A319508 -id:A319508 - cp's OEIS frontend

A319509 a(n) = n! * [x^n] 1/(1 - n + exp(x)(exp(nx) - 1)/(exp(x) - 1)).

1, -1, 13, -828, 145046, -53306325, 35351663831, -38335940184976, 63385171527442332, -151639317344211911505, 503956292395339783686325, -2252032996384696958326480356, 13175456854397460097168816336930, -98695402553214372025148083384255381

Offset: 0

Ilya Gutkovskiy, Sep 21 2018

sign

G. C. Greubel, Table of n, a(n) for n = 0..167

Cf. A319508.

Table[n! SeriesCoefficient[1/(1 - n + Exp[x] (Exp[n x] - 1)/(Exp[x] - 1)), {x, 0, n}], {n, 0, 13}]

default(seriesprecision, 101); {a(n) = n!*polcoeff((1/(1-n+exp(x)*(exp(n*x)-1)/(exp(x)-1)) + O(x^(n+1))), n)};
for(n=0, 25, print1(a(n), ", ")) \\ G. C. Greubel, Oct 09 2018

a(n) = n! * [x^n] 1/(1 - n + exp(x) + exp(2*x) + exp(3*x) + ... + exp(n*x)).

A320253 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k - Sum_{j=1..k} exp(j*x)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 6, 23, 13, 0, 1, 10, 86, 261, 75, 0, 1, 15, 230, 1836, 3947, 541, 0, 1, 21, 505, 7900, 52250, 74613, 4683, 0, 1, 28, 973, 25425, 361754, 1858716, 1692563, 47293, 0, 1, 36, 1708, 67473, 1706629, 20706700, 79345346, 44794221, 545835, 0

Offset: 0

Ilya Gutkovskiy, Oct 08 2018

			E.g.f. of column k: A_k(x) = 1 + (1/2)*k*(k + 1)*x/1! + (1/6)*k*(3*k^3 + 8*k^2 + 6*k + 1)*x^2/2! + (1/4)*k^2*(k + 1)^2*(3*k^2 + 7*k + 3)*x^3/3! + (1/30)*k*(45*k^7 + 270*k^6 + 635*k^5 + 741*k^4 + 440*k^3 + 115*k^2 + 5*k - 1)*x^4/4! + ...
Square array begins:
  1,    1,      1,        1,         1,          1,  ...
  0,    1,      3,        6,        10,         15,  ...
  0,    3,     23,       86,       230,        505,  ...
  0,   13,    261,     1836,      7900,      25425,  ...
  0,   75,   3947,    52250,    361754,    1706629,  ...
  0,  541,  74613,  1858716,  20706700,  143195025,  ...

Columns k=0..10 give A000007, A000670, A004700, A004701, A004702, A004703, A004704, A004705, A004706, A004707, A004708.

Main diagonal gives A319508.

Table[Function[k, n! SeriesCoefficient[1/(1 + k - Sum[Exp[i x], {i, 1, k}]), {x, 0, n}]][j - n], {j, 0, 9}, {n, 0, j}] // Flatten
Table[Function[k, n! SeriesCoefficient[1/(1 + k - Exp[x] (Exp[k x] - 1)/(Exp[x] - 1)), {x, 0, n}]][j - n], {j, 0, 9}, {n, 0, j}] // Flatten

E.g.f. of column k: 1/(1 + k - exp(x)*(exp(k*x) - 1)/(exp(x) - 1)).

A320288 a(n) = n! * [x^n] exp(exp(x)(exp(nx) - 1)/(exp(x) - 1) - n).

Original entry on oeis.org

1, 1, 14, 504, 35054, 4004100, 680823583, 161337142848, 50830272555828, 20549783554154775, 10370522690234157175, 6390016526512315766520, 4721172172018812127424546, 4119920939845363203406535407, 4192465334819134111336349480680, 4920767556196547768620408273728000

Offset: 0

Ilya Gutkovskiy, Oct 09 2018

nonn

G. C. Greubel, Table of n, a(n) for n = 0..218

Cf. A103438, A319508.

Table[n! SeriesCoefficient[Exp[Exp[x] (Exp[n x] - 1)/(Exp[x] - 1) - n], {x, 0, n}], {n, 0, 15}]

a(n)={my(A=O(x^(n+2))); n!*polcoef((exp(exp(x + A)*(exp(n*x + A) - 1)/(exp(x + A) - 1) - n)), n)}; \\ Andrew Howroyd, Nov 04 2018

a(n) = n! * [x^n] exp(exp(x) + exp(2*x) + exp(3*x) + ... + exp(n*x) - n).

a(n) ~ c * exp(n*exp(1) - 3*n) * n^(2*n), where c = exp((exp(1) - 1)/2) / sqrt(exp(1) - 1) = 1.801245710492990660565773944914841332489711300610532... - Vaclav Kotesovec, Jul 02 2022, updated Mar 18 2024

A331340 a(n) = n! * [x^n] 1 / (1 + Sum_{k=1..n} log(1 - k*x)).

Original entry on oeis.org

1, 1, 23, 1872, 371524, 147316050, 102823452318, 115685840003328, 196669439127051840, 480847207762313690400, 1626231663646322798946000, 7372321556702072183715972096, 43653032698484678876818157764224, 330351436922959495109028135649934640

Offset: 0

Ilya Gutkovskiy, Jan 14 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..167

Cf. A007840, A319508, A319509, A331341.

