A006252 -id:A006252 - cp's OEIS frontend

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

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

A097397 Coefficients in asymptotic expansion of normal probability function.

Original entry on oeis.org

1, 1, 1, 5, 9, 129, 57, 9141, -36879, 1430049, -15439407, 418019205, -7404957255, 196896257505, -4656470025015, 134136890777205, -3845524501226655, 123250625100419265, -4085349586734306015, 145973136800663973765

Offset: 0

Michael Somos, Aug 13 2004

sign

a(0) + a(1)*x/(1-2*x) + a(2)*x^2/((1-2*x)*(1-4*x)) + ... = 1 + x + 3*x^2 + 15*x^3 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 932.

Vaclav Kotesovec, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A006252, A352070, A352073.

Table[Sum[2^(n - 2*k)*(2*k)!/k! * SeriesCoefficient[(1 - n + x)*Pochhammer[2 - n + x, -1 + n], {x, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 10 2019 *)

a(n)=sum(k=0,n, 2^(n-2*k)*(2*k)!/k!* polcoeff(prod(i=0,n-1,x-i),k))

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1-log(1+2*x)))) \\ Seiichi Manyama, Mar 05 2022

E.g.f.: 1/sqrt(1 - log(1 + 2*x)). - Seiichi Manyama, Mar 05 2022
a(n) ~ n! * (-1)^(n+1) * 2^(n-1) / (log(n)^(3/2) * n) * (1 - 3*(gamma + 1)/(2*log(n)) + 15*(1 + 2*gamma + gamma^2 - Pi^2/6) / (8*log(n)^2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 05 2022
From Seiichi Manyama, Nov 18 2023: (Start)
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 * (1/2 * k/n - 1) * (k-1)! * binomial(n,k) * a(n-k). (End)

A317172 a(n) = n! * [x^n] 1/(1 - n*log(1 + x)).

Original entry on oeis.org

1, 1, 6, 114, 4168, 248870, 21966768, 2685571560, 434202400896, 89679267601632, 23032451508686400, 7199033431349412576, 2690461258552995849216, 1184680716090974803461072, 606986901206377433194091520, 358023049940533240478842992000, 240858598980174362552808566194176

Offset: 0

Ilya Gutkovskiy, Jul 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A006252, A088501, A094420, A242817, A317171.

Main diagonal of A320080.

Table[n! SeriesCoefficient[1/(1 - n Log[1 + x]), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[StirlingS1[n, k] n^k k!, {k, n}], {n, 16}]]

{a(n) = sum(k=0, n, k!*n^k*stirling(n, k, 1))} \\ Seiichi Manyama, Jun 12 2020

a(n) = Sum_{k=0..n} Stirling1(n,k)*n^k*k!.

a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / exp(n + 1/2). - Vaclav Kotesovec, Jul 23 2018

A347022 Expansion of e.g.f. 1 / (1 - 5 * log(1 + x))^(1/5).

Original entry on oeis.org

1, 1, 5, 50, 720, 13650, 320370, 8967720, 291538080, 10795026840, 448484788680, 20658543923280, 1044915105622800, 57572197848878400, 3432143603792520000, 220109018869587398400, 15110184224165199667200, 1105545474191480800492800, 85881534014930659599571200

Offset: 0

Ilya Gutkovskiy, Aug 11 2021

nonn

Cf. A006252, A008548, A320343, A346984, A346987, A347020, A347021, A347023.

nmax = 18; CoefficientList[Series[1/(1 - 5 Log[1 + x])^(1/5), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] 5^k Pochhammer[1/5, k], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} Stirling1(n,k) * A008548(k).
a(n) ~ n! * exp(1/25) / (Gamma(1/5) * 5^(1/5) * n^(4/5) * (exp(1/5) - 1)^(n + 1/5)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (5 - 4*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

A088819 Expansion of e.g.f. (1+x)^(1/(1-log(1+x))).

