A078944 -id:A078944 - cp's OEIS frontend

A129333 Fourth column of PE^4.

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

A292860 Square array A(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)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 5, 0, 1, 4, 12, 22, 15, 0, 1, 5, 20, 57, 94, 52, 0, 1, 6, 30, 116, 309, 454, 203, 0, 1, 7, 42, 205, 756, 1866, 2430, 877, 0, 1, 8, 56, 330, 1555, 5428, 12351, 14214, 4140, 0, 1, 9, 72, 497, 2850, 12880, 42356, 88563, 89918, 21147, 0

Offset: 0

Seiichi Manyama, Sep 25 2017

			Square array begins:
   1,   1,    1,     1,     1,      1,      1, ...
   0,   1,    2,     3,     4,      5,      6, ...
   0,   2,    6,    12,    20,     30,     42, ...
   0,   5,   22,    57,   116,    205,    330, ...
   0,  15,   94,   309,   756,   1555,   2850, ...
   0,  52,  454,  1866,  5428,  12880,  26682, ...
   0, 203, 2430, 12351, 42356, 115155, 268098, ...

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

Columns k=0-10 give: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263, A221159, A276506, A276507.
Rows n=0..2 give A000012, A001477, A002378.
Main diagonal gives A242817.
Same array, different indexing is A189233.
Cf. A292861.

A:= proc(n, k) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 25 2017

A[0, ] = 1; A[n /; n >= 0, k_ /; k >= 0] := A[n, k] = k*Sum[Binomial[n-1, j]*A[j, k], {j, 0, n-1}]; A[, ] = 0;
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 13 2021 *)
A292860[n_, k_] := BellB[n, k]; Table[A292860[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Peter Luschny, Dec 23 2021 *)

A(0,k) = 1 and A(n,k) = k * Sum_{j=0..n-1} binomial(n-1,j) * A(j,k) for n > 0.
A(n,k) = Sum_{j=0..n} k^j * Stirling2(n,j). - Seiichi Manyama, Jul 27 2019
A(n,k) = BellPolynomial(n, k). - Peter Luschny, Dec 23 2021

A357598 Expansion of e.g.f. sinh(2 * (exp(x)-1)) / 2.

Original entry on oeis.org

0, 1, 1, 5, 25, 117, 601, 3509, 22457, 153141, 1105561, 8453557, 68339833, 581495605, 5184047961, 48259748533, 468040609593, 4719817792565, 49396003390489, 535526127566773, 6004124908829177, 69509047405180213, 829801009239621849, 10202835010223731893

Offset: 0

Seiichi Manyama, Oct 05 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..558
Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A024429, A264037, A357572.

Cf. A065143, A078944, A357599.

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sinh(2*(exp(x)-1))/2)))

a(n) = sum(k=0, (n-1)\2, 4^k*stirling(n, 2*k+1, 2));

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = round((Bell_poly(n, 2)-Bell_poly(n, -2)))/4;

a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * Stirling2(n,2*k+1).
a(n) = ( Bell_n(2) - Bell_n(-2) )/4, where Bell_n(x) is n-th Bell polynomial.
a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A065143(k).

A221176 a(n) = Sum_{i=0..n} Stirling2(n,i)*2^(4i).

Original entry on oeis.org

1, 16, 272, 4880, 91920, 1810192, 37142288, 791744272, 17490370320, 399558315792, 9421351690000, 228916588400400, 5723078052339472, 147025755978698512, 3876566243300318992, 104789417805394595600, 2901159958960121863952, 82188946843192555474704, 2380551266738846355103504, 70441182699006212824911632

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 16 labeled boxes. - Peter Bala, Mar 23 2013

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, A221159. A027710, A144180, A144223, A144263, A189233.

With[{nn=20},CoefficientList[Series[Exp[16 (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 19 2024 *)

E.g.f. exp(16*(exp(x) - 1)). - Peter Bala, Mar 23 2013

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.

A309085 a(n) = exp(4) * Sum_{k>=0} (-4)^k*k^n/k!.

Original entry on oeis.org

1, -4, 12, -20, -20, 172, 108, -2388, -3220, 47532, 161900, -1062740, -8532628, 13623212, 431041132, 1206169260, -17833021588, -169685043796, 180187176044, 13462762665132, 79377664422252, -553096696140884, -11670986989785492, -44371854928405844, 829755609457185644

Offset: 0

Ilya Gutkovskiy, Jul 11 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..564

Column k = 4 of A292861.

