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A130126 Number of partitions of n in which each even part has odd multiplicity. 7
1, 1, 2, 3, 4, 6, 10, 13, 17, 24, 33, 43, 58, 75, 98, 127, 161, 205, 262, 328, 414, 517, 641, 794, 982, 1205, 1475, 1803, 2197, 2664, 3230, 3896, 4693, 5640, 6754, 8077, 9647, 11479, 13637, 16178, 19152, 22624, 26695, 31426, 36948, 43372, 50819, 59463, 69490 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: Product_{n>=1} (1+q^(2n)-q^(4n))/((1-q^(2n-1))(1-q^(4n))).

a(n) ~ sqrt(Pi^2/2 + 4*log(phi)^2) * exp(sqrt((Pi^2 + 8*log(phi)^2)*(n/2))) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

EXAMPLE

a(5) = 6 because we have 5, 41, 32, 311, 2111 and 11111 (221 does not qualify).

MAPLE

g:=product((1+q^(2*n)-q^(4*n))/((1-q^(2*n-1))*(1-q^(4*n))), n=1..50): gser:= series(g, q=0, 45): seq(coeff(gser, q, n), n=0..42); # Emeric Deutsch, Aug 24 2007

# second Maple program:

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+

      add(`if`(irem(i, 2)=0 and irem(j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)))

    end:

a:= n-> b(n, n):

seq(a(n), n=0..50);  # Alois P. Heinz, Feb 27 2013

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + Sum[If[Mod[i, 2] == 0 && Mod[j, 2] == 0, 0, b[n - i*j, i - 1]], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-Fran├žois Alcover, Jan 24 2014, after Alois P. Heinz *)

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k) - x^(4*k))/((1-x^(2*k-1)) * (1-x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)

CROSSREFS

Cf. A131942.

Sequence in context: A241547 A273542 A061018 * A288338 A121152 A229863

Adjacent sequences:  A130123 A130124 A130125 * A130127 A130128 A130129

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Aug 01 2007

EXTENSIONS

More terms from Emeric Deutsch, Aug 24 2007

STATUS

approved

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Last modified July 8 01:48 EDT 2020. Contains 335502 sequences. (Running on oeis4.)