A346985 -id:A346985 - cp's OEIS frontend

A094419 Generalized ordered Bell numbers Bo(6,n).

1, 6, 78, 1518, 39390, 1277646, 49729758, 2258233998, 117196187550, 6842432930766, 443879517004638, 31674687990494478, 2465744921215207710, 207943837884583262286, 18885506918597311159518, 1837699347783655374914958, 190743171535070652261555870, 21035482423625416328497024206

Offset: 0

Ralf Stephan, May 02 2004

nonn

Sixth row of array A094416, which has more information.

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

Cf. A032033, A094416, A094417, A094418, A131689.
Cf. A346985, A354252, A365555, A365556, A365557.
Cf. A384514, A384521, A384522, A384523.

A094416:= func< n,k | (&+[Factorial(j)*n^j*StirlingSecond(k,j): j in [0..k]]) >;
A094419:= func< k | A094416(6,k) >;
[A094419(n): n in [0..30]]; // G. C. Greubel, Jan 12 2024

t = 30; Range[0, t]! CoefficientList[Series[1/(7 - 6 Exp[x]),{x, 0, t}], x] (* Vincenzo Librandi, Mar 16 2014 *)

my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(7-6*exp(x)))) \\ Joerg Arndt, Jan 15 2024

a(n) = (-1)^(n+1)*polylog(-n, 7/6)/7; \\ Seiichi Manyama, Jun 01 2025

def A094416(n,k): return sum(factorial(j)*n^j*stirling_number2(k,j) for j in range(k+1)) # array
def A094419(k): return A094416(6,k)
[A094419(n) for n in range(31)] # G. C. Greubel, Jan 12 2024

E.g.f.: 1/(7 - 6*exp(x)).
a(n) = Sum_{k=0..n} A131689(n,k) * 6^k. - Philippe Deléham, Nov 03 2008
a(n) ~ n! / (7*(log(7/6))^(n+1)). - Vaclav Kotesovec, Mar 14 2014
a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020
a(0) = 1; a(n) = 6 * a(n-1) - 7 * Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023
From Seiichi Manyama, Jun 01 2025: (Start)
a(n) = (-1)^(n+1)/7 * Li_{-n}(7/6), where Li_{n}(x) is the polylogarithm function.
a(n) = (1/7) * Sum_{k>=0} k^n * (6/7)^k.
a(n) = (6/7) * Sum_{k=0..n} 7^k * (-1)^(n-k) * A131689(n,k) for n > 0. (End)

A305404 Expansion of Sum_{k>=0} (2k - 1)!!x^k/Product_{j=1..k} (1 - j*x).

Original entry on oeis.org

1, 1, 4, 25, 217, 2416, 32839, 527185, 9761602, 204800551, 4801461049, 124402647370, 3529848676237, 108859319101261, 3625569585663484, 129689000146431205, 4958830249864725997, 201834650901695603296, 8712774828941647677019, 397596632650906687905565

Offset: 0

Ilya Gutkovskiy, May 31 2018

Stirling transform of A001147.

Alois P. Heinz, Table of n, a(n) for n = 0..391
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000670, A001147, A004123, A305405, A346982 - A346985.

b:= proc(n, m) option remember;
     `if`(n=0, doublefactorial(2*m-1), 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 = 19; CoefficientList[Series[Sum[(2 k - 1)!! x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 19; CoefficientList[Series[1/Sqrt[3 - 2 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] (2 k - 1)!!, {k, 0, n}], {n, 0, 19}]

E.g.f.: 1/sqrt(3 - 2*exp(x)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*(2*k - 1)!!.
a(n) ~ sqrt(2/3) * n^n / ((log(3/2))^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Jul 01 2018
Conjecture: a(n) = Sum_{k>=0} k^n * binomial(2*k,k) / (2^k * 3^(k + 1/2)). - Diego Rattaggi, Oct 11 2020
O.g.f. conjectural: 1/(1 - x/(1 - 3*x/(1 - 3*x/(1 - 6*x/(1 - 5*x/(1 - 9*x/(1 - 7*x/(1 - ... - (2*n-1)*x/(1 - 3*n*x/(1 - ... )))))))))) - a continued fraction of Stieltjes-type. - Peter Bala, Dec 06 2020
a(0) = 1; a(n) = Sum_{k=1..n} (2 - k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = a(n-1) - 3*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

A346982 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^(1/3).

