A318160 Number of compositions of n into exactly n nonnegative parts with largest part ceiling(n/2).

1, 1, 1, 6, 18, 50, 195, 392, 1652, 2970, 12825, 22022, 96030, 160888, 705341, 1162800, 5116200, 8335338, 36773397, 59366450, 262462010, 420630210, 1862790699, 2967563040, 13160496684, 20861295000, 92624149475, 146203657992, 649794035142, 1021964428880

Offset: 0

Author

Alois P. Heinz, Aug 19 2018

Keywords

• nonn

Examples

			a(3) = 6: 012, 021, 102, 120, 201, 210.
a(4) = 18: 0022, 0112, 0121, 0202, 0211, 0220, 1012, 1021, 1102, 1120, 1201, 1210, 2002, 2011, 2020, 2101, 2110, 2200.
a(5) = 50: 00023, 00032, 00113, 00131, 00203, 00230, 00302, 00311, 00320, 01013, 01031, 01103, 01130, 01301, 01310, 02003, 02030, 02300, 03002, 03011, 03020, 03101, 03110, 03200, 10013, 10031, 10103, 10130, 10301, 10310, 11003, 11030, 11300, 13001, 13010, 13100, 20003, 20030, 20300, 23000, 30002, 30011, 30020, 30101, 30110, 30200, 31001, 31010, 31100, 32000.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2409

Crossrefs

Bisections give: A318161 (even part), A318162 (odd part).

Cf. A180281.

Programs

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0, 0, Sum[b[n - j, i - 1, k], {j, 0, Min[n, k]}]]];
a[n_] := If[n == 0, 1, b[n, n, Ceiling[n/2]] - b[n, n, Ceiling[n/2] - 1]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 28 2022, after Alois P. Heinz in A180281 *)

Formula

a(n) = A180281(n,ceiling(n/2)).

a(n) = 3^(3*n/2 - 7/4 + (-1)^n/4) * sqrt(n/Pi) / 2^(n - 3/2). - Vaclav Kotesovec, Sep 21 2019