A348476 Number of compositions of n into exactly n nonnegative parts such that all positive parts are odd.

1, 1, 1, 4, 13, 36, 106, 323, 981, 2992, 9196, 28392, 87946, 273287, 851579, 2659764, 8324357, 26100560, 81969496, 257800532, 811862268, 2559731360, 8079294664, 25525787344, 80719066698, 255466082911, 809138591431, 2564605664428, 8134003910311, 25813957574292

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1968

Formula

a(n) ~ c * d^n / sqrt(Pi*n), where d = 3.22870495109450172934784925586... is largest positive root of the equation 4*d^4 - 12*d^3 + 4*d^2 - 24*d + 5 = 0 and c = 0.4302331663731241127284415754... is positive root of the equation 5824*c^8 - 32*c^4 - 4*c^2 - 5 = 0. - Vaclav Kotesovec, Nov 01 2021

Example

a(0) = 1: [].
a(1) = 1: [1].
a(2) = 1: [1,1].
a(3) = 4: [3,0,0], [0,3,0], [0,0,3], [1,1,1].
a(4) = 13: [3,1,0,0], [3,0,1,0], [3,0,0,1], [1,3,0,0], [0,3,1,0], [0,3,0,1],[1,0,3,0], [0,1,3,0], [0,0,3,1], [1,0,0,3], [0,1,0,3], [0,0,1,3], [1,1,1,1].

Maple

b:= proc(n, t) option remember; `if`(t=0, 1-signum(n),
      add(`if`(j=0 or j::odd, b(n-j, t-1), 0), j=0..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

Mathematica

b[n_, t_] := b[n, t] = If[t == 0, 1 - Sign[n],
     Sum[If[j == 0 || OddQ[j], b[n - j, t - 1], 0], {j, 0, n}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)

Crossrefs

Cf. A001700, A088218, A165817, A305161, A324969.

Sequence in context: A031506 A251701 A291702 * A065297 A067635 A222425

Adjacent sequences: A348473 A348474 A348475 * A348477 A348478 A348479

Keyword

nonn

Author

Alois P. Heinz, Oct 19 2021

Status

approved