%I #14 Jul 11 2020 11:02:48

%S 0,0,0,1,12,120,1140,10815,104496,1037484,10627560,112508550,

%T 1231481460,13933510734,162864103584,1965078765195,24453461392080,

%U 313549334233440,4138796594051568,56188737057169593,783876449182595400

%N Fourth column of PE^3.

%C Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.

%F PE=exp(matpascal(5))/exp(1); A = PE^3; a(n)= A[ n,4 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^3; a(n)=A[ n,4]

%F E.g.f.: (x^3/6) * exp(3 * (exp(x) - 1)). - _Ilya Gutkovskiy_, Jul 11 2020

%p A056857 := proc(n,c) combinat[bell](n-1-c)*binomial(n-1,c) ; end: A078937 := proc(n,c) add( A056857(n,k)*A056857(k+1,c),k=0..n) ; end: A078938 := proc(n,c) add( A078937(n,k)*A056857(k+1,c),k=0..n) ; end: A129329 := proc(n) A078938(n+1,3) ; end: seq(A129329(n),n=0..27) ; # _R. J. Mathar_, May 30 2008

%t A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]];

%t A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}];

%t A078938[n_, c_] := Sum[A078937[n, k] A056857[k + 1, c], {k, 0, n}];

%t a[n_] := A078938[n + 1, 3];

%t a /@ Range[0, 20] (* _Jean-François Alcover_, Mar 24 2020, after _R. J. Mathar_ *)

%Y Cf. A056857, A078937, A078938, A078944, A078945, A000110.

%Y Cf. A078937, A078938, A129323, A129324, A129325, A027710.

%Y Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

%K nonn,easy

%O 0,5

%A _Gottfried Helms_, Apr 08 2007

%E More terms from _R. J. Mathar_, May 30 2008