A350464 Table read by rows. Interpolating the swinging factorial (A056040) and the double factorial (A001147).

1, 0, 1, 0, 1, 3, 0, 2, 15, 15, 0, 6, 91, 210, 105, 0, 6, 690, 2835, 3150, 945, 0, 30, 5214, 42405, 79695, 51975, 10395, 0, 20, 44772, 666666, 2057055, 2207205, 945945, 135135, 0, 140, 384756, 11274900, 54879825, 90090000, 62432370, 18918900, 2027025

Offset

0,6

Links

Table of n, a(n) for n=0..44.

Formula

The partial Bell polynomials Y_{2*n, k}(Z) applied to the list Z of the aerated swinging factorials (A056040).

Example

Triangle starts:
[0] 1;
[1] 0,  1;
[2] 0,  1,   3;
[3] 0,  2,   15,     15;
[4] 0,  6,   91,     210,     105;
[5] 0,  6,   690,    2835,    3150,     945;
[6] 0,  30,  5214,   42405,   79695,    51975,    10395;
[7] 0,  20,  44772,  666666,  2057055,  2207205,  945945,  135135;

Mathematica

Swing[n_] := n! / Floor[n/2]!^2;
Z[n_] := Flatten[Table[{0, Swing[j]}, {j, 0, n}]];
T[n_, k_] := BellY[2 n, k, Z[n - k]];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten

Crossrefs

Cf. A350465 (row sums), A350466 (alternating row sums).

Cf. A056040, A001147, A001880.

Sequence in context: A302381 A303102 A302953 * A247706 A361527 A247704

Adjacent sequences: A350461 A350462 A350463 * A350465 A350466 A350467

Keyword

nonn,tabl

Author

Peter Luschny, Mar 13 2022

Status

approved