A078940 -id:A078940 - cp's OEIS frontend

A027710 Number of ways of placing n labeled balls into n unlabeled (but 3-colored) boxes.

1, 3, 12, 57, 309, 1866, 12351, 88563, 681870, 5597643, 48718569, 447428856, 4318854429, 43666895343, 461101962108, 5072054649573, 57986312752497, 687610920335610, 8442056059773267, 107135148331162767, 1403300026585387686, 18946012544520590991

Offset: 0

George Yuhasz (gyuhasz(AT)vt.edu) and John W. Layman

nonn

Binomial transform of this sequence is A078940 and a(n+1) = 3*A078940(n). - Paul D. Hanna, Dec 08 2003
First column of the cube of the matrix exp(P)/exp(1) given in A011971. - Gottfried Helms, Mar 30 2007. Base matrix in A011971, second power in A078937, third power in A078938, fourth power in A078939.
The number of ways of putting n labeled balls into a set of bags and then putting the bags into 3 labeled boxes. - Peter Bala, Mar 23 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Amit Kumar Singh, Akash Kumar and Thambipillai Srikanthan, Accelerating Throughput-aware Run-time Mapping for Heterogeneous MPSoCs, ACM Transactions on Design Automation of Electronic Systems, 2012. - From _N. J. A. Sloane_, Dec 24 2012
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.

Cf. A000110, A001861, A056857, A078937, A078938, A078940, A078944, A078945, A129323, A129324, A129325, A129327, A129328, A129329, A129331, A129332, A129333, A144180, A144223, A144263, A189233, A221159, A221176.

b:= proc(n, m) option remember; `if`(n=0,
      1, m*b(n-1, m)+3*b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Aug 03 2021

colors=3; Array[ bell, 25 ]; For[ x=1, x<=25, x++, bell[ x ]=0 ]; bell[ 1 ]=colors;
Print[ "1 ", colors ]; For[ n=2, n<=25, n++, bell[ n ]=colors*bell[ n-1 ];
For[ i=1, n-i>1, i++, bell[ n-i ]=bell[ n-i ]*(n-i)+colors*bell[ n-i-1 ] ];
bellsum=0; For[ t=0, tVaclav Kotesovec, Mar 12 2014 *)

a(n)=if(n<0,0,n!*polcoeff(exp(3*(exp(x+x*O(x^n))-1)),n))

from sage.combinat.expnums import expnums2
expnums(22, 3) # Zerinvary Lajos, Jun 26 2008

E.g.f.: exp {3(e^x-1)}. - Michael Somos, Oct 18 2002
a(n) = exp(-3)*Sum_{k>=0} 3^k*k^n/k!. - Benoit Cloitre, Sep 25 2003
G.f.: 3*(x/(1-x))*A(x/(1-x)) = A(x) - 1; thrice the binomial transform equals the sequence shifted one place left. - Paul D. Hanna, Dec 08 2003
a(n) = Sum_{k = 0..n} 3^k*A048993(n, k); A048993: Stirling2 numbers. - Philippe Deléham, May 09 2004
PE=exp(matpascal(5))/exp(1); A = PE^3; a(n)= A[ n,1 ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^3; a(n)=A[ n,1]. - Gottfried Helms, Apr 08 2007
G.f.: (G(0) - 1)/(x-1)/3 where G(k) =  1 - 3/(1-k*x)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
G.f.: T(0)/(1-3*x), where T(k) = 1 - 3*x^2*(k+1)/( 3*x^2*(k+1) - (1-3*x-x*k)*(1-4*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 24 2013
a(n) ~ n^n * exp(n/LambertW(n/3)-3-n) / (sqrt(1+LambertW(n/3)) * LambertW(n/3)^n). - Vaclav Kotesovec, Mar 12 2014
G.f.: Sum_{j>=0} 3^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 07 2019

Entry revised by N. J. A. Sloane, Apr 25 2007

A078945 Row sums of A078939.

Original entry on oeis.org

1, 5, 29, 189, 1357, 10589, 88909, 797085, 7583373, 76179037, 804638925, 8904557341, 102929260813, 1239432543709, 15511264432973, 201330839371421, 2705249923950477, 37567754666530141, 538369104335121869

Offset: 0

Paul D. Hanna, Dec 18 2002

nonn

Equals A078944(n+1)/4.

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

Column k=4 of A335975.

Cf. A078939, A078944, A000110, A035009, A078940.

