A004212 -id:A004212 - cp's OEIS frontend

A355164 a(n) = exp(-1/3) * Sum_{k>=0} (3k + 2)^n / (3^k k!).

1, 3, 12, 63, 405, 3024, 25515, 239355, 2465478, 27600669, 333051669, 4303119330, 59202612693, 863285928327, 13288589222508, 215177742933579, 3654114236490393, 64902307993517160, 1202782377224829015, 23207417212751493327, 465302639045308247262, 9677171073270491712513, 208434297638273958963225

Offset: 0

Ilya Gutkovskiy, Jun 22 2022

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..484
Adam Buck, Jennifer Elder, Azia A. Figueroa, Pamela E. Harris, Kimberly Harry, and Anthony Simpson, Flattened Stirling Permutations, arXiv:2306.13034 [math.CO], 2023. See p. 14.

Cf. A003575, A004212, A284864, A355165, A355167.

nmax = 22; CoefficientList[Series[Exp[2 x + (Exp[3 x] - 1)/3], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
Table[Sum[Binomial[n, k] 2^(n - k) 3^k BellB[k, 1/3], {k, 0, n}], {n, 0, 22}]

E.g.f.: exp(2*x + (exp(3*x) - 1) / 3).
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 3^(k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * A004212(k).
a(n) ~ 3^(n + 2/3) * n^(n + 2/3) * exp(n/LambertW(3*n) - n - 1/3) / (sqrt(1 + LambertW(3*n)) * LambertW(3*n)^(n + 2/3)). - Vaclav Kotesovec, Jun 27 2022

A367744 Expansion of e.g.f. exp(1 - x - exp(3*x)).

Original entry on oeis.org

1, -4, 7, 17, -14, -637, -2951, 14126, 333205, 2076245, -12283700, -423234511, -4163106203, 8148184700, 952894223755, 15568620884189, 69314620864450, -2816256959131561, -83397946135434515, -1025683419252783946, 4726361848234575553, 525779836596438636689, 12363747028673287330948, 112888493670408785796989

Offset: 0

Ilya Gutkovskiy, Nov 29 2023

sign

Cf. A000587, A004212, A109747, A284859, A284860, A317996, A367743.

nmax = 23; CoefficientList[Series[Exp[1 - x - Exp[3 x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] - Sum[Binomial[n - 1, k - 1] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 3^k BellB[k, -1], {k, 0, n}], {n, 0, 23}]

a(n) = exp(1) * Sum_{k>=0} (-1)^k * (3*k-1)^n / k!.
a(0) = 1; a(n) = -a(n-1) - Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 3^k * A000587(k).

A111673 Triangle, generated from A111579.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 15, 11, 4, 1, 1, 1, 52, 49, 19, 5, 1, 1, 1, 203, 257, 109, 29, 6, 1, 1, 1, 877, 1539, 742, 201, 41, 7, 1, 1, 1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1, 1, 21147, 75905, 51193, 15821, 3176, 505, 71, 9, 1, 1, 1, 115975, 609441, 498118, 170389, 35451, 5497, 729, 89, 10, 1, 1

Offset: 0

Gary W. Adamson, Aug 14 2005

Columns are inverse binomial transforms of columns (k>0) of A111579.

			First few rows of the triangle are:
  1,
  1, 1,
  1, 1, 1,
  1, 2, 1, 1,
  1, 5, 3, 1, 1,
  1, 15, 11, 4, 1, 1,
  1, 52, 49, 19, 5, 1, 1,
  1, 203, 257, 109, 29, 6, 1, 1,
  1, 877, 1539, 742, 201, 41, 7, 1, 1,
  1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1,
  ...
Inverse binomial transform of column 2 of A111579 (1, 2, 5, 15, 52, 203...) = column 2 (1, 1, 2, 5, 15, 52...).

A. Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.
A. Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]

Cf. A111579, A008277.
For two other versions of this triangle see A241578, A241579.
Rows and columns give A000110, A004211, A004212, A004213, A005011, A005012, A028387, A241577.

More terms from N. J. A. Sloane, Apr 29 2014

A375174 Expansion of e.g.f. exp( (1/(1 - 9*x)^(1/3) - 1)/3 ).

Original entry on oeis.org

1, 1, 13, 289, 9073, 367681, 18249661, 1071805393, 72684954049, 5588943933313, 480445784729101, 45656401249018561, 4752397230972673393, 537724197016879848769, 65711109523289467682173, 8624825762253351871394161, 1210085772867351648907603201

Offset: 0

Seiichi Manyama, Aug 02 2024

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A374882, A375176.

Cf. A004212.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-9*x)^(1/3)-1)/3)))

a(n) = Sum_{k=0..n} 9^(n-k) * |Stirling1(n,k)| * A004212(k) = 9^n * Sum_{k=0..n} (1/3)^k * |Stirling1(n,k)| * Bell_k(1/3), where Bell_n(x) is n-th Bell polynomial.
From Vaclav Kotesovec, Aug 02 2024: (Start)
a(n) = 36*(n-2)*a(n-1) - 18*(27*n^2 - 135*n + 172)*a(n-2) + (2916*n^3 - 26244*n^2 + 79056*n - 79703)*a(n-3) - 729*(n-4)*(n-3)*(3*n - 11)*(3*n - 10)*a(n-4).
a(n) ~ 3^(2*n - 1/4) * n^(n - 3/8) / (2*exp(n - 4*n^(1/4)/3^(3/2) + 1/3)) * (1 - 35/(32*sqrt(3)*n^(1/4))). (End)

A375176 Expansion of e.g.f. exp( (exp( (exp(9*x) - 1)/3 ) - 1)/3 ).

