A352117 -id:A352117 - 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

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

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

A352118 Expansion of e.g.f. 1/(2 - exp(3*x))^(1/3).

Original entry on oeis.org

1, 1, 7, 73, 1063, 20041, 464167, 12752713, 405439783, 14641740361, 592050220327, 26499885031753, 1300723181304103, 69470729022993481, 4010891467932629287, 248920020505516389193, 16525139232054244298023, 1168557027163488299171401

Offset: 0

Seiichi Manyama, Mar 05 2022

nonn

Cf. A000670, A352117, A352119.

Cf. A352070.

m = 17; Range[0, m]! * CoefficientList[Series[(2 - Exp[3*x])^(-1/3), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)

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

a(n) = sum(k=0, n, 3^(n-k)*prod(j=0, k-1, 3*j+1)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} 3^(n-k) * (Product_{j=0..k-1} (3*j+1)) * Stirling2(n,k).
a(n) ~ n! * 3^n / (2^(1/3) * Gamma(1/3) * n^(2/3) * log(2)^(n + 1/3)). - Vaclav Kotesovec, Mar 05 2022
From Seiichi Manyama, Nov 18 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} 3^k * (1 - 2/3 * k/n) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = a(n-1) - 2*Sum_{k=1..n-1} (-3)^k * binomial(n-1,k) * a(n-k). (End)

A352119 Expansion of e.g.f. 1/(2 - exp(4*x))^(1/4).

Original entry on oeis.org

1, 1, 9, 121, 2289, 56401, 1713849, 61939081, 2595199329, 123690992161, 6608289658089, 391154820258841, 25408740616159569, 1797051730819428721, 137463201511019813529, 11308020549364112399401, 995455518982520306979009, 93373681491447943767190081

Offset: 0

Seiichi Manyama, Mar 05 2022

nonn

Cf. A000670, A352117, A352118.

Cf. A352073.

m = 17; Range[0, m]! * CoefficientList[Series[(2 - Exp[4*x])^(-1/4), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(4*x))^(1/4)))

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

a(n) = Sum_{k=0..n} 4^(n-k) * (Product_{j=0..k-1} (4*j+1)) * Stirling2(n,k).
a(n) ~ n! * 2^(2*n - 1/4) / (Gamma(1/4) * n^(3/4) * log(2)^(n + 1/4)). - Vaclav Kotesovec, Mar 05 2022
From Seiichi Manyama, Nov 18 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} 4^k * (1 - 3/4 * k/n) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = a(n-1) - 2*Sum_{k=1..n-1} (-4)^k * binomial(n-1,k) * a(n-k). (End)

A194453 E.g.f. satisfies: A(x) = exp(x) - sqrt(1 - A(x)^2).

Original entry on oeis.org

1, 2, 7, 44, 421, 5342, 83707, 1556984, 33495721, 817880282, 22341817807, 675009140324, 22347321835021, 804481291160822, 31286388389010307, 1307157133950142064, 58390601701376026321, 2776992745284738150962, 140092142842449580093207

Offset: 1

Paul D. Hanna, Aug 24 2011

nonn

Compare e.g.f. to the identity: cosh(x) = exp(x) - sqrt(cosh(x)^2 - 1).

			E.g.f.: A(x) = x + 2*x^2/2! + 7*x^3/3! + 44*x^4/4! + 421*x^5/5! +...
where A( log(sqrt(1-x^2) + x) ) = x and
log(sqrt(1-x^2) + x) = x - 2*x^2/2! + 5*x^3/3! - 24*x^4/4! + 129*x^5/5! - 960*x^6/6! +...+ -(-1)^n*A194349(n)*x^n/n! +...

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A194349.

Partial sums of A352117.

a:= n-> n!*coeff(series(RootOf(A=exp(x)-sqrt(1-A^2), A), x, n+1), x, n):
seq(a(n), n=1..20);  # Alois P. Heinz, Sep 21 2013

Rest[CoefficientList[Series[(E^x-Sqrt[2-E^(2*x)])/2, {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 22 2013 *)

{a(n)=n!*polcoeff(serreverse( log(sqrt(1-x^2 +O(x^(n+2)))+x)),n)}

{a(n)=n!*polcoeff((exp(x+x*O(x^n))-sqrt(2-exp(2*x+x*O(x^n))))/2,n)}

E.g.f.: A(x) = (exp(x) - sqrt(2 - exp(2*x))) / 2.
E.g.f. A(x) satisfies: A( log(sqrt(1-x^2) + x) ) = x; thus, e.g.f. A(x) is a signed series reversion of the e.g.f. of A194349.
E.g.f. A(x) satisfies: A(x) = sinh(x) + exp(-x)*A(x)^2. - Paul D. Hanna, Aug 29 2018
a(n) ~ (2*n)^(n-1) / (exp(n) * log(2)^(n-1/2)). - Vaclav Kotesovec, Sep 22 2013

A367423 Expansion of e.g.f. 1 / sqrt(1 + log(1 - 2*x)).

Original entry on oeis.org

1, 1, 5, 41, 465, 6729, 118437, 2455809, 58630401, 1584058161, 47783202213, 1591924168185, 58055219617425, 2300356943749305, 98409722434170885, 4520749198158270225, 221954573405993807745, 11598560660172502840545, 642753897983638032821445

Offset: 0

Seiichi Manyama, Nov 18 2023

nonn

Cf. A097397, A352117.

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

a(n) = Sum_{k=0..n} 2^(n-k) * (Product_{j=0..k-1} (2*j+1)) * |Stirling1(n,k)|.
a(0) = 1; a(n) = Sum_{k=1..n} 2^k * (1 - 1/2 * k/n) * (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ 2^(n + 1/2) * n^n / (exp(1) - 1)^(n + 1/2). - Vaclav Kotesovec, Jun 09 2025

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

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

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A352118 Expansion of e.g.f. 1/(2 - exp(3*x))^(1/3).

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A352119 Expansion of e.g.f. 1/(2 - exp(4*x))^(1/4).

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A194453 E.g.f. satisfies: A(x) = exp(x) - sqrt(1 - A(x)^2).

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