A063170 -id:A063170 - cp's OEIS frontend

A332408 a(n) = Sum_{k=0..n} binomial(n,k) * k! * k^n.

1, 1, 10, 213, 8284, 513105, 46406286, 5772636373, 945492503320, 197253667623681, 51069324556151290, 16067283861476491941, 6037615013420387657844, 2670812587802323522405393, 1373842484756310928089102022, 813119045938378747809030359445

Offset: 0

Ilya Gutkovskiy, Apr 23 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..232

Cf. A000522, A006153, A031971, A063170, A072034, A256016, A277452, A332627.

Join[{1}, Table[Sum[Binomial[n, k] k! k^n, {k, 0, n}], {n, 1, 15}]]

a(n) = sum(k=0, n, binomial(n,k) * k! * k^n); \\ Michel Marcus, Apr 24 2020

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x*exp(x))^k))) \\ Seiichi Manyama, Feb 19 2022

G.f.: Sum_{k>=0} k! * k^k * x^k / (1 - k*x)^(k+1).
a(n) = n! * Sum_{k=0..n} k^n / (n-k)!.
a(n) ~ c * n! * n^n, where c = A073229 = exp(exp(-1)). - Vaclav Kotesovec, Feb 20 2021
E.g.f.: Sum_{k>=0} (k*x*exp(x))^k. - Seiichi Manyama, Feb 19 2022

A277510 E.g.f.: -1/(1-LambertW(-x))^2.

Original entry on oeis.org

-1, 2, -2, 6, 8, 170, 1872, 29246, 519808, 10698642, 248787200, 6458737142, 185138721792, 5808233422394, 197952647108608, 7283047491096750, 287705410381709312, 12145740050403520034, 545696709922799419392, 25998534614835587104742, 1309210567403228200960000

Offset: 0

Vaclav Kotesovec, Oct 18 2016

sign

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A063170, A134095, A277458, A277490, A308506.

CoefficientList[Series[-1/(1-LambertW[-x])^2, {x, 0, 20}], x] * Range[0, 20]!

x='x+O('x^50); Vec(serlaplace(-1/(1 - lambertw(-x))^2)) \\ G. C. Greubel, Nov 08 2017

a(n) ~ n^(n-1) / 4.

A277490 E.g.f.: -1/(1+LambertW(-x)^2).

Original entry on oeis.org

-1, 0, 2, 12, 72, 520, 5040, 67284, 1156736, 23655888, 549676800, 14216252380, 405068387328, 12624364306008, 427599019108352, 15646376279614500, 615155126821355520, 25861820048469628576, 1157706908035331457024, 54977324662490442177708, 2760439046217459138560000

Offset: 0

Vaclav Kotesovec, Oct 17 2016

sign

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A063170, A134095.

CoefficientList[Series[-1/(1+LambertW[-x]^2), {x, 0, 20}], x] * Range[0, 20]!

x='x+O('x^50); Vec(serlaplace(-1/(1 + lambertw(-x)^2))) \\ G. C. Greubel, Nov 08 2017

a(n) ~ n^(n-1) / 2.

A295098 a(n) = n! * [x^n] exp(nx)(1 + exp(x^2/2)x(1 + sqrt(Pi/2)*erf(x/sqrt(2)))).

Original entry on oeis.org

1, 2, 10, 75, 760, 9715, 150060, 2719017, 56556480, 1328337117, 34773226340, 1003998156293, 31696623421488, 1086258754644505, 40161805428662876, 1593475984997421525, 67534151717002711296, 3044989873158805787409, 145537456143562934305860, 7350253384336351186239341, 391132792671917087054081200

Offset: 0

Ilya Gutkovskiy, Nov 14 2017

nonn

The n-th term of the n-th binomial transform of A006882.

G. C. Greubel, Table of n, a(n) for n = 0..250
N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers

Cf. A006882, A063170, A263529, A295099, A295100.

Table[n! SeriesCoefficient[Exp[n x] (1 + Exp[x^2/2] x (1 + Sqrt[Pi/2] Erf[x/Sqrt[2]])), {x, 0, n}], {n, 0, 20}]

a(n) ~ c * n^n, where c = 1 + exp(1/2) * (1 + sqrt(Pi/2) * erf(1/sqrt(2))) = 4.0594074053425761445394754992332... - Vaclav Kotesovec, Aug 21 2018

A295099 a(n) = n! * [x^n] exp(nx)/sqrt(1 - 2x).