A078945 := 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=4,%),66)) end:
seq(A078945(n),n=0..18); # Peter Luschny, Mar 30 2011

Table[n!, {n, 0, 20}]CoefficientList[Series[E^(4E^x-4+x), {x, 0, 20}], x]
Table[1/E^4/4*Sum[m^n/m!*4^m,{m,0,Infinity}],{n,1,20}] (* Vaclav Kotesovec, Mar 12 2014 *)
Table[BellB[n+1, 4]/4, {n, 0, 20}] (* Vaclav Kotesovec, Jun 26 2022 *)

E.g.f.: exp(4*(exp(x)-1)+x).
Stirling transform of [1, 4, 4^2, 4^3, ...]. - 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^{-4}*f_n(4). - Milan Janjic, May 30 2008
G.f.: 1/(Q(0) - 4*x) where Q(k) =  1 - x*(k+1)/( 1 - 4*x/Q(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 22 2013
G.f.: T(0)/(1-5*x), where T(k) = 1 - 4*x^2*(k+1)/( 4*x^2*(k+1) - (1-5*x-x*k)*(1-6*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 28 2013
a(n) = exp(-4) * Sum_{k>=0} (k + 1)^n * 4^k / k!. - Ilya Gutkovskiy, Apr 20 2020
a(n) ~ n^(n+1) * exp(n/LambertW(n/4) - n - 4) / (4 * sqrt(1 + LambertW(n/4)) * LambertW(n/4)^(n+1)). - Vaclav Kotesovec, Jun 26 2022
a(0) = 1; a(n) = a(n-1) + 4 * 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

A078938 Cube of lower triangular matrix of A056857 (successive equalities in set partitions of n).

Original entry on oeis.org

1, 3, 1, 12, 6, 1, 57, 36, 9, 1, 309, 228, 72, 12, 1, 1866, 1545, 570, 120, 15, 1, 12351, 11196, 4635, 1140, 180, 18, 1, 88563, 86457, 39186, 10815, 1995, 252, 21, 1, 681870, 708504, 345828, 104496, 21630, 3192, 336, 24, 1, 5597643, 6136830, 3188268

Offset: 0

Paul D. Hanna, Dec 18 2002

Cube of the matrix exp(P)/exp(1) given in A011971. - Gottfried Helms, Apr 08 2007. Base matrix in A011971, second power in A129321, third power in this entry, fourth power in A078939
First column gives A027710. Row sums give A078940.
Riordan array [exp(3*exp(x)-3),x], whose production matrix has e.g.f. exp(x*t)(t+3*exp(x)). [From Paul Barry, Nov 26 2008]

			Rows:
1,
3,1,
12,6,1,
57,36,9,1,
309,228,72,12,1,
1866,1545,570,120,15,1,
12351,11196,4635,1140,180,18,1,
...

Cf. A056857, A078937, A078939, A078940, A027710.
Cf. A078938, A078944, A078945, A000110.
Cf. A129321, A129323, A129324, A129325, A027710.
Cf. A129327, A129328, A129329, A078944, A129331, A129332, A129333.

m=matpascal(5)-matid(6); pe=matid(6)+m/1! + m^2/2!+m^3/3!+m^4/4!+m^5/5! ; A = pe^3 - Gottfried Helms, Apr 08 2007

PE=exp(matpascal(5))/exp(1); A = PE^3; a(n)= A[ n,sequentially read ] with exact integer arithmetic: PE=exp(matpascal(5)-matid(6)); A = PE^3; a(n)=A[ n,sequentially read] - Gottfried Helms, Apr 08 2007

Exponential function of 3*Pascal's triangle (taken as a lower triangular matrix) divided by e^3: [A078938] = (1/e^3)*exp(3*[A007318]) = [A056857]^3.

Entry revised by N. J. A. Sloane, Apr 25 2007

A343975 a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(n,k) * a(k-1).

Original entry on oeis.org

1, 3, 15, 81, 489, 3237, 23211, 178707, 1467051, 12768345, 117263829, 1131901521, 11444383251, 120847326879, 1329303053391, 15197269729689, 180211641841353, 2212525627591533, 28078380387448515, 367782119667874083, 4965441830591976339, 69014083524412401873, 986364827548578356421

Offset: 0

Ilya Gutkovskiy, Jun 07 2021

nonn

Cf. A027710, A032033, A040027, A078940, A343523, A344735, A344840, A345077, A345078, A345081.

a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 22}]
nmax = 22; A[] = 0; Do[A[x] = 1 + 3 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 + 3 * x * A(x/(1 - x)) / (1 - x)^2.

A355254 Expansion of e.g.f. exp(3*(exp(x) - 1) - x).

Original entry on oeis.org

1, 2, 7, 29, 142, 785, 4813, 32240, 233449, 1812161, 14980768, 131174939, 1211111629, 11745451658, 119255234371, 1264050651953, 13952113296766, 160006824960725, 1902825936046105, 23423342243273696, 297982102750214605, 3911917977005948453, 52926119656555824520

Offset: 0

Vaclav Kotesovec, Jun 26 2022

nonn

Inverse binomial transform of A027710.
In general, if m >= 1 and e.g.f. = exp(m*exp(x) + r*x + s) then
a(n) ~ n^(n+r) * exp(n/LambertW(n/m) - n + s) / (m^r * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n+r)).
Equivalently, a(n) ~ n! * (n/m)^r * exp(n/LambertW(n/m) + s) / (sqrt(2*Pi*n * (1 + LambertW(n/m))) * LambertW(n/m)^(n+r)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..545

Cf. A000296, A027710, A078940, A217924.

nmax = 25; CoefficientList[Series[Exp[3*Exp[x]-3-x], {x, 0, nmax}], x] * Range[0, nmax]!

my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) - x))) \\ Michel Marcus, Dec 04 2023

a(n) ~ 3 * n^(n-1) * exp(n/LambertW(n/3) - n - 3) / (sqrt(1 + LambertW(n/3)) * LambertW(n/3)^(n-1)).

a(0) = 1; a(n) = -a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - Ilya Gutkovskiy, Dec 04 2023

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!.

