A305404 -id:A305404 - cp's OEIS frontend

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

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

A346985 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(1/6).

Original entry on oeis.org

1, 1, 8, 113, 2325, 62896, 2109143, 84403033, 3924963750, 207976793991, 12369246804853, 815880360117978, 59107920881218525, 4665585774576259261, 398534278371999103888, 36627974592437584634573, 3603954453161886215458025, 377983931878997401821759456, 42095013846928585982896180123

Offset: 0

Ilya Gutkovskiy, Aug 09 2021

nonn

Stirling transform of A008542.

In general, for k >= 1, if e.g.f. = 1 / (k + 1 - k*exp(x))^(1/k), then a(n) ~ n! / (Gamma(1/k) * (k+1)^(1/k) * n^(1 - 1/k) * log(1 + 1/k)^(n + 1/k)). - Vaclav Kotesovec, Aug 14 2021

Alois P. Heinz, Table of n, a(n) for n = 0..343

Cf. A000670, A008542, A094419, A305404, A346982, A346983, A346984, A352117, A352118, A352119.

Cf. A094419, A354252, A365555, A365556, A365557.

g:= proc(n) option remember; `if`(n<2, 1, (6*n-5)*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/(7 - 6 Exp[x])^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 18}]

a[n]:=if n=0 then 1 else (1/n)*sum(binomial(n,k)*(n+5*k)*a[k],k,0,n-1);
makelist(a[n],n,0,50); /* Tani Akinari, Aug 22 2023 */

a(n) = Sum_{k=0..n} Stirling2(n,k) * A008542(k).
a(n) ~ n! / (Gamma(1/6) * 7^(1/6) * n^(5/6) * log(7/6)^(n + 1/6)). - Vaclav Kotesovec, Aug 14 2021
For n > 0, a(n) = (1/n)*Sum_{k=0..n-1} binomial(n,k)*(n+5*k)*a(k). - Tani Akinari, Aug 22 2023
O.g.f. (conjectural): 1/(1 - x/(1 - 7*x/(1 - 7*x/(1 - 14*x/(1 - 13*x/(1 - 21*x/(1 - ... - (6*n-5)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction). - Peter Bala, Aug 25 2023
a(0) = 1; a(n) = a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

A354242 Expansion of e.g.f. 1/sqrt(5 - 4 * exp(x)).

Original entry on oeis.org

1, 2, 14, 158, 2486, 50222, 1239254, 36126638, 1214933846, 46299580142, 1971815255894, 92809525295918, 4784166929982806, 268050260650705262, 16219498558371118934, 1054102762745609325998, 73229184033780135425366, 5415407651703010175897582

Offset: 0

Seiichi Manyama, May 20 2022

nonn

From Peter Bala, Jul 07 2022: (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 16 we obtain the sequence [1, 2, 14, 14, 6, 14, 6, 14, 6, ...], with an apparent period of 2 beginning at a(3). Cf. A354253.
More generally, we conjecture that the same property holds for integer sequences having an e.g.f. of the form G(exp(x) - 1), where G(x) is an integral power series. (End)

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

Cf. A305404, A354252, A354253.

Cf. A000670, A001813, A094417, A316747, A354240, A354241.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(5-4*exp(x))))

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

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

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (exp(x) - 1)^k.
a(n) = Sum_{k=0..n} (2*k)! * Stirling2(n,k)/k!.
a(n) ~ sqrt(2/5) * n^n / (exp(n) * log(5/4)^(n + 1/2)). - Vaclav Kotesovec, Jun 04 2022
Conjectural o.g.f. as a continued fraction of Stieltjes type: 1/(1 - 2*x/(1 - 5*x/(1 - 6*x/(1 - 10*x/(1 - 10*x/(1 - 15*x/(1 - ... - (4*n-2)*x/(1 - 5*n*x/(1 - ...))))))))). - Peter Bala, Jul 07 2022
a(0) = 1; a(n) = Sum_{k=1..n} (4 - 2*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = 2*a(n-1) - 5*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

A346978 Expansion of e.g.f. 1 / sqrt(1 + 2 * log(1 - x)).

