A004123 -id:A004123 - cp's OEIS frontend

A301921 Expansion of e.g.f. 1/(1 - (exp(x) - 1)/(1 - (exp(x) - 1)^2/(1 - (exp(x) - 1)^3/(1 - ...)))), a continued fraction.

1, 1, 3, 19, 159, 1651, 21303, 324619, 5653119, 110909251, 2424648903, 58430418619, 1537673312079, 43860906193651, 1347852526593303, 44392923532503019, 1560023977386027039, 58259266750803410851, 2303999137417453606503, 96188099015599819297819, 4227325636692027926037999

Offset: 0

Ilya Gutkovskiy, Jun 19 2018

nonn

From Peter Bala, Aug 19 2025: (Start)

Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 9 we obtain the sequence [0, 1, 3, 1, 6, 4, 0, 7, 3, 1, 6, 4, 0, 7, 3, 1, 6, 4, 0, 7, ...] with an apparent period of 6 = phi(9) beginning at n = 2. Cf. A004123. (End)

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 159*x^4/4! + 1651*x^5/5! + 21303*x^6/6! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A005169, A019538, A301923.

nmax = 20; CoefficientList[Series[1/(1 + ContinuedFractionK[-(Exp[x] - 1)^k, 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
b[n_] := b[n] = SeriesCoefficient[1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, n}]), {x, 0, n}]; a[n_] := a[n] = Sum[StirlingS2[n, k] b[k] k!, {k, 0, n}]; Table[a[n], {n, 0, 20}]

a(n) = Sum_{k=0..n} Stirling2(n,k)*A005169(k)*k!.

a(n) ~ c * d^n * n!, where d = 2.19787763261059933075080498218168228... and c = 0.250957960982243982921501085974065... - Vaclav Kotesovec, Dec 20 2018

A354290 Expansion of e.g.f. exp(f(x) - 1) where f(x) = 1/(3 - 2*exp(x)).

Original entry on oeis.org

1, 2, 14, 142, 1878, 30494, 585398, 12946910, 323717622, 9020101470, 276940926646, 9283709731806, 337237965060982, 13191050077634654, 552593521885522486, 24677110613547498718, 1169994350288769049334, 58684818937875321715038

Offset: 0

Seiichi Manyama, May 23 2022

nonn

Cf. A075729, A354291.

Cf. A004123, A354286, A354288.

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, 2^k*k!*stirling(j, k, 2))*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A004123(k+1) * binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} 2^k * A000262(k) * Stirling2(n,k).
a(n) ~ exp(1/(6*log(3/2)) - 5/6 + 2*sqrt(n)/sqrt(3*log(3/2)) - n) * (n^(n - 1/4) / (sqrt(2) * 3^(1/4) * log(3/2)^(n + 1/4))). - Vaclav Kotesovec, May 23 2022

A375948 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^(3/2).

Original entry on oeis.org

1, 3, 18, 153, 1683, 22698, 362403, 6683463, 139787568, 3269240883, 84535585263, 2394699999948, 73749495626253, 2453332830142743, 87667856626175298, 3349116499958627733, 136209377351085310863, 5875794769594996985778, 267968680043585007829383

Offset: 0

Seiichi Manyama, Sep 03 2024

nonn

Cf. A005649, A375949.
Cf. A004123, A367470, A367471, A375954.
Cf. A001147.

nmax=18; CoefficientList[Series[1 / (3 - 2 * Exp[x])^(3/2),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)

a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k+1)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} A001147(k+1) * Stirling2(n,k).

a(n) ~ 2^(3/2) * n^(n+1) / (3^(3/2) * log(3/2)^(n + 3/2) * exp(n)). - Vaclav Kotesovec, May 20 2025

A238466 Generalized ordered Bell numbers Bo(9,n).

