A078945 -id:A078945 - cp's OEIS frontend

A129328 Third column of PE^3.

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

A078940 Row sums of A078938.

Original entry on oeis.org

1, 4, 19, 103, 622, 4117, 29521, 227290, 1865881, 16239523, 149142952, 1439618143, 14555631781, 153700654036, 1690684883191, 19328770917499, 229203640111870, 2814018686591089, 35711716110387589, 467766675528462562

Offset: 0

Paul D. Hanna, Dec 18 2002

nonn

Divide by 3^n and insert an initial 1 to get sequence that shifts left one place under 1/3 order binomial transformation. - Franklin T. Adams-Watters, Jul 13 2006

Binomial transform of A027710. - Vaclav Kotesovec, Jun 26 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Column k=3 of A335975.

Cf. A027710, A035009, A078938, A078945, A355254.

A078940 := proc(n) local a,b,i;
a := [seq(2,i=1..n)]; b := [seq(1,i=1..n)];
exp(-x)*hypergeom(a,b,x); round(evalf(subs(x=3,%),66)) end:
seq(A078940(n),n=0..19); # Peter Luschny, Mar 30 2011

Table[n!, {n, 0, 20}]CoefficientList[Series[E^(3E^x-3+x), {x, 0, 20}], x]
Table[1/E^3/3*Sum[m^n/m!*3^m,{m,0,Infinity}],{n,1,20}] (* Vaclav Kotesovec, Mar 12 2014 *)
Table[BellB[n+1, 3]/3, {n, 0, 20}] (* Vaclav Kotesovec, Jan 15 2016 *)
nmax = 20; Clear[g]; g[nmax+1] = 1; g[k_] := g[k] = 1 - (k+4)*x - 3*(k+1)*x^2/g[k+1]; CoefficientList[Series[1/g[0], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 15 2016, after Sergei N. Gladkovskii *)

E.g.f.: exp(3*(exp(x)-1)+x).
Stirling transform of [1, 3, 3^2, 3^3, ...]. - Gerald McGarvey, Jun 01 2005
Define f_1(x), f_2(x), ... such that f_1(x)=e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n)=e^{-3}*f_n(3). - Milan Janjic, May 30 2008
G.f.: 1/T(0), where T(k) = 1 - (k+4)*x - 3*(k+1)*x^2/T(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2016
a(n) = exp(-3) * Sum_{k>=0} (k + 1)^n * 3^k / k!. - Ilya Gutkovskiy, Apr 20 2020
a(n) ~ n^(n+1) * exp(n/LambertW(n/3) - n - 3) / (3 * sqrt(1 + LambertW(n/3)) * LambertW(n/3)^(n+1)). - Vaclav Kotesovec, Jun 26 2022
a(0) = 1; a(n) = a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Dec 05 2023

More terms from Robert G. Wilson v, Dec 19 2002

A344735 a(0) = 1; a(n) = 4 * Sum_{k=1..n} binomial(n,k) * a(k-1).

Original entry on oeis.org

1, 4, 24, 156, 1120, 8740, 73384, 657900, 6259184, 62876852, 664134968, 7349666684, 84956020864, 1023006054980, 12802727760840, 166174971580684, 2232866214809360, 31007771007956948, 444360490882720344, 6562410784684023452, 99749853821538893216, 1558780425524233360740

Offset: 0

Ilya Gutkovskiy, Jun 07 2021

nonn

Cf. A040027, A078944, A078945, A094417, A343523, A343975, A344840, A345077, A345078, A345081.

a[0] = 1; a[n_] := a[n] = 4 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 21}]
nmax = 21; A[] = 0; Do[A[x] = 1 + 4 x A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + 4 * x * A(x/(1 - x)) / (1 - x)^2.

A335975 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1) + x).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 15, 1, 1, 5, 19, 47, 52, 1, 1, 6, 29, 103, 227, 203, 1, 1, 7, 41, 189, 622, 1215, 877, 1, 1, 8, 55, 311, 1357, 4117, 7107, 4140, 1, 1, 9, 71, 475, 2576, 10589, 29521, 44959, 21147, 1, 1, 10, 89, 687, 4447, 23031, 88909, 227290, 305091, 115975, 1

Offset: 0

Seiichi Manyama, Jul 03 2020

			Square array begins:
  1,   1,    1,     1,     1,      1,      1, ...
  1,   2,    3,     4,     5,      6,      7, ...
  1,   5,   11,    19,    29,     41,     55, ...
  1,  15,   47,   103,   189,    311,    475, ...
  1,  52,  227,   622,  1357,   2576,   4447, ...
  1, 203, 1215,  4117, 10589,  23031,  44683, ...
  1, 877, 7107, 29521, 88909, 220341, 478207, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-4 give: A000012, A000110(n+1), A035009(n+1), A078940, A078945.
Main diagonal gives A334240.
Cf. A292860, A335977.

T[0, k_] := 1; T[n_, k_] := T[n - 1, k] + k * Sum[T[j, k] * Binomial[n - 1, j], {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Jul 03 2020 *)

T(0,k) = 1 and T(n,k) = T(n-1,k) + k * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

T(n,k) = exp(-k) * Sum_{j>=0} (j + 1)^n * k^j / j!.

A335982 Expansion of e.g.f. exp(4 * (1 - exp(-x)) + x).

Original entry on oeis.org

1, 5, 21, 69, 149, 69, -619, -187, 9365, -3515, -193643, 453957, 4704917, -29425595, -83918443, 1640246085, -3184430955, -74516517307, 604223657877, 1324972362053, -52526078298475, 264984579390533, 2477371363954069, -44206576595187899, 133280843118435477

Offset: 0

Ilya Gutkovskiy, Jul 03 2020

sign

Cf. A078944, A078945, A109747, A309085, A335868, A335980, A335981.

nmax = 24; CoefficientList[Series[Exp[4 (1 - Exp[-x]) + x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = a[n - 1] + 4 Sum[(-1)^(n - k - 1) Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]

a(n) = exp(4) * (-1)^n * Sum_{k>=0} (-4)^k * (k - 1)^n / k!.

a(0) = 1; a(n) = a(n-1) + 4 * Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n-1,k) * a(k).

A320433 Expansion of e.g.f. exp(4 * (1 - exp(x)) + x).

Original entry on oeis.org

1, -3, 5, 5, -43, -27, 597, 805, -11883, -40475, 265685, 2133157, -3405803, -107760283, -301542315, 4458255397, 42421260949, -45046794011, -3365690666283, -19844416105563, 138274174035221, 2917746747446373, 11092963732101461, -207438902364296411, -3205301465165742187

Offset: 0

Seiichi Manyama, Jul 06 2020

sign

Column k=4 of A335977.

Cf. A078945, A335982.

m = 24; Range[0, m]! * CoefficientList[Series[Exp[4 * (1 - Exp[x]) + x], {x, 0, m}], x] (* Amiram Eldar, Jul 06 2020 *)
Table[Sum[Binomial[n, k] * BellB[k, -4], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 06 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(exp(4*(1-exp(x))+x)))

a(0) = 1 and a(n) = a(n-1) - 4 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.
a(n) = exp(4) * Sum_{k>=0} (k + 1)^n * (-4)^k / k!.
a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k, -4). - Vaclav Kotesovec, Jul 06 2020

cp's OEIS Frontend

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.

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A129333 Fourth column of PE^4.

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A078940 Row sums of A078938.

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A344735 a(0) = 1; a(n) = 4 * Sum_{k=1..n} binomial(n,k) * a(k-1).

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A335975 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(k*(exp(x) - 1) + x).

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