A335981 Expansion of e.g.f. exp(3 * (1 - exp(-x)) + x).

Original entry on oeis.org

1, 4, 13, 31, 40, -23, -95, 490, 823, -8393, 3766, 174775, -658787, -2751404, 34033297, -55552037, -1170734432, 9362348365, 3277050925, -562286419646, 3848880970147, 8815342530739, -356804325202730, 2389771436686339, 8677476137729929, -302470260552857660

Offset: 0

Ilya Gutkovskiy, Jul 03 2020

sign

Cf. A027710, A078940, A109747, A309084, A335868, A335980, A335982.

nmax = 25; CoefficientList[Series[Exp[3 (1 - Exp[-x]) + x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = a[n - 1] + 3 Sum[(-1)^(n - k - 1) Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 25}]

a(n) = exp(3) * (-1)^n * Sum_{k>=0} (-3)^k * (k - 1)^n / k!.

a(0) = 1; a(n) = a(n-1) + 3 * Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n-1,k) * a(k).

A367889 Expansion of e.g.f. exp(3(exp(x) - 1) + 2x).

Original entry on oeis.org

1, 5, 28, 173, 1165, 8468, 65923, 546197, 4791214, 44301143, 430158397, 4372004546, 46381674085, 512328076385, 5879362011436, 69958289731457, 861605015493073, 10965899141265500, 144018319806024991, 1949190279770578145, 27153595018237222774

Offset: 0

Ilya Gutkovskiy, Dec 04 2023

nonn

Cf. A000110, A005493, A035009, A078940, A355247, A367888.

nmax = 20; CoefficientList[Series[Exp[3 (Exp[x] - 1) + 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + 3 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n, k] 2^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 20}]

my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) + 2*x))) \\ Michel Marcus, Dec 04 2023

G.f. A(x) satisfies: A(x) = 1 + x * ( 2 * A(x) + 3 * A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-3) * Sum_{k>=0} 3^k * (k+2)^n / k!.
a(0) = 1; a(n) = 2 * a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * A027710(k).

A320432 Expansion of e.g.f. exp(3 * (1 - exp(x)) + x).

Original entry on oeis.org

1, -2, 1, 7, -8, -65, 37, 1024, 1351, -19001, -92618, 232513, 4087189, 9953926, -123909155, -1170342533, -676144160, 62840385619, 490129709977, -551829062288, -40624407525941, -305175084654341, 698633855671510, 34571970743398621, 278738497423756153, -663168571756087538

Offset: 0

Seiichi Manyama, Jul 06 2020

sign

Column k=3 of A335977.

Cf. A078940, A335981.

m = 25; Range[0, m]! * CoefficientList[Series[Exp[3 * (1 - Exp[x]) + x], {x, 0, m}], x] (* Amiram Eldar, Jul 06 2020 *)
Table[Sum[Binomial[n, k] * BellB[k, -3], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 06 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(exp(3*(1-exp(x))+x)))

a(0) = 1 and a(n) = a(n-1) - 3 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.
a(n) = exp(3) * Sum_{k>=0} (k + 1)^n * (-3)^k / k!.
a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k, -3). - Vaclav Kotesovec, Jul 06 2020

A367925 Expansion of e.g.f. 1/(4 - x - 3*exp(x)).

Original entry on oeis.org

1, 4, 35, 459, 8025, 175383, 4599507, 140728437, 4920898317, 193579534155, 8461200381111, 406815231899409, 21337866382711521, 1212458502624643719, 74193773349948903483, 4864422156647044661949, 340191752483516373189621, 25278147388666498256368323

Offset: 0

Seiichi Manyama, Dec 05 2023

nonn

Cf. A006155, A367924.

Cf. A032033, A078940, A367831, A367836, A367923.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+3*sum(j=1, i, binomial(i, j)*v[i-j+1])); v;

a(0) = 1; a(n) = n * a(n-1) + 3 * Sum_{k=1..n} binomial(n,k) * a(n-k).

cp's OEIS Frontend

A027710 Number of ways of placing n labeled balls into n unlabeled (but 3-colored) boxes.

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A078945 Row sums of A078939.

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A078938 Cube of lower triangular matrix of A056857 (successive equalities in set partitions of n).

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A343975 a(0) = 1; a(n) = 3 * Sum_{k=1..n} binomial(n,k) * a(k-1).

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A355254 Expansion of e.g.f. exp(3*(exp(x) - 1) - x).

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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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A335981 Expansion of e.g.f. exp(3 * (1 - exp(-x)) + x).

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A367889 Expansion of e.g.f. exp(3*(exp(x) - 1) + 2*x).

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A367889 Expansion of e.g.f. exp(3(exp(x) - 1) + 2x).