Original entry on oeis.org

1, 1, 2, 6, 22, 100, 518, 3122, 20676, 154524, 1238952, 11030448, 103376832, 1068000024, 11407673496, 134352996744, 1603035004368, 21276244952784, 278535036773856, 4141886572833888, 58405909554175776, 973789956270781056, 14462380128843907680

Offset: 0

Vladeta Jovovic, Nov 22 2003

sign

a(34) is the first negative term.

Georg Fischer, Table of n, a(n) for n = 0..100

Row sums of A079639.

Cf. A000262, A006252.

CoefficientList[Series[(1+x)^(1/(1-Log[1+x])), {x, 0, 100}], x]* Range[0, 100]! (* Georg Fischer, Feb 17 2019 *)

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, k!*stirling(j, k, 1))*binomial(i-1, j-1)*v[i-j+1])); v; \\ Seiichi Manyama, May 23 2022

a(n) = Sum_{k=0..n} Stirling1(n, k)*A000262(k). - Vladeta Jovovic, Nov 26 2003

a(0) = 1; a(n) = Sum_{k=1..n} A006252(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, May 23 2022

A306042 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k).

Original entry on oeis.org

1, 1, 3, 8, 50, 94, 2446, -9024, 297216, -3183264, 64191984, -1041792192, 22098943632, -478805234064, 11856288460272, -308662348027008, 8575865689645440, -248582819381690880, 7556655091130023680, -240521346554744194560, 8049494171497089265920, -283469026458500121634560

Offset: 0

Ilya Gutkovskiy, Jun 17 2018

sign

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000041, A006252, A053529, A167137, A320349.

a:=series(mul(1/(1-log(1+x)^k),k=1..100),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # Paolo P. Lava, Mar 26 2019

nmax = 21; CoefficientList[Series[Product[1/(1 - Log[1 + x]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] Log[1 + x]^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] PartitionsP[k] k!, {k, 0, n}], {n, 0, 21}]

E.g.f.: exp(Sum_{k>=1} sigma(k)*log(1 + x)^k/k).

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000041(k)*k!.

A351133 a(n) = Sum_{k=0..n} k! * k^(2n) Stirling1(n,k).

Original entry on oeis.org

1, 1, 31, 3992, 1342294, 932514674, 1161340476698, 2356863300156504, 7278091701243797640, 32477694155566998880608, 201155980661221409458717152, 1674230688936725338278370413264, 18235249164492209082483584810706528

Offset: 0

Seiichi Manyama, Feb 02 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..162

Cf. A006252, A320083, A351134.

Cf. A229260, A351135, A351136.

a[0] = 1; a[n_] := Sum[k! * k^(2*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 13, 0] (* Amiram Eldar, Feb 02 2022 *)

a(n) = sum(k=0, n, k!*k^(2*n)*stirling(n, k, 1));

first(n)=my(x='x+O('x^(n+1))); Vec(serlaplace(sum(k=0, n, log(1+k^2*x)^k)))

E.g.f.: Sum_{k>=0} log(1 + k^2*x)^k.

a(n) ~ c * d^n * n^(3*n + 1/2), where d = 0.3417329834649268103028466896966197580428514873775849996969994420891... and c = 2.92355271092039591960355156784704285135358... - Vaclav Kotesovec, Feb 03 2022

A352070 Expansion of e.g.f. 1/(1 - log(1 + 3*x))^(1/3).

Original entry on oeis.org

1, 1, 1, 10, 10, 604, -1844, 107344, -1201400, 42193576, -875584376, 29853569008, -880141783184, 32865860907424, -1216481572723616, 51296026356128512, -2244334822166729600, 106984479644794783360, -5358207684820194270080, 286466413246622566048000

Offset: 0

Seiichi Manyama, Mar 05 2022

sign

Cf. A006252, A097397, A352073.

