A244585 -id:A244585 - cp's OEIS frontend

A317855 Decimal expansion of a constant related to the asymptotics of A122400.

3, 1, 6, 1, 0, 8, 8, 6, 5, 3, 8, 6, 5, 4, 2, 8, 8, 1, 3, 8, 3, 0, 1, 7, 2, 2, 0, 2, 5, 8, 8, 1, 3, 2, 4, 9, 1, 7, 2, 6, 3, 8, 2, 7, 7, 4, 1, 8, 8, 5, 5, 6, 3, 4, 1, 6, 2, 7, 2, 7, 8, 2, 0, 7, 5, 3, 7, 6, 9, 7, 0, 5, 9, 2, 1, 9, 3, 0, 4, 6, 1, 1, 2, 1, 9, 7, 5, 7, 4, 6, 8, 5, 4, 9, 7, 8, 4, 5, 9, 3, 2, 4, 2, 2, 7

Offset: 1

Vaclav Kotesovec, Aug 09 2018

			3.161088653865428813830172202588132491726382774188556341627278...

Cf. A121886, A122399, A122400, A122418, A122419, A122420, A227619, A232192, A243802, A244585, A248798, A317340, A326010.

Cf. A301584, A301585, A301586, A303056, A303057, A303654.

r = r /. FindRoot[E^(1/r)/r + (1 + E^(1/r)) * ProductLog[-E^(-1/r)/r] == 0, {r, 3/4}, WorkingPrecision -> 120]; RealDigits[(1 + Exp[1/r])*r^2][[1]]

r=solve(r=.8,1,exp(1/r)/r + (1+exp(1/r))*lambertw(-exp(-1/r)/r))
(1+exp(1/r))*r^2 \\ Charles R Greathouse IV, Jun 15 2021

Equals (1+exp(1/r))*r^2, where r = 0.873702433239668330496568304720719298213992... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0.

A242228 a(n) = Sum_{k=1..n} k^(2n-1) k! * Stirling2(n,k).

Original entry on oeis.org

1, 17, 1651, 473741, 300257371, 355743405917, 706872713310331, 2182548723605418941, 9894910566488309801851, 63052832687428562206049117, 545439670961897317869306191611, 6226501736967631584015448186252541, 91619831483112536750163352484302283131

Offset: 1

Vaclav Kotesovec, May 08 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..160

Cf. A000629, A220179, A229260, A244585.

Table[Sum[k^(2*n-1) * k! * StirlingS2[n,k], {k,1,n}], {n,1,20}]

a(n) ~ c * d^n * (n!)^3 / n^2, where d = r^3*(1+exp(2/r)) = 7.8512435106631367719817991716164612615296980032514..., r = 0.94520217245242431308104743874492469552738... is the root of the equation (1+exp(-2/r))*LambertW(-exp(-1/r)/r) = -1/r, and c = 0.15095210978787998524366903417512193343948127919...

E.g.f.: Sum_{k>=1} (exp(k^2*x) - 1)^k / k. - Seiichi Manyama, Jun 19 2024

A243802 E.g.f.: exp( Sum_{n>=1} (exp(n*x) - 1)^n / n ).

Original entry on oeis.org

1, 1, 6, 95, 3043, 167342, 14175447, 1715544861, 280986929888, 59828264507385, 16056622678756319, 5300955907062294008, 2110872493413444115109, 997542435957462115205773, 551887323312314977683048334, 353334615697796170374209624907, 259179558930246734075836153918127

Offset: 0

Paul D. Hanna, Aug 21 2014

nonn

Compare to: exp( Sum_{n>=1} (exp(x) - 1)^n/n ) = 1/(2-exp(x)), the e.g.f. of Fubini numbers (A000670).

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 95*x^3/3! + 3043*x^4/4! + 167342*x^5/5! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A244585, A244437.

{a(n) = n!*polcoeff( exp( sum(m=1,n+1, (exp(m*x +x*O(x^n)) - 1)^m / m) ), n)}
for(n=0,20,print1(a(n),", "))

a(n) ~ c * d^n * (n!)^2 / n^(3/2), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491..., r = 0.873702433239668330496568304720719298... is the root of the equation exp(1/r)/r + (1+exp(1/r)) * LambertW(-exp(-1/r)/r) = 0, and c = 0.37498840921734807101035131780130551... . - Vaclav Kotesovec, Aug 21 2014

A373855 a(n) = Sum_{k=1..n} k! * k^(n-1) * |Stirling1(n,k)|.