Original entry on oeis.org

1, 9, 171, 4869, 184851, 8772309, 499559571, 33190014069, 2520110222451, 215270320769109, 20431783142389971, 2133148392099721269, 242954208655633344051, 29977118969127060357909, 3983272698956314883956371, 567091857051921058649396469

Offset: 0

Vincenzo Librandi, Mar 18 2014

nonn

Row 9 of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..250

Cf. A000670, A004123, A032033, A094417, A094418, A094419, A238464.

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(10 - 9*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]];

t=30; Range[0, t]! CoefficientList[Series[1/(10 - 9 Exp[x]), {x, 0, t}], x]

E.g.f.: 1/(10 - 9*exp(x)).
a(n) ~ n! / (10*(log(10/9))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 9*a(n-1) - 10*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A238467 Generalized ordered Bell numbers Bo(10,n).

Original entry on oeis.org

1, 10, 210, 6610, 277410, 14553010, 916146210, 67285818610, 5647734061410, 533307215001010, 55954905981282210, 6457903731351210610, 813080459351919805410, 110901542660769629769010, 16290196917457939734258210

Offset: 0

Vincenzo Librandi, Mar 18 2014

nonn

Row 10 of array A094416, which has more information.

Vincenzo Librandi, Table of n, a(n) for n = 0..250

Cf. A000670, A004123, A032033, A094417, A094418, A094419, A238464.

m:=20; R:=LaurentSeriesRing(RationalField(), m); b:=Coefficients(R!(1/(11 - 10*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]];

t=30; Range[0, t]! CoefficientList[Series[1/(11 - 10 Exp[x]), {x, 0, t}], x]

E.g.f.: 1/(11 - 10*exp(x)).
a(n) ~ n! / (11*(log(11/10))^(n+1)). - Vaclav Kotesovec, Mar 20 2014
a(0) = 1; a(n) = 10*a(n-1) - 11*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A305405 Expansion of Sum_{k>=0} k!!x^k/Product_{j=1..k} (1 - jx).

Original entry on oeis.org

1, 1, 3, 10, 41, 201, 1126, 7043, 48603, 366298, 2987189, 26163501, 244654150, 2430411335, 25539609327, 282834656434, 3290175964577, 40089424302657, 510340938343270, 6772086558823547, 93481666812344979, 1339885322519303434, 19907413622297965373, 306126204811557339045

Offset: 0

Ilya Gutkovskiy, May 31 2018

nonn

Stirling transform of A006882.

Alois P. Heinz, Table of n, a(n) for n = 0..517
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000670, A004123, A006882, A305404.

b:= proc(n, m) option remember;
     `if`(n=0, doublefactorial(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 04 2021

nmax = 23; CoefficientList[Series[Sum[k!! x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[1 + Exp[(E^x - 1)^2/2] (Exp[x] - 1) (1 + Sqrt[Pi/2] Erf[(Exp[x] - 1)/Sqrt[2]]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] k!!, {k, 0, n}], {n, 0, 23}]

E.g.f.: 1 + exp((exp(x) - 1)^2/2)*(exp(x) - 1)*(1 + sqrt(Pi/2)*erf((exp(x) - 1)/sqrt(2))).

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

A367486 Expansion of e.g.f. 1/(3 - 2*exp(x))^x.

Original entry on oeis.org

1, 0, 4, 18, 168, 1830, 24540, 388122, 7084560, 146650446, 3395460900, 86962122786, 2441210321880, 74542218945558, 2459830123779756, 87236196407090730, 3308881779086345760, 133667058288336876894, 5729380391745420070068

Offset: 0

Seiichi Manyama, Nov 19 2023

nonn

Cf. A004123, A367470, A367471.
Cf. A354413, A367488.
Cf. A367489.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*sum(k=1, j-1, 2^k*(k-1)!*stirling(j-1, k, 2))*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A367489(k) * binomial(n-1,k-1) * a(n-k).

A368319 Expansion of e.g.f. exp(2x) / (3 - 2exp(x)).