Original entry on oeis.org

1, 2, 11, 96, 1145, 17320, 317547, 6843872, 169603793, 4752704160, 148631984075, 5132717953792, 194022218612169, 7969667589513344, 353510496652374635, 16842274069331520000, 857827370723082312737, 46516913938434654949888, 2675772791433589181094027, 162742831545094476694814720

Offset: 0

Ilya Gutkovskiy, Nov 14 2017

nonn

The n-th term of the n-th binomial transform of A001147.

G. C. Greubel, Table of n, a(n) for n = 0..250
N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers

Cf. A001147, A063170, A081367, A084262, A295098, A295100.

Table[n! SeriesCoefficient[Exp[n x]/Sqrt[1 - 2 x], {x, 0, n}], {n, 0, 19}]

a(n) ~ 2^(n+1) * n^n / exp(n/2). - Vaclav Kotesovec, Nov 14 2017

A295100 a(n) = n! * [x^n] exp(nx)/(1 - 2x).

Original entry on oeis.org

1, 3, 20, 201, 2688, 44815, 894528, 20792205, 551518208, 16438822587, 543934387200, 19783668211153, 784536321392640, 33689132092480839, 1557397919735103488, 77117362592836807125, 4072280214605427376128, 228441851811771488284915, 13566762607790788699226112, 850372121882700252639269337

Offset: 0

Ilya Gutkovskiy, Nov 14 2017

nonn

The n-th term of the n-th binomial transform of A000165.

Robert Israel, Table of n, a(n) for n = 0..375
N. J. A. Sloane, Transforms
Index entries for sequences related to factorial numbers

Cf. A000165, A010844, A063170, A082032, A295098, A295099.

S:= series(exp(n*x)/(1-2*x),x,51):
seq(n!*coeff(S,x,n),n=0..50); # Robert Israel, Nov 14 2017

Table[n! SeriesCoefficient[Exp[n x]/(1 - 2 x), {x, 0, n}], {n, 0, 19}]

a(n) ~ 2^n * exp(n/2) * n!. - Vaclav Kotesovec, Nov 14 2017

a(n) = n! * Sum_{k=0..n} n^k*2^(n-k)/k! = 2^n*Gamma(n+1, n/2)*exp(n/2). - Robert Israel, Nov 14 2017

A331690 a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * n^(n - k).

Original entry on oeis.org

1, 1, 4, 33, 456, 9445, 272448, 10386817, 503758720, 30202999821, 2189000524800, 188349613075393, 18954958449853440, 2203304642871358741, 292675996808408743936, 44022321302156791898625, 7438113993194856900034560, 1401876939543892434209075581

Offset: 0

Ilya Gutkovskiy, Jan 24 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..271

Cf. A000670, A063170, A086914, A094420, A122704, A122778, A229234, A255927, A301419, A326323, A326324.

Join[{1}, Table[Sum[StirlingS2[n, k] k! n^(n - k), {k, 0, n}], {n, 1, 17}]]
Table[SeriesCoefficient[Sum[k! x^k/Product[(1 - n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 17}]
Join[{1}, Table[n^(n + 1) PolyLog[-n, 1/(n + 1)]/(n + 1), {n, 1, 17}]]

a(n) = sum(k=0, n, stirling(n, k, 2)*k!*n^(n-k)); \\ Michel Marcus, Jan 24 2020

a(n) = [x^n] Sum_{k>=0} k! * x^k / Product_{j=1..k} (1 - n*j*x).
a(n) = n! * [x^n] n / (1 + n - exp(n*x)) for n > 0.
a(n) = n^(n + 1) * Sum_{k>=1} k^n / (n + 1)^(k + 1) for n > 0.
a(n) ~ n! * n^(n+1) / ((n+1) * log(n+1)^(n+1)). - Vaclav Kotesovec, Jun 06 2022

A336949 a(n) = n! * [x^n] 1 / (exp(-n*x) - x).