Original entry on oeis.org

0, 1, 5, 80, 2690, 155074, 13658386, 1706098008, 286888266696, 62485391828448, 17112247116585744, 5755236604915060944, 2331975856351260982848, 1120439648590390138640304, 629855675998212293917375344, 409557081242059531918330384896

Offset: 0

Seiichi Manyama, Jun 19 2024

nonn

Cf. A003713, A373856.
Cf. A092552, A244585, A373857.
Cf. A320096.

nmax=15; Range[0,nmax]!CoefficientList[Series[Sum[(-Log[1 - k*x])^k / k,{k,nmax}],{x,0,nmax}],x] (* Stefano Spezia, Jun 19 2024 *)

a(n) = sum(k=1, n, k!*k^(n-1)*abs(stirling(n, k, 1)));

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

A373871 a(n) = Sum_{k=1..n} k! * k^(n-3) * Stirling2(n,k).

Original entry on oeis.org

0, 1, 2, 13, 233, 8311, 495437, 44495263, 5619239453, 949995402271, 207228784973597, 56681221280785663, 19000392210559326173, 7661410911700580500831, 3658694812581483750630557, 2042247041839449013948374463, 1317554928647608644852032652893

Offset: 0

Seiichi Manyama, Jun 20 2024

nonn

Cf. A373869, A373870, A373872.

Cf. A122399, A244585, A373873.

a(n) = sum(k=1, n, k!*k^(n-3)*stirling(n, k, 2));

E.g.f.: Sum_{k>=1} (exp(k*x) - 1)^k / k^3.

A373857 a(n) = Sum_{k=1..n} k! * k^(n-1) * Stirling1(n,k).

Original entry on oeis.org

0, 1, 3, 32, 734, 28994, 1752046, 150262104, 17356844088, 2597710341600, 488957612319984, 113044488306692304, 31490845086661001664, 10403092187976909854640, 4021236906890850070201488, 1798052050351216209712206336, 920859156623446912386646303104

Offset: 0

Seiichi Manyama, Jun 19 2024

nonn

Cf. A092552, A244585, A373855.

Cf. A320082, A320083.

nmax=16; Range[0,nmax]!CoefficientList[Series[Sum[(Log[1 + k*x])^k / k,{k,nmax}],{x,0,nmax}],x] (* Stefano Spezia, Jun 19 2024 *)

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

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

A373873 a(n) = Sum_{k=1..n} k! * k^(n-2) * Stirling2(n,k).

Original entry on oeis.org

0, 1, 3, 31, 765, 34651, 2502213, 263824891, 38248036725, 7298877611371, 1773652375115973, 534749297993098651, 195883403209280580885, 85687658454617655817291, 44120264185381411695106533, 26413555571018242181844978811

Offset: 0

Seiichi Manyama, Jun 20 2024

nonn

Cf. A122399, A244585, A373871.

Cf. A220181, A373874, A373875.

Table[Sum[k! k^(n-2) StirlingS2[n,k],{k,n}],{n,0,20}] (* Harvey P. Dale, Jul 13 2025 *)

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

E.g.f.: Sum_{k>=1} (exp(k*x) - 1)^k / k^2.

cp's OEIS Frontend

A317855 Decimal expansion of a constant related to the asymptotics of A122400.

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A242228 a(n) = Sum_{k=1..n} k^(2*n-1) * k! * Stirling2(n,k).

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A243802 E.g.f.: exp( Sum_{n>=1} (exp(n*x) - 1)^n / n ).

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PARI

Formula

A373855 a(n) = Sum_{k=1..n} k! * k^(n-1) * |Stirling1(n,k)|.

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Formula

A373871 a(n) = Sum_{k=1..n} k! * k^(n-3) * Stirling2(n,k).

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PARI

Formula

A373857 a(n) = Sum_{k=1..n} k! * k^(n-1) * Stirling1(n,k).

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Programs

Mathematica

PARI

Formula

A373873 a(n) = Sum_{k=1..n} k! * k^(n-2) * Stirling2(n,k).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A242228 a(n) = Sum_{k=1..n} k^(2n-1) k! * Stirling2(n,k).