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A130780 Number of partitions of n such that number of odd parts is greater than or equal to number of even parts. 39
1, 1, 1, 3, 3, 6, 8, 12, 16, 23, 32, 42, 58, 75, 102, 131, 173, 220, 288, 363, 466, 587, 743, 929, 1164, 1448, 1797, 2224, 2738, 3368, 4122, 5042, 6133, 7466, 9035, 10941, 13184, 15888, 19064, 22876, 27343 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
a(n) = A108950(n) + A045931(n) = A000041(n) - A108949(n). - Reinhard Zumkeller, Jan 21 2010
a(n) = Sum_{k=0..n} A240009(n,k). - Alois P. Heinz, Mar 30 2014
LINKS
FORMULA
G.f.: Sum_{k>=0} x^k/Product_{i=1..k} (1-x^(2*i))^2.
EXAMPLE
a(5)=6 because we have 5,41,32,311,211 and 11111 (221 does not qualify).
MAPLE
g:=sum(x^k/(product((1-x^(2*i))^2, i=1..k)), k=0..50): gser:=series(g, x=0, 50): seq(coeff(gser, x, n), n=1..40); # Emeric Deutsch, Aug 24 2007
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0,
`if`(t>=0, 1, 0), `if`(i<1, 0, b(n, i-1, t)+
`if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1)))))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80); # Alois P. Heinz, Mar 30 2014
MATHEMATICA
$RecursionLimit = 1000; b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t >= 0, 1, 0], If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t + (2*Mod[i, 2]-1)]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, May 12 2015, after Alois P. Heinz *)
opgQ[n_]:=Module[{len=Length[n], op}, op=Length[Select[n, OddQ]]; op>= len-op]; Table[Count[IntegerPartitions[n], _?(opgQ)], {n, 0, 50}] (* Harvey P. Dale, Dec 12 2021 *)
CROSSREFS
Cf. A171966, A171967. - Reinhard Zumkeller, Jan 21 2010
Sequence in context: A241390 A241831 A239946 * A174524 A143592 A280197
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Aug 19 2007
EXTENSIONS
More terms from Emeric Deutsch, Aug 24 2007
STATUS
approved

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Last modified March 2 19:19 EST 2024. Contains 370498 sequences. (Running on oeis4.)