Original entry on oeis.org

1, 1, 5, 41, 477, 7201, 133685, 2945881, 75145677, 2177900241, 70687244965, 2539879312521, 100086803174077, 4291845333310081, 198954892070938645, 9914294755149067961, 528504758009562261677, 30010032597449931644721, 1808359960001658961070725

Offset: 0

Ilya Gutkovskiy, Aug 09 2021

Stirling transform of A007559.

Seiichi Manyama, Table of n, a(n) for n = 0..373

Cf. A000670, A007559, A305404, A346983, A346984, A346985, A352117, A352118, A352119.

Cf. A032033, A365558.

g:= proc(n) option remember; `if`(n<2, 1, (3*n-2)*g(n-1)) end:
b:= proc(n, m) option remember;
     `if`(n=0, g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 09 2021

nmax = 18; CoefficientList[Series[1/(4 - 3 Exp[x])^(1/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} Stirling2(n,k) * A007559(k).
a(n) ~ n! / (Gamma(1/3) * 2^(2/3) * n^(2/3) * log(4/3)^(n + 1/3)). - Vaclav Kotesovec, Aug 14 2021
From  Peter Bala, Aug 22 2023: (Start)
O.g.f. (conjectural): 1/(1 - x/(1 - 4*x/(1 - 4*x/(1 - 8*x/(1 - 7*x/(1 - 12*x/(1 - ... - (3*n-2)*x/(1 - 4*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction).
More generally, it appears that the o.g.f. of the sequence whose e.g.f. is equal to 1/(r+1 - r*exp(s*x))^(m/s) corresponds to the S-fraction 1/(1 - r*m*x/(1 - s*(r+1)*x/(1 - r*(m+s)*x/(1 - 2*s(r+1)*x/(1 - r*(m+2*s)*x/(1 - 3*s(r+1)*x/( 1 - ... ))))))). This is the case r = 3, s = 1, m = 1/3. (End)
a(0) = 1; a(n) = Sum_{k=1..n} (3 - 2*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = a(n-1) - 4*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

A346984 Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(1/5).

Original entry on oeis.org

1, 1, 7, 85, 1495, 34477, 983983, 33476437, 1322441575, 59492222077, 3002578396255, 168005805229285, 10321907081030167, 690761732852321677, 50015387402165694607, 3895721046926471861365, 324805103526730206129607, 28861947117644330678207389, 2722944810091827410698112959

Offset: 0

Ilya Gutkovskiy, Aug 09 2021

Stirling transform of A008548.

Seiichi Manyama, Table of n, a(n) for n = 0..351

Cf. A000670, A008548, A094418, A305404, A346982, A346983, A346985, A352117, A352118, A352119.

g:= proc(n) option remember; `if`(n<2, 1, (5*n-4)*g(n-1)) end:
b:= proc(n, m) option remember;
     `if`(n=0, g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 09 2021

nmax = 18; CoefficientList[Series[1/(6 - 5 Exp[x])^(1/5), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] 5^k Pochhammer[1/5, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} Stirling2(n,k) * A008548(k).
a(n) ~ n! / (Gamma(1/5) * 6^(1/5) * n^(4/5) * log(6/5)^(n + 1/5)). - Vaclav Kotesovec, Aug 14 2021
O.g.f. (conjectural): 1/(1 - x/(1 - 6*x/(1 - 6*x/(1 - 12*x/(1 - 11*x/(1 - 18*x/(1 - ... - (5*n-4)*x/(1 - 6*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type. - Peter Bala, Aug 22 2023
a(0) = 1; a(n) = Sum_{k=1..n} (5 - 4*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = a(n-1) - 6*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

A354252 Expansion of e.g.f. 1/sqrt(7 - 6 * exp(x)).

Original entry on oeis.org

1, 3, 30, 489, 11127, 325218, 11612595, 489926559, 23846152332, 1315294430043, 81078316924035, 5523729981650004, 412148874577007037, 33425421047034028743, 2927620572178735480350, 275410244285003264624949, 27695140477706524122414867

Offset: 0

Seiichi Manyama, May 21 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..343

Cf. A305404, A354242, A354253.
Cf. A094419, A346985, A354252, A365556, A365557.
Cf. A011781.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(7-6*exp(x))))

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, binomial(2*k, k)*(3*(exp(x)-1)/2)^k)))

a(n) = sum(k=0, n, (3/2)^k*(2*k)!*stirling(n, k, 2)/k!);

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (3 * (exp(x) - 1)/2)^k.
a(n) = Sum_{k=0..n} (3/2)^k * (2*k)! * Stirling2(n,k)/k!.
a(n) ~ sqrt(2/7) * n^n / (exp(n) * log(7/6)^(n + 1/2)). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (6 - 3*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = 3*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A346983 Expansion of e.g.f. 1 / (5 - 4 * exp(x))^(1/4).

