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%I #16 Jul 11 2020 11:03:21
%S 0,0,1,6,36,220,1410,9534,68040,511704,4046310,33560010,291244668,
%T 2638581972,24901833866,244333004790,2487900487440,26245651191600,
%U 286408960814862,3228529392965250,37544229610105220,449858650676764140
%N Third column of PE^2.
%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^2; a(n)=A[n,3]; with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,3].
%F E.g.f.: (x^2/2) * exp(2 * (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: A129324 := proc(n) A078937(n+1,2) ; end: seq(A129324(n),n=0..23) ; # _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 a[n_] := A078937[n + 1, 2];
%t a /@ Range[0, 21] (* _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 0,4
%A _Gottfried Helms_, Apr 08 2007
%E More terms from _R. J. Mathar_, May 30 2008
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