Original entry on oeis.org

1, 1, 4, 26, 234, 2694, 37812, 626352, 11962164, 258787812, 6255195168, 167072685240, 4886611129320, 155335056242040, 5332298685827760, 196590247328769120, 7747254471910795920, 324986515253994589200, 14458392906960271354560, 679977065168639138610720

Offset: 0

Ilya Gutkovskiy, Aug 09 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..390

Cf. A001147, A007840, A088500, A305404, A320343.

nmax = 19; CoefficientList[Series[1/Sqrt[1 + 2 Log[1 - x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] (2 k - 1)!!, {k, 0, n}], {n, 0, 19}]

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * (2*k-1)!!.
a(n) ~ n^n / (exp(n/2) * (exp(1/2) - 1)^(n + 1/2)). - Vaclav Kotesovec, Aug 09 2021
a(0) = 1; a(n) = Sum_{k=1..n} (2 - k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023

A320343 Expansion of e.g.f. 1/sqrt(1 - 2*log(1 + x)).

Original entry on oeis.org

1, 1, 2, 8, 42, 294, 2472, 24828, 286164, 3751428, 54864408, 887989200, 15731200680, 303068103480, 6304498706880, 140890167340560, 3365469544248720, 85585469309951760, 2308349518803845280, 65819488298810181120, 1978202007765686904480, 62505106242073569018720, 2071320752120227622985600

Offset: 0

Ilya Gutkovskiy, Jan 22 2019

nonn

Cf. A001147, A006252, A048994, A088501, A305404, A346978.

seq(n!*coeff(series(1/sqrt(1-2*log(1+x)),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 29 2019

nmax = 22; CoefficientList[Series[1/Sqrt[1 - 2 Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] (2 k - 1)!!, {k, 0, n}], {n, 0, 22}]

a(n) = Sum_{k=0..n} Stirling1(n,k)*A001147(k).
a(n) ~ n^n / ((exp(1/2) - 1)^(n + 1/2) * exp(n - 1/4)). - Vaclav Kotesovec, Jan 29 2019
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (2 - k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 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

A354253 Expansion of e.g.f. 1/sqrt(9 - 8 * exp(x)).

Original entry on oeis.org

1, 4, 52, 1108, 32980, 1261204, 58928212, 3253363348, 207225008980, 14958174725524, 1206698072485972, 107589343503498388, 10505997552329149780, 1115087729794287434644, 127819745001180490920532, 15736779719362919373550228, 2071062794354825889656471380

Offset: 0

Seiichi Manyama, May 21 2022

nonn

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 7 we obtain the sequence [1, 4, 3, 2, 3, 0, 0, 4, 3, 2, 3, 0, 0, 4, 3, 2, 3, 0, 0, ...], with a preperiod of length 1 and an apparent period thereafter of 6 = phi(7). Cf. A354242. - Peter Bala, Jul 08 2022

Seiichi Manyama, Table of n, a(n) for n = 0..331

Cf. A305404, A354242, A354252.

Cf. A144828, A238465.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(9-8*exp(x))))

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

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

E.g.f.: Sum_{k>=0} binomial(2*k,k) * (2 * (exp(x) - 1))^k.
a(n) = Sum_{k=0..n} 2^k * (2*k)! * Stirling2(n,k)/k!.
a(n) ~ sqrt(2) * n^n / (3 * exp(n) * log(9/8)^(n + 1/2)). - Vaclav Kotesovec, Jun 04 2022
Conjectural o.g.f. as a continued fraction of Stieltjes type: 1/(1 - 4*x/(1 - 9*x/(1 - 12*x/(1 - 18*x/(1 - 20*x/(1 - 27*x/(1 - ... - (8*n-4)*x/(1 - 9*n*x/(1 - ...))))))))). - Peter Bala, Jul 08 2022
a(0) = 1; a(n) = Sum_{k=1..n} (8 - 4*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = 4*a(n-1) - 9*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

cp's OEIS Frontend

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

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A346985 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(1/6).

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A354242 Expansion of e.g.f. 1/sqrt(5 - 4 * exp(x)).

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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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A346978 Expansion of e.g.f. 1 / sqrt(1 + 2 * log(1 - x)).

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A320343 Expansion of e.g.f. 1/sqrt(1 - 2*log(1 + x)).

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

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