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