Original entry on oeis.org

1, 1, 6, 61, 891, 16996, 400251, 11217781, 364638336, 13486045291, 559192836771, 25691965808026, 1295521405067181, 71131584836353861, 4224255395774155566, 269791923787785076921, 18439806740525320993551, 1342957106015632474616956, 103824389511747541791086511

Offset: 0

Ilya Gutkovskiy, Aug 09 2021

Stirling transform of A007696.

Seiichi Manyama, Table of n, a(n) for n = 0..360

Cf. A000670, A007696, A305404, A346982, A346984, A346985, A352117, A352118, A352119.

Cf. A094417, A354242, A365567.

g:= proc(n) option remember; `if`(n<2, 1, (4*n-3)*g(n-1)) end:
b:= proc(n, m) option remember;
     `if`(n=0, g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..18);  # Alois P. Heinz, Aug 09 2021

nmax = 18; CoefficientList[Series[1/(5 - 4 Exp[x])^(1/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] 4^k Pochhammer[1/4, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} Stirling2(n,k) * A007696(k).
a(n) ~ n! / (Gamma(1/4) * 5^(1/4) * n^(3/4) * log(5/4)^(n + 1/4)). - Vaclav Kotesovec, Aug 14 2021
O.g.f. (conjectural): 1/(1 - x/(1 - 5*x/(1 - 5*x/(1 - 10*x/(1 - 9*x/(1 - 15*x/(1 - ... - (4*n-3)*x/(1 - 5*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type. - Peter Bala, Aug 22 2023
a(0) = 1; a(n) = Sum_{k=1..n} (4 - 3*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = a(n-1) - 5*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

A352117 Expansion of e.g.f. 1/sqrt(2 - exp(2*x)).

Original entry on oeis.org

1, 1, 5, 37, 377, 4921, 78365, 1473277, 31938737, 784384561, 21523937525, 652667322517, 21672312694697, 782133969325801, 30481907097849485, 1275870745561131757, 57083444567425884257, 2718602143583362124641, 137315150097164841942245

Offset: 0

Seiichi Manyama, Mar 05 2022

nonn

Cf. A000670, A097397, A216794, A352118, A352119, A305404, A346982 - A346985.

m = 18; Range[0, m]! * CoefficientList[Series[(2 - Exp[2*x])^(-1/2), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)

a[n]:=if n=0 then 1 else sum(a[n-k]*(1-k/n/2)*binomial(n,k)*2^k,k,1,n);
makelist(a[n],n,0,50); /* Tani Akinari, Sep 06 2023 */

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

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

a(n) = Sum_{k=0..n} 2^(n-k) * (Product_{j=0..k-1} (2*j+1)) * Stirling2(n,k).
a(n) ~ 2^n * n^n / (log(2)^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Mar 05 2022
Conjectural o.g.f. as a continued fraction of Stieltjes type: 1/(1 - x/(1 - 4*x/(1 - 3*x/(1 - 8*x/(1 - ... - (2*n-1)*x/(1 - 4*n*x/(1 - ... ))))))). Cf. A346982.  - Peter Bala, Aug 22 2023
For n > 0, a(n) = Sum_{k=1..n} a(n-k)*(1-k/n/2)*binomial(n,k)*2^k. - Tani Akinari, Sep 06 2023
a(0) = 1; a(n) = a(n-1) - 2*Sum_{k=1..n-1} (-2)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 18 2023

A365556 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(2/3).

Original entry on oeis.org

1, 4, 44, 764, 18204, 551644, 20291804, 877970524, 43680345564, 2456429581404, 154072160204764, 10663000409493084, 807124301044917724, 66329628496719183964, 5881222650127663682524, 559616682597652939940444, 56879286407092006924382684

Offset: 0

Seiichi Manyama, Sep 09 2023

nonn

Cf. A094419, A346985, A354252, A365555, A365557.

a[n_] := Sum[Product[6*j + 4, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 11 2023 *)

a(n) = sum(k=0, n, prod(j=0, k-1, 6*j+4)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} 2^k * (Product_{j=0..k-1} (3*j+2)) * Stirling2(n,k) = Sum_{k=0..n} (Product_{j=0..k-1} (6*j+4)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (6 - 2*k/n) * binomial(n,k) * a(n-k).
O.g.f. (conjectural): 1/(1 - 4*x/(1 - 7*x/(1 - 10*x/(1 - 14*x/(1 - 16*x/(1 - 21*x/(1 - ... - (6*n - 2)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction). - Peter Bala, Sep 24 2023
a(n) ~ Gamma(1/3) * sqrt(3) * n^(n + 1/6) / (sqrt(2*Pi) * 7^(2/3) * exp(n) * log(7/6)^(n + 2/3)). - Vaclav Kotesovec, Nov 11 2023
a(0) = 1; a(n) = 4*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A365557 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(5/6).

