A190905 Euler transform of the swinging factorial A056040.

1, 1, 3, 9, 18, 60, 117, 371, 747, 2199, 4697, 12735, 28571, 72815, 169176, 412440, 978086, 2316754, 5547293, 12909723, 30966639, 71357601, 170636159, 391242623, 930120982, 2128073530, 5023630830, 11486060090, 26918052717, 61539213693, 143227189518

Offset: 0

Author

Peter Luschny, Jul 06 2011

Keywords

• nonn

Links

• M. Bernstein and N. J. A. Sloane, Some Canonical Integer Sequences, (arXiv:0205301v1), May 28 2002. [Link to arXiv version]
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Crossrefs

Cf. A107895.

Programs

Maple

EulerTrans := proc(p) local b; b := proc(n) option remember; local d, j;
`if`(n=0,1,add(add(d*p(d),d=numtheory[divisors](j))*b(n-j),j=1..n)/n) end end:
A190905 := EulerTrans(n->n!/iquo(n,2)!^2): seq( A190905(n),n=0..30); # After Alois P. Heinz, A000335.

Mathematica

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; EulerTrans[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; a = EulerTrans[sf]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 29 2013, after Maple *)