Original entry on oeis.org

1, 2, 14, 195, 4440, 147745, 6698448, 394852577, 29250137472, 2652483234033, 288363456748800, 36952298766628465, 5504130616452258816, 941845623036360908489, 183298110723156455921664, 40221612394630225987208625, 9876429434585097671993032704

Offset: 0

Ilya Gutkovskiy, Aug 08 2020

nonn

Cf. A063170, A072597, A235328, A336947, A336948.

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

a(n)={n!*polcoef(1/(exp(-n*x + O(x*x^n)) - x), n)} \\ Andrew Howroyd, Aug 08 2020

a(n) = n! * Sum_{k=0..n} (n * (n-k+1))^k / k!.

A219546 Schenker primes.

Original entry on oeis.org

5, 13, 23, 31, 37, 41, 43, 47, 53, 59, 61, 71, 79, 101, 103, 107, 109, 127, 137, 149, 157, 163, 173, 179, 181, 191, 197, 199, 211, 223, 229, 241, 251, 257, 263, 271, 277, 283, 293, 311, 317, 337, 349, 353, 359, 367, 383, 397, 401, 409, 419, 421, 431, 439, 461

Offset: 1

Jonathan Sondow, Nov 22 2012

nonn

Amdeberhan, Callan, and Moll (2012) call a prime p a Schenker prime if p divides A063170(r) (the r-th Schenker sum with n-th term) for some r < p.

For any non-Schenker prime p, Amdeberhan, Callan, and Moll (2012) give a formula for the p-adic valuation of any Schenker sum with n-th term.

			5 is a Schenker prime because 2 < 5 and 5 divides A063170(2) = 10.
17 is not a Schenker prime because 17 is not a factor of A063170(1) = 2, or of A063170(2) = 10, . . . , or of A063170(16) = 105224992014096760832.

Vaclav Kotesovec, Table of n, a(n) for n = 1..783
T. Amdeberhan, D. Callan, and V. Moll, p-adic analysis and combinatorics of truncated exponential sums, preprint, 2012.
T. Amdeberhan, D. Callan and V. Moll, Valuations and combinatorics of truncated exponential sums, INTEGERS 13 (2013), #A21.

Cf. A063170.

pmax = 300; A063170 = Table[n!*Sum[n^k/k!, {k, 0, n}], {n, 1, pmax}]; Rest[Select[Table[If[PrimeQ[j] && SelectFirst[Range[j], Divisible[A063170[[#]], j] &] != j, j, 0], {j, 1, pmax}], # != 0 &]] (* Vaclav Kotesovec, Nov 30 2017 *)

More terms from Vaclav Kotesovec, Nov 30 2017

A308330 a(n) = n! * [x^n] exp(exp(n*x)/(1 - x) - 1).

Original entry on oeis.org

1, 2, 19, 346, 10217, 441226, 26023123, 1998840586, 193094418161, 22841006706928, 3239088790361491, 541309430523114804, 105106521730010262745, 23431755937256853296514, 5936989025261397848036755, 1694791457312643753292004446, 540937403928198054978670965089

Offset: 0

Ilya Gutkovskiy, May 20 2019

nonn

Cf. A063170, A292914, A308331, A321974.

Table[n! SeriesCoefficient[Exp[Exp[n x]/(1 - x) - 1], {x, 0, n}], {n, 0, 16}]

cp's OEIS Frontend

A332408 a(n) = Sum_{k=0..n} binomial(n,k) * k! * k^n.

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A277510 E.g.f.: -1/(1-LambertW(-x))^2.

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A277490 E.g.f.: -1/(1+LambertW(-x)^2).

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A295098 a(n) = n! * [x^n] exp(n*x)*(1 + exp(x^2/2)*x*(1 + sqrt(Pi/2)*erf(x/sqrt(2)))).

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A295099 a(n) = n! * [x^n] exp(n*x)/sqrt(1 - 2*x).

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A295100 a(n) = n! * [x^n] exp(n*x)/(1 - 2*x).

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A331690 a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * n^(n - k).

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A336949 a(n) = n! * [x^n] 1 / (exp(-n*x) - x).

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A295098 a(n) = n! * [x^n] exp(nx)(1 + exp(x^2/2)x(1 + sqrt(Pi/2)*erf(x/sqrt(2)))).

A295099 a(n) = n! * [x^n] exp(nx)/sqrt(1 - 2x).

A295100 a(n) = n! * [x^n] exp(nx)/(1 - 2x).