Original entry on oeis.org

1, 4, 22, 166, 1642, 20254, 299722, 5174446, 102094042, 2266154014, 55890234922, 1516265078926, 44874837768442, 1438774580904574, 49678366226498122, 1837828899444250606, 72522300447277154842, 3040654011774599283934, 134985159308312666889322

Offset: 0

Seiichi Manyama, Dec 21 2023

nonn

Cf. A004123, A201339, A368320, A368321.

b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));
a(n, m=2, t=2) = my(u=1+1/t); u^m*b(n, t)-(1/t)*sum(j=0, m-1, u^j*(m-1-j)^n);

a(n) = 2^n + 2 * Sum_{k=1..n} binomial(n,k) * a(n-k).

a(n) = (9/4)*A004123(n+1) - (1/2)*(1 + (3/2)*0^n).

A368320 Expansion of e.g.f. exp(3x) / (3 - 2exp(x)).

Original entry on oeis.org

1, 5, 31, 245, 2455, 30365, 449551, 7761605, 153140935, 3399230765, 83835351871, 2274397617365, 67312256650615, 2158161871352765, 74517549339738991, 2756743349166359525, 108783450670915699495, 4560981017661898860365, 202477738962469000202911

Offset: 0

Seiichi Manyama, Dec 21 2023

nonn

Cf. A004123, A201339, A368319, A368321.

b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));
a(n, m=3, t=2) = my(u=1+1/t); u^m*b(n, t)-(1/t)*sum(j=0, m-1, u^j*(m-1-j)^n);

a(n) = 3^n + 2 * Sum_{k=1..n} binomial(n,k) * a(n-k).

a(n) = (27/8)*A004123(n+1) - (1/2)*(2^n + 3/2 + (9/4)*0^n).

A368321 Expansion of e.g.f. exp(4x) / (3 - 2exp(x)).

Original entry on oeis.org

1, 6, 42, 354, 3642, 45426, 673962, 11641314, 229708122, 5098836306, 125752998282, 3411596337474, 100968384710202, 3237242806231986, 111776324007217002, 4135115023742364834, 163175176006352025882, 6841471526492783720466, 303716608443703306594122

Offset: 0

Seiichi Manyama, Dec 21 2023

nonn

Cf. A004123, A201339, A368319, A368320.

b(n, t) = sum(k=0, n, t^k*k!*stirling(n, k, 2));
a(n, m=4, t=2) = my(u=1+1/t); u^m*b(n, t)-(1/t)*sum(j=0, m-1, u^j*(m-1-j)^n);

a(n) = 4^n + 2 * Sum_{k=1..n} binomial(n,k) * a(n-k).

a(n) = (81/16)*A004123(n+1) - (1/2)*(3^n + (3/2)*2^n + 9/4 + (27/8)*0^n).

cp's OEIS Frontend

A301921 Expansion of e.g.f. 1/(1 - (exp(x) - 1)/(1 - (exp(x) - 1)^2/(1 - (exp(x) - 1)^3/(1 - ...)))), a continued fraction.

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A354290 Expansion of e.g.f. exp(f(x) - 1) where f(x) = 1/(3 - 2*exp(x)).

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A375948 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^(3/2).

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Mathematica

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A238466 Generalized ordered Bell numbers Bo(9,n).

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Magma

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A238467 Generalized ordered Bell numbers Bo(10,n).

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Magma

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A305405 Expansion of Sum_{k>=0} k!!*x^k/Product_{j=1..k} (1 - j*x).

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Maple

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A367486 Expansion of e.g.f. 1/(3 - 2*exp(x))^x.

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A368319 Expansion of e.g.f. exp(2*x) / (3 - 2*exp(x)).

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A368320 Expansion of e.g.f. exp(3*x) / (3 - 2*exp(x)).

A305405 Expansion of Sum_{k>=0} k!!x^k/Product_{j=1..k} (1 - jx).

A368319 Expansion of e.g.f. exp(2x) / (3 - 2exp(x)).

A368320 Expansion of e.g.f. exp(3x) / (3 - 2exp(x)).

A368321 Expansion of e.g.f. exp(4x) / (3 - 2exp(x)).