A027710 -id:A027710 - cp's OEIS frontend

A129323 Second column of PE^2.

0, 1, 4, 18, 88, 470, 2724, 17010, 113712, 809262, 6101820, 48540778, 405935688, 3557404838, 32577733972, 310987560930, 3087723669600, 31823217868318, 339845199259500, 3754422961010522, 42843681016834680, 504339820818380694

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: A129323 := proc(n) A078937(n+1,1) ; end: seq(A129323(n),n=0..23) ; # R. J. Mathar, May 30 2008

Table[Sum[BellB[n, 2], {i, 0, n}], {n, -1, 20}] (* Zerinvary Lajos, Jul 16 2009 *)

PE=exp(matpascal(5))/exp(1); A = PE^2; a(n)=A[n,2] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,2]

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

A129324 Third column of PE^2.

Original entry on oeis.org

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

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: A129324 := proc(n) A078937(n+1,2) ; end: seq(A129324(n),n=0..23) ; # 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}];
a[n_] := A078937[n + 1, 2];
a /@ Range[0, 21] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

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

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

A129325 Fourth column of PE^2.

Original entry on oeis.org

0, 0, 0, 1, 8, 60, 440, 3290, 25424, 204120, 1705680, 14836470, 134240040, 1262060228, 12313382536, 124509169330, 1303109358880, 14098102762160, 157473907149600, 1813923418494126, 21523529286435000, 262809607270736540

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: A129325 := proc(n) A078937(n+1,3) ; end: seq(A129325(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}];
a[n_] := A078937[n + 1, 3];
a /@ Range[0, 21] (* Jean-François Alcover, Mar 24 2020, after R. J. Mathar *)

m=matpascal(30)-matid(31); pe=matid(31)+sum(i=1,30,m^i/i!); A=pe^2; A[,4] \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008

PE=exp(matpascal(5))/exp(1); A = PE^2; a(n)=A[n,4] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,4]

More terms from R. J. Mathar and Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2008

A129327 Second column of PE^3.

Original entry on oeis.org

0, 1, 6, 36, 228, 1545, 11196, 86457, 708504, 6136830, 55976430, 535904259, 5369146272, 56145107577, 611336534802, 6916529431620, 81152874393168, 985767316792449, 12376996566040980, 160399065135692073

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: A129327 := proc(n) A078938(n+1,1) ; end: seq(A129327(n),n=0..27) ; # R. J. Mathar, May 30 2008

Table[Sum[BellB[n, 3], {i, 0, n}], {n, -1, 18}] (* Zerinvary Lajos, Jul 16 2009 *)

PE=exp(matpascal(5))/exp(1); A = PE^3; a(n)= A[ n,2 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^3; a(n)=A[ n,2]

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

A129328 Third column of PE^3.

Original entry on oeis.org

0, 0, 1, 9, 72, 570, 4635, 39186, 345828, 3188268, 30684150, 307870365, 3215425554, 34899450768, 393015753039, 4585024011015, 55332235452960, 689799432341928, 8871905851132041, 117581467377389310, 1603990651356920730

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: A129328 := proc(n) A078938(n+1,2) ; end: seq(A129328(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, 2];
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,3 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^3; a(n)=A[ n,3]

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

A129329 Fourth column of PE^3.

Original entry on oeis.org

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

A129331 Second column of PE^4.

Original entry on oeis.org

0, 1, 8, 60, 464, 3780, 32568, 296492, 2845088, 28695060, 303334920, 3351877628, 38622668400, 463036981732, 5764038605528, 74365952622540, 992720923710272, 13690497077256628, 194777994524434344, 2855149354656290716

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: A129331 := proc(n) A078939(n+1,1) ; end: seq(A129331(n),n=0..25) ; # R. J. Mathar, May 30 2008

Table[Sum[BellB[n, 4], {i, 0, n}], {n, -1, 18}] (* Zerinvary Lajos, Jul 16 2009 *)

PE=exp(matpascal(5))/exp(1); A = PE^4; a(n)= A[ n,2 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^4; a(n)=A[ n,2]

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

A221159 a(n) = Sum_{i=0..n} Stirling2(n,i)*2^(3i).

Original entry on oeis.org

1, 8, 72, 712, 7624, 87496, 1067976, 13781448, 187104200, 2661876168, 39549629384, 611918940616, 9834596715464, 163824830616008, 2823080829871048, 50238768569014728, 921839901090823112, 17416746966515278280, 338394913332895863752, 6753431112631087835592, 138296031340416209103816

Offset: 0

N. J. A. Sloane, Jan 04 2013

nonn

The number of ways of putting n labeled balls into a set of bags and then putting the bags into 8 labeled boxes. - Peter Bala, Mar 23 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Frank Simon, Algebraic Methods for Computing the Reliability of Networks, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012. See Table 5.1.

Cf. A000110, A001861, A078944, A221176, A027710, A144180, A144223, A144263, A189233.

Table[BellB[n,8],{n,0,20}] (* Vaclav Kotesovec, Mar 12 2014 *)

E.g.f.: exp(8*(exp(x) - 1)). - Peter Bala, Mar 23 2013
a(n) ~ n^n * exp(n/LambertW(n/8)-8-n) / (sqrt(1+LambertW(n/8)) * LambertW(n/8)^n). - Vaclav Kotesovec, Mar 12 2014
G.f.: Sum_{j>=0} 8^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 11 2019

cp's OEIS Frontend

A129323 Second column of PE^2.

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A129324 Third column of PE^2.

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A129325 Fourth column of PE^2.

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A129327 Second column of PE^3.

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A129328 Third column of PE^3.

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A129329 Fourth column of PE^3.

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A129331 Second column of PE^4.

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A129332 Third column of PE^4.