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A032021 Number of compositions (ordered partitions) of n into distinct odd parts. 7
1, 1, 0, 1, 2, 1, 2, 1, 4, 7, 4, 7, 6, 13, 6, 19, 32, 25, 32, 31, 58, 43, 82, 49, 132, 181, 156, 193, 230, 325, 278, 457, 376, 715, 448, 967, 1290, 1345, 1386, 1723, 2276, 2341, 3116, 2959, 4750, 3823, 6358, 4681, 9480, 10945, 11832, 12169, 16442, 18793, 21002, 25537, 27820, 37687 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

C. G. Bower, Transforms (2)

FORMULA

"AGK" (ordered, elements, unlabeled) transform of 1, 0, 1, 0...(odds)

G.f.: sum(k>=0, k! * x^(k^2) / prod(j=1..k, 1-x^(2*j) ) ). - Vladeta Jovovic, Aug 05 2004

MAPLE

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)

       ->x+y, b(n, i-2), [0, `if`(i>n, [], b(n-i, i-2))[]], 0)))

    end:

a:= proc(n) local l; l:= b(n, n-1+irem(n, 2));

      a(n):= add(l[i]*(i-1)!, i=1..nops(l))

    end:

seq(a(n), n=0..100);  # Alois P. Heinz, Nov 09 2012

MATHEMATICA

b[n_, i_] := If[n == 0, {1}, If[i<1, {}, Plus @@ PadRight[{b[n, i-2], Join[{0}, If[i>n, {}, b[n-i, i-2]]]}]]]; a[n_] := Module[{l}, l = b[n, n-1+Mod[n, 2]]; Sum[l[[i]]*(i-1)!, {i, 1, Length[l]}]]; Table[a[n], {n, 0, 100}] (* Jean-Fran├žois Alcover, Jan 30 2014, after Alois P. Heinz *)

PROG

(PARI)

N=66;  q='q+O('q^N);

gf=sum(k=0, N, k! * q^(k^2) / prod(j=1, k, 1-q^(2*j) ) );

Vec(gf)

/* Joerg Arndt, Sep 17 2012 */

CROSSREFS

Cf. A032020, A000700.

Sequence in context: A058511 A106380 A076198 * A295686 A246996 A260612

Adjacent sequences:  A032018 A032019 A032020 * A032022 A032023 A032024

KEYWORD

nonn

AUTHOR

Christian G. Bower

EXTENSIONS

Prepended a(0)=1, Joerg Arndt, Oct 20 2012

STATUS

approved

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Last modified February 22 13:09 EST 2018. Contains 299454 sequences. (Running on oeis4.)