A129324 Third column of PE^2.

0, 0, 1, 6, 36, 220, 1410, 9534, 68040, 511704, 4046310, 33560010, 291244668, 2638581972, 24901833866, 244333004790, 2487900487440, 26245651191600, 286408960814862, 3228529392965250, 37544229610105220, 449858650676764140

Offset: 0

Author

Gottfried Helms, Apr 08 2007

Keywords

• nonn
• easy

Comments

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

Crossrefs

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

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

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

Programs

Maple

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

Mathematica

A056857[n_, c_] := If[n <= c, 0, BellB[n - 1 - c] Binomial[n - 1, c]];
A078937[n_, c_] := Sum[A056857[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078937[n + 1, 2];
a /@ Range[0, 21] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

Formula

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].

E.g.f.: (x^2/2) * exp(2 * (exp(x) - 1)). - Ilya Gutkovskiy, Jul 11 2020

Extensions

More terms from R. J. Mathar, May 30 2008