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Third column of PE^3.
14

%I #10 Mar 24 2020 08:03:15

%S 0,0,1,9,72,570,4635,39186,345828,3188268,30684150,307870365,

%T 3215425554,34899450768,393015753039,4585024011015,55332235452960,

%U 689799432341928,8871905851132041,117581467377389310,1603990651356920730

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

%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: A129328 := proc(n) A078938(n+1,2) ; end: seq(A129328(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, 2];

%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 0,4

%A _Gottfried Helms_, Apr 08 2007

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