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A129333 Fourth column of PE^4.

%I A129333 #10 Mar 24 2020 08:30:47
%S A129333 0,0,0,1,16,200,2320,26460,303968,3557904,42676320,526076100,
%T A129333 6673368240,87148818328,1171554274800,16206294360620,230561544221120,
%U A129333 3371256518888480,50628767109223872,780358333403627796
%N A129333 Fourth column of PE^4.
%C A129333 Base matrix is in A011971; second power is in A078937; third power is in A078938; fourth power is in A078939.
%F A129333 PE=exp(matpascal(5))/exp(1); A = PE^4; a(n)= A[ n,4 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^4; a(n)=A[ n,4]
%p A129333 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: A078939 := proc(n,c) add( A078938(n,k)*A056857(k+1,c),k=0..n) ; end: A129333 := proc(n) A078939(n+1,3) ; end: seq(A129333(n),n=0..25) ; # _R. J. Mathar_, May 30 2008
%t A129333 A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]];
%t A129333 A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}];
%t A129333 A078938[n_, c_] := Sum[A078937[n, k] A056857[k + 1, c], {k, 0, n}];
%t A129333 A078939[n_, c_] := Sum[A078938[n, k] A056857[k + 1, c], {k, 0, n}];
%t A129333 a[n_] := A078939[n + 1, 3];
%t A129333 a /@ Range[0, 19] (* _Jean-François Alcover_, Mar 24 2020, after _R. J. Mathar_ *)
%Y A129333 Cf. A056857, A078937, A078938, A078944, A078945, A000110.
%Y A129333 Cf. A078937, A078938, A129323, A129324, A129325, A027710.
%Y A129333 Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.
%K A129333 nonn,easy
%O A129333 0,5
%A A129333 _Gottfried Helms_, Apr 08 2007
%E A129333 More terms from _R. J. Mathar_, May 30 2008