Cf. A007559, A133480, A352113, A352118.

m = 19; Range[0, m]! * CoefficientList[Series[(1 - Log[1 + 3*x])^(-1/3), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)
Table[Sum[3^(n-k) * Product[3*j+1, {j,0,k-1}] * StirlingS1[n,k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Sep 07 2023 *)

a[n]:=if n=0 then 1 else n!*sum(a[n-k]*(2/n/3-1/k)*(-3)^k/(n-k)!,k,1,n);
makelist(a[n],n,0,50); /* Tani Akinari, Sep 07 2023 */

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

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

a(n) = Sum_{k=0..n} 3^(n-k) * (Product_{j=0..k-1} (3*j+1)) * Stirling1(n,k).
For n > 0, a(n) = n!*Sum_{k=1..n} a(n-k)*(2/n/3-1/k)*(-3)^k/(n-k)!. - Tani Akinari, Sep 07 2023
a(n) ~ -(-1)^n * 3^(n-1) * n! / (n * log(n)^(4/3)) * (1 - 4*(1+gamma)/(3*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 07 2023

A352073 Expansion of e.g.f. 1/(1 - log(1 + 4*x))^(1/4).

Original entry on oeis.org

1, 1, 1, 17, 1, 1889, -12415, 631665, -11224575, 461864385, -13754112255, 596055636945, -24148300842495, 1181210529292065, -59009709972278655, 3297137505670374705, -193318225258785780735, 12263541239089421903745, -820804950905249837195775

Offset: 0

Seiichi Manyama, Mar 05 2022

sign

Cf. A006252, A097397, A352070.

Cf. A007696, A008545, A352119.

m = 18; Range[0, m]! * CoefficientList[Series[(1 - Log[1 + 4*x])^(-1/4), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)

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

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

a(n) = Sum_{k=0..n} 4^(n-k) * (Product_{j=0..k-1} (4*j+1)) * Stirling1(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-4)^k * (3/4 * k/n - 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 18 2023

A192554 a(n) = Sum_{k=0..n} abs(Stirling1(n,k))(-1)^(n-k)k!^2.

Original entry on oeis.org

1, 1, 3, 26, 398, 9724, 344236, 16663968, 1056631824, 84962783664, 8446120969104, 1016998946575776, 145848462866589600, 24562489788256472064, 4799789988678066147840, 1077128972416478325901824, 275111625956753684599202304

Offset: 0

Emanuele Munarini, Jul 04 2011

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..253

Cf. A006252, A320502, A351280.

Cf. A064618.

Table[Sum[Abs[StirlingS1[n,k]](-1)^(n-k)k!^2,{k,0,n}],{n,0,100}]
Table[Sum[StirlingS1[n,k] * k!^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 05 2021 *)

makelist(sum(abs(stirling1(n,k))*(-1)^(n-k)*k!^2,k,0,n),n,0,24);

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, k!*log(1+x)^k))) \\ Seiichi Manyama, Apr 22 2022

a(n) = Sum_{k=0..n} Stirling1(n,k) * k!^2. - Vaclav Kotesovec, Jul 05 2021
a(n) ~ exp(-1/2) * n!^2. - Vaclav Kotesovec, Jul 05 2021
E.g.f.: Sum_{k>=0} k! * log(1+x)^k. - Seiichi Manyama, Apr 22 2022

cp's OEIS Frontend

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

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A097397 Coefficients in asymptotic expansion of normal probability function.

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A317172 a(n) = n! * [x^n] 1/(1 - n*log(1 + x)).

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A347022 Expansion of e.g.f. 1 / (1 - 5 * log(1 + x))^(1/5).

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A088819 Expansion of e.g.f. (1+x)^(1/(1-log(1+x))).

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A306042 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k).

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A351133 a(n) = Sum_{k=0..n} k! * k^(2*n) * Stirling1(n,k).

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A352070 Expansion of e.g.f. 1/(1 - log(1 + 3*x))^(1/3).

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A351133 a(n) = Sum_{k=0..n} k! * k^(2n) Stirling1(n,k).

A192554 a(n) = Sum_{k=0..n} abs(Stirling1(n,k))(-1)^(n-k)k!^2.