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

%I A350464 #7 Apr 09 2022 14:40:42
%S A350464 1,0,1,0,1,3,0,2,15,15,0,6,91,210,105,0,6,690,2835,3150,945,0,30,5214,
%T A350464 42405,79695,51975,10395,0,20,44772,666666,2057055,2207205,945945,
%U A350464 135135,0,140,384756,11274900,54879825,90090000,62432370,18918900,2027025
%N A350464 Table read by rows. Interpolating the swinging factorial (A056040) and the double factorial (A001147).
%F A350464 The partial Bell polynomials Y_{2*n, k}(Z) applied to the list Z of the aerated swinging factorials (A056040).
%e A350464 Triangle starts:
%e A350464 [0] 1;
%e A350464 [1] 0,  1;
%e A350464 [2] 0,  1,   3;
%e A350464 [3] 0,  2,   15,     15;
%e A350464 [4] 0,  6,   91,     210,     105;
%e A350464 [5] 0,  6,   690,    2835,    3150,     945;
%e A350464 [6] 0,  30,  5214,   42405,   79695,    51975,    10395;
%e A350464 [7] 0,  20,  44772,  666666,  2057055,  2207205,  945945,  135135;
%t A350464 Swing[n_] := n! / Floor[n/2]!^2;
%t A350464 Z[n_] := Flatten[Table[{0, Swing[j]}, {j, 0, n}]];
%t A350464 T[n_, k_] := BellY[2 n, k, Z[n - k]];
%t A350464 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
%Y A350464 Cf. A350465 (row sums), A350466 (alternating row sums).
%Y A350464 Cf. A056040, A001147, A001880.
%K A350464 nonn,tabl
%O A350464 0,6
%A A350464 _Peter Luschny_, Mar 13 2022