Original entry on oeis.org

1, 1, 13, 208, 4132, 99328, 2799073, 90310006, 3281661436, 132615087517, 5897867191525, 286140731152972, 15031839986716483, 849637058684740030, 51389339196926149645, 3310400979718767433801, 226189040323182011660827, 16333609964679285918346633

Offset: 0

Seiichi Manyama, Aug 02 2024

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A374882, A375174.

Cf. A004212.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((exp((exp(9*x)-1)/3)-1)/3)))

a(n) = Sum_{k=0..n} 9^(n-k) * Stirling2(n,k) * A004212(k) = 9^n * Sum_{k=0..n} (1/3)^k * Stirling2(n,k) * Bell_k(1/3), where Bell_n(x) is n-th Bell polynomial.

A369785 Expansion of e.g.f. exp( (exp(3*(exp(x)-1))-1)/3 ).

Original entry on oeis.org

1, 1, 5, 32, 252, 2368, 25865, 321310, 4461684, 68329293, 1142114917, 20663072796, 401891071075, 8355591197398, 184796601094141, 4329517995684305, 107060130166069859, 2785248872828731497, 76017344650249268158, 2171058618712177987046

Offset: 0

Seiichi Manyama, Feb 01 2024

nonn

Cf. A000258, A369784.

Cf. A004212, A049425.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((exp(3*(exp(x)-1))-1)/3)))

a(n) = Sum_{k=0..n} Stirling2(n,k) * A004212(k).

A380259 Expansion of e.g.f. exp( (1/(1-2*x)^(3/2) - 1)/3 ).

Original entry on oeis.org

1, 1, 6, 51, 561, 7566, 120711, 2221311, 46269126, 1075249881, 27560477331, 771948530046, 23446574573841, 767288588019201, 26905482997736526, 1006166248423254171, 39962774633459923881, 1679677496419394133846, 74471142324541556576151

Offset: 0

Seiichi Manyama, Jan 18 2025

nonn

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A049377, A380212.

Table[Sum[3^k * 2^(n-k) * Abs[StirlingS1[n,k]] * BellB[k, 1/3], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 23 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-2*x)^(3/2)-1)/3)))

a(n) = Sum_{k=0..n} 2^(n-k) * |Stirling1(n,k)| * A004212(k) = Sum_{k=0..n} 3^k * 2^(n-k) * |Stirling1(n,k)| * Bell_k(1/3), where Bell_n(x) is n-th Bell polynomial.
a(n) = (1/exp(1/3)) * (-2)^n * n! * Sum_{k>=0} binomial(-3*k/2,n)/(3^k * k!).
a(n) ~ 2^(n + 3/10) * n^(n - 1/5) * exp(-1/3 + 2^(1/5)*n^(1/5)/4 + 5*2^(3/5)*n^(3/5)/6 - n) / sqrt(5) * (1 + 2^(4/5) / (30 * n^(1/5))). - Vaclav Kotesovec, Jan 23 2025

A380260 Expansion of e.g.f. exp( ((1+2*x)^(3/2) - 1)/3 ).

Original entry on oeis.org

1, 1, 2, 3, 9, 6, 111, -573, 7638, -95751, 1450431, -24643134, 468589617, -9843336567, 226448287794, -5662061186949, 152892006728841, -4434211761771978, 137468475061977663, -4536657554920874181, 158788359466681092966, -5875324355407515077439, 229142457698060305226367

Offset: 0

Seiichi Manyama, Jan 18 2025

sign

Eric Weisstein's World of Mathematics, Bell Polynomial.

Cf. A049425, A380259.

With[{nn=30},CoefficientList[Series[Exp[((1+2x)^(3/2)-1)/3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 29 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(((1+2*x)^(3/2)-1)/3)))

a(n) = Sum_{k=0..n} 2^(n-k) * Stirling1(n,k) * A004212(k) = Sum_{k=0..n} 3^k * 2^(n-k) * Stirling1(n,k) * Bell_k(1/3), where Bell_n(x) is n-th Bell polynomial.

a(n) = (1/exp(1/3)) * 2^n * n! * Sum_{k>=0} binomial(3*k/2,n)/(3^k * k!).

cp's OEIS Frontend

A355164 a(n) = exp(-1/3) * Sum_{k>=0} (3*k + 2)^n / (3^k * k!).

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A367744 Expansion of e.g.f. exp(1 - x - exp(3*x)).

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A111673 Triangle, generated from A111579.

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A375174 Expansion of e.g.f. exp( (1/(1 - 9*x)^(1/3) - 1)/3 ).

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A375176 Expansion of e.g.f. exp( (exp( (exp(9*x) - 1)/3 ) - 1)/3 ).

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A369785 Expansion of e.g.f. exp( (exp(3*(exp(x)-1))-1)/3 ).

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A380259 Expansion of e.g.f. exp( (1/(1-2*x)^(3/2) - 1)/3 ).

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A380260 Expansion of e.g.f. exp( ((1+2*x)^(3/2) - 1)/3 ).

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Mathematica

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Formula

A355164 a(n) = exp(-1/3) * Sum_{k>=0} (3k + 2)^n / (3^k k!).