Table[n! SeriesCoefficient[1/(1 + Sum[Log[1 - k x], {k, 1, n}]), {x, 0, n}], {n, 0, 13}]
Table[n! SeriesCoefficient[1/(1 + Log[Sum[StirlingS1[n + 1, n - k + 1] x^k, {k, 0, n}]]), {x, 0, n}], {n, 0, 13}]

a(n) = n! * [x^n] 1 / (1 + log(Sum_{k=0..n} Stirling1(n+1,n-k+1) * x^k)).

a(n) ~ sqrt(Pi) * n^(3*n + 1/2) / (2^(n - 1/2) * exp(n - 5/3)). - Vaclav Kotesovec, Jan 28 2020

A331341 a(n) = n! * [x^n] 1 / (1 - Sum_{k=1..n} log(1 + k*x)).

Original entry on oeis.org

1, 1, 13, 864, 151276, 55463850, 36662614458, 39635566403328, 65354864056231104, 155978053040893370400, 517297066212058929642000, 2307448887344816064221408256, 13478142770116878179295616074624, 100820731073923375628659569173854704

Offset: 0

Ilya Gutkovskiy, Jan 14 2020

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..167

Cf. A006252, A319508, A319509, A331340.

Table[n! SeriesCoefficient[1/(1 - Sum[Log[1 + k x], {k, 1, n}]), {x, 0, n}], {n, 0, 13}]
Table[n! SeriesCoefficient[1/(1 - Log[Sum[Abs[StirlingS1[n + 1, n - k + 1]] x^k, {k, 0, n}]]), {x, 0, n}], {n, 0, 13}]

a(n) = n! * [x^n] 1 / (1 - log(Sum_{k=0..n} |Stirling1(n+1,n-k+1)| * x^k)).

a(n) ~ sqrt(Pi) * n^(3*n + 1/2) / (2^(n - 1/2) * exp(n - 1/3)). - Vaclav Kotesovec, Jan 28 2020

A355428 a(n) = n! * [x^n] 1/(1 - Sum_{k=1..n} (exp(k*x) - 1)/k).

Original entry on oeis.org

1, 1, 11, 284, 13564, 1037479, 116171621, 17916010524, 3640962169776, 942959405612913, 303168464105203113, 118474395231479349050, 55306932183983923942940, 30397993745996492901617435, 19429788681469866219869997285

Offset: 0

Seiichi Manyama, Jul 01 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..226

Main diagonal of A355427.

Cf. A319508.

Table[n! * SeriesCoefficient[1/(1 + HarmonicNumber[n] + E^((n + 1)*x) * LerchPhi[E^x, 1, n + 1] + Log[1 - E^x]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 02 2022 *)

a(n) = n!*polcoef(1/(1-sum(k=1, n, (exp(k*x+x*O(x^n))-1)/k)), n);

a(n) ~ c * d^n * n^(2*n + 1/2), where d = 0.4573611067742364103005235654624761643997061199669064548746966610712579358... and c = 2.41592773370058066984975000807924527905758896927935098069320182397... - Vaclav Kotesovec, Jul 02 2022

A331582 a(n) = n! * [x^n] exp(n - exp(x) * (exp(n*x) - 1) / (exp(x) - 1)).

Original entry on oeis.org

1, -1, 4, 0, -1654, 102750, -4079389, -178722208, 83191059372, -14561829897345, 1115121827539325, 403631463559529040, -251989999508801085674, 76158421344845152140737, -3994730250899559184766830, -13162858116922635098226480000, 10823217968258750568539067678392

Offset: 0

Ilya Gutkovskiy, Jan 21 2020

sign

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

Cf. A103438, A319508, A319509, A320288.

Table[n! SeriesCoefficient[Exp[n - Exp[x] (Exp[n x] - 1)/(Exp[x] - 1)], {x, 0, n}], {n, 0, 16}]
b[n_, k_] := b[n, k] = If[n == 0, 1, -Sum[Binomial[n - 1, j - 1] Sum[i^j, {i, 1, k}] b[n - j, k], {j, 1, n}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 16}]

a(n) = n! * [x^n] exp(n - exp(x) - exp(2*x) - exp(3*x) - ... - exp(n*x)).

cp's OEIS Frontend

A319509 a(n) = n! * [x^n] 1/(1 - n + exp(x)*(exp(n*x) - 1)/(exp(x) - 1)).

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A320253 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k - Sum_{j=1..k} exp(j*x)).

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A320288 a(n) = n! * [x^n] exp(exp(x)*(exp(n*x) - 1)/(exp(x) - 1) - n).

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A331340 a(n) = n! * [x^n] 1 / (1 + Sum_{k=1..n} log(1 - k*x)).

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A331341 a(n) = n! * [x^n] 1 / (1 - Sum_{k=1..n} log(1 + k*x)).

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A355428 a(n) = n! * [x^n] 1/(1 - Sum_{k=1..n} (exp(k*x) - 1)/k).

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Mathematica

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A331582 a(n) = n! * [x^n] exp(n - exp(x) * (exp(n*x) - 1) / (exp(x) - 1)).

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Mathematica

Formula

A319509 a(n) = n! * [x^n] 1/(1 - n + exp(x)(exp(nx) - 1)/(exp(x) - 1)).

A320288 a(n) = n! * [x^n] exp(exp(x)(exp(nx) - 1)/(exp(x) - 1) - n).