A129331 -id:A129331 - cp's OEIS frontend

A129329 Fourth column of PE^3.

0, 0, 0, 1, 12, 120, 1140, 10815, 104496, 1037484, 10627560, 112508550, 1231481460, 13933510734, 162864103584, 1965078765195, 24453461392080, 313549334233440, 4138796594051568, 56188737057169593, 783876449182595400

Offset: 0

Gottfried Helms, Apr 08 2007

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

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, 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: A129329 := proc(n) A078938(n+1,3) ; end: seq(A129329(n),n=0..27) ; # R. J. Mathar, May 30 2008

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}];
A078938[n_, c_] := Sum[A078937[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078938[n + 1, 3];
a /@ Range[0, 20] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

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]

E.g.f.: (x^3/6) * exp(3 * (exp(x) - 1)). - Ilya Gutkovskiy, Jul 11 2020

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

A129332 Third column of PE^4.

Original entry on oeis.org

0, 0, 1, 12, 120, 1160, 11340, 113988, 1185968, 12802896, 143475300, 1668342060, 20111265768, 251047344600, 3241258872124, 43230289541460, 594927620980320, 8438127851537312, 123214473695309652, 1850390947982126268

Offset: 0

Gottfried Helms, Apr 08 2007

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

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, 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: A129332 := proc(n) A078939(n+1,2) ; end: seq(A129332(n),n=0..25) ; # R. J. Mathar, May 30 2008

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}];
A078938[n_, c_] := Sum[A078937[n, k] A056857[k + 1, c], {k, 0, n}];
A078939[n_, c_] := Sum[A078938[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078939[n + 1, 2];
a /@ Range[0, 19] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

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

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

A129333 Fourth column of PE^4.

Original entry on oeis.org

0, 0, 0, 1, 16, 200, 2320, 26460, 303968, 3557904, 42676320, 526076100, 6673368240, 87148818328, 1171554274800, 16206294360620, 230561544221120, 3371256518888480, 50628767109223872, 780358333403627796

Offset: 0

Gottfried Helms, Apr 08 2007

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

Cf. A056857, A078937, A078938, A078944, A078945, A000110.
Cf. A078937, A078938, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, 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

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}];
A078938[n_, c_] := Sum[A078937[n, k] A056857[k + 1, c], {k, 0, n}];
A078939[n_, c_] := Sum[A078938[n, k] A056857[k + 1, c], {k, 0, n}];
a[n_] := A078939[n + 1, 3];
a /@ Range[0, 19] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

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]

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

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A129329 Fourth column of PE^3.

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A129332 Third column of PE^4.

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A129333 Fourth column of PE^4.

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