1..infinity).

%F a(n) ~ exp(sqrt(n/6)*Pi) * Pi^(1/4) * Gamma(1/4) / (2^(31/8) * 3^(5/8) * n^(9/8)). - _Vaclav Kotesovec_, Mar 07 2016

%e a(26)=3 because we have [21,5],[17,9] and [13,13].

%p g:=1/product(1-x^(4*i+1),i=1..50): gser:=series(g,x=0,93): seq(coeff(gser,x,n),n=0..88);

%t nmax = 100; CoefficientList[Series[Product[1/(1-x^(4*k+1)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 07 2016 *)

%Y Cf. A035451, A035462.

%K nonn

%O 0,19

%A _Emeric Deutsch_, Apr 05 2006

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Last modified April 26 12:20 EDT 2019. Contains 322472 sequences. (Running on oeis4.)