Cf. A000587, A078944, A213170, A309084, A318179.

[1] cat [(&+[((-4)^k*StirlingSecond(m,k)):k in [0..m]]):m in [1..24]]; // Marius A. Burtea, Jul 11 2019

Table[Exp[4] Sum[(-4)^k k^n/k!, {k, 0, Infinity}], {n, 0, 24}]
Table[BellB[n, -4], {n, 0, 24}]
nmax = 24; CoefficientList[Series[Sum[(-4)^j x^j/Product[(1 - k x), {k, 1, j}] , {j, 0, nmax}], {x, 0, nmax}], x]
nmax = 24; CoefficientList[Series[Exp[4 (1 - Exp[x])], {x, 0, nmax}], x] Range[0, nmax]!

a(n) = sum(k=0, n, (-4)^k * stirling(n,k,2)); \\ Michel Marcus, Jul 12 2019

G.f.: Sum_{j>=0} (-4)^j*x^j / Product_{k=1..j} (1 - k*x).
E.g.f.: exp(4*(1 - exp(x))).
a(n) = Sum_{k=0..n} (-4)^k * Stirling2(n,k).

A367891 Expansion of e.g.f. exp(4*(exp(x) - 1 - x)).

Original entry on oeis.org

1, 0, 4, 4, 52, 164, 1364, 7620, 60148, 449252, 3831700, 33811716, 320082228, 3178774564, 33234163668, 363535920196, 4153091085172, 49406896240996, 610777358429204, 7830140410294148, 103914148870277556, 1425254885630973604, 20173671034640405588

Offset: 0

Ilya Gutkovskiy, Dec 04 2023

nonn

Cf. A000296, A078944, A194689, A346739, A367890.

nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 4 Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 22}]
Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k, 4], {k, 0, n}], {n, 0, 22}]

my(x='x+O('x^30)); Vec(serlaplace(exp(4*(exp(x) - 1 - x)))) \\ Michel Marcus, Dec 04 2023

G.f. A(x) satisfies: A(x) = 1 - 4 * x * ( A(x) - A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-4) * Sum_{k>=0} 4^k * (k-4)^n / k!.
a(0) = 1; a(n) = 4 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * A078944(k).

A276506 E.g.f.: exp(9*(exp(x)-1)).

Original entry on oeis.org

1, 9, 90, 981, 11511, 144108, 1911771, 26730981, 392209380, 6016681467, 96202473183, 1599000785730, 27563715220509, 491777630207037, 9064781481234546, 172346601006842337, 3375007346801025099, 67983454804021156548, 1406921223577401454239, 29881379179971835132761

Offset: 0

Vincenzo Librandi, Sep 17 2016

nonn

Number of ways of placing n labeled balls into n unlabeled (but 9-colored) boxes.

Alois P. Heinz, Table of n, a(n) for n = 0..514

Cf. similar sequences with e.g.f. exp(k*(exp(x)-1)): A001861 (k=2), A027710 (k=3), A078944 (k=4), A144180 (k=5) A144223 (k=6), A144263 (k=7), A221159 (k=8), this sequence (k=9), A276507 (k=10).

a:= proc(n) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, k-1)*a(n-k), k=1..n-1))*9)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 25 2017

Mathematica
```
Table[BellB[n, 9], {n, 0, 30}]
```

my(x='x+O('x^99)); Vec(serlaplace(exp(9*(exp(x)-1)))) \\ Altug Alkan, Sep 17 2016

G.f.: A(x) satisfies 9*(x/(1-x))*A(x/(1-x)) = A(x)-1; nine times the binomial transform equals this sequence shifted one place left.

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).

cp's OEIS Frontend

A129333 Fourth column of PE^4.

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

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A292860 Square array A(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)).

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A357598 Expansion of e.g.f. sinh(2 * (exp(x)-1)) / 2.

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PARI

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A221176 a(n) = Sum_{i=0..n} Stirling2(n,i)*2^(4i).

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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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A309085 a(n) = exp(4) * Sum_{k>=0} (-4)^k*k^n/k!.

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A367891 Expansion of e.g.f. exp(4*(exp(x) - 1 - x)).

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