Original entry on oeis.org

1, 5, 60, 1105, 27505, 862900, 32665935, 1448431605, 73618245530, 4219213176975, 269178309769385, 18919087590749230, 1452439246800583805, 120926788470961893425, 10852505784073190637460, 1044349665968997385498605, 107273533723839304683589205

Offset: 0

Seiichi Manyama, Sep 09 2023

nonn

Cf. A094419, A346985, A354252, A365555, A365556.

a[n_] := Sum[Product[6*j + 5, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 11 2023 *)

a(n) = sum(k=0, n, prod(j=0, k-1, 6*j+5)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (6*j+5)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (6 - k/n) * binomial(n,k) * a(n-k).
O.g.f. (conjectural): 1/(1 - 5*x/(1 - 7*x/(1 - 11*x/(1 - 14*x/(1 - 17*x/(1 - 21*x/(1 - ... - (6*n - 1)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction). - Peter Bala, Sep 24 2023
a(n) ~ Gamma(1/3)^2 * sqrt(3) * n^(n + 1/3) / (14^(5/6) * Pi * exp(n) * log(7/6)^(n + 5/6)). - Vaclav Kotesovec, Nov 11 2023
a(0) = 1; a(n) = 5*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A365555 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(1/3).

Original entry on oeis.org

1, 2, 18, 274, 5938, 167122, 5786418, 237857874, 11319677618, 612109819602, 37069480301618, 2485356833141074, 182753029186750898, 14623552941626800082, 1265002802597606144818, 117633823750542653153874, 11701922865351577653913778

Offset: 0

Seiichi Manyama, Sep 09 2023

nonn

Cf. A094419, A346985, A354252, A365556, A365557.

a[n_] := Sum[Product[6*j + 2, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 11 2023 *)
With[{nn=20},CoefficientList[Series[1/(7-6Exp[x])^(1/3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 21 2024 *)

a(n) = sum(k=0, n, prod(j=0, k-1, 6*j+2)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} 2^k * (Product_{j=0..k-1} (3*j+1)) * Stirling2(n,k) = Sum_{k=0..n} (Product_{j=0..k-1} (6*j+2)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (6 - 4*k/n) * binomial(n,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n - 1/6) / (Gamma(1/3) * 7^(1/3) * exp(n) * log(7/6)^(n + 1/3)). - Vaclav Kotesovec, Sep 09 2023
From Peter Bala, Sep 24 2023: (Start)
O.g.f. (conjectural): 1/(1 - 2*x/(1 - 7*x/(1 - 8*x/(1 - 14*x/(1 - 14*x/(1 - 21*x/(1 - ... - (6*n - 4)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction).
More generally, it appears that the o.g.f. of the sequence whose e.g.f. is 1/(r+1 - r*exp(s*x))^(m/s) corresponds to the S-fraction 1/(1 - r*m*x/(1 - s*(r+1)*x/(1 - r*(m+s)*x/(1 - 2*s(r+1)*x/(1 - r*(m+2*s)*x/(1 - 3*s(r+1)*x/( 1 - ... ))))))). This is the case r = 6, s = 1, m = 1/3. (End)
a(0) = 1; a(n) = 2*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

cp's OEIS Frontend

A094419 Generalized ordered Bell numbers Bo(6,n).

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A305404 Expansion of Sum_{k>=0} (2*k - 1)!!*x^k/Product_{j=1..k} (1 - j*x).

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A346982 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^(1/3).

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A346984 Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(1/5).

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A354252 Expansion of e.g.f. 1/sqrt(7 - 6 * exp(x)).

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PARI

PARI

PARI

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A346983 Expansion of e.g.f. 1 / (5 - 4 * exp(x))^(1/4).

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Maple

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A352117 Expansion of e.g.f. 1/sqrt(2 - exp(2*x)).

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Mathematica

Maxima

PARI

PARI

A305404 Expansion of Sum_{k>=0} (2k - 1)!!x^k/Product_{j=1..k} (1 - j*x).