A282190 -id:A282190 - cp's OEIS frontend

A305819 Expansion of e.g.f. 1/(1 + LambertW(-log(1 + x))).

1, 1, 3, 17, 132, 1334, 16442, 239994, 4041776, 77183328, 1647541632, 38877352392, 1004869488048, 28234217634024, 856830099396840, 27930093941047464, 973269467390922240, 36104568839480990400, 1420556927968241858880, 59088101641333114906944, 2590680379402887359111424

Offset: 0

Ilya Gutkovskiy, Aug 18 2018

nonn

Inverse Stirling transform of A000312.

Seiichi Manyama, Table of n, a(n) for n = 0..397
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A000312, A052807, A277489, A282190, A305981.

a:=series(1/(1+LambertW(-log(1+x))),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019

nmax = 20; CoefficientList[Series[1/(1 + LambertW[-Log[1 + x]]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] k^k, {k, n}], {n, 20}]]

a(n) = sum(k=0, n, k^k*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 05 2022

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-log(1+x))))) \\ Seiichi Manyama, Feb 05 2022

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

a(n) ~ n^n / ((exp(exp(-1)) - 1)^(n + 1/2) * exp(n + (1 - exp(-1))/2)). - Vaclav Kotesovec, Aug 18 2018

A308490 a(0) = 1, a(n) = Sum_{k=1..n} stirling2(n,k) * k^(2*k).

Original entry on oeis.org

1, 1, 17, 778, 70023, 10439451, 2327592658, 725325847443, 301054612941037, 160546901676583432, 106969402879501806589, 87079496403914056543799, 85043317211453886535179728, 98135961356804028347727824541, 132097548629285541942722646521053

Offset: 0

Vaclav Kotesovec, May 31 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..214

Cf. A229261, A282190, A308491, A316747, A323280, A351182.

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^2*(exp(x)-1))^k/k!))) \\ Seiichi Manyama, Feb 04 2022

a(n) ~ exp(exp(-2)/2) * n^(2*n).

E.g.f.: Sum_{k>=0} (k^2 * (exp(x) - 1))^k / k!. - Seiichi Manyama, Feb 04 2022

A305981 Expansion of e.g.f. 1/(1 + LambertW(log(1 - x))).

Original entry on oeis.org

1, 1, 5, 41, 468, 6854, 122582, 2589978, 63129392, 1743732192, 53827681152, 1836453542472, 68620052332752, 2786929842106344, 122241516227220504, 5758920745460806824, 290017142065771138560, 15547326972257789803200, 883974436758296523437760, 53131928820278417749940544, 3366145488853852112016117504

Offset: 0

Ilya Gutkovskiy, Aug 18 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..376
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A000312, A052807, A277489, A282190, A305819.

a:=series(1/(1+LambertW(log(1-x))),x=0,21): seq(n!*coeff(a,x,n),n=0..20); # Paolo P. Lava, Mar 26 2019

nmax = 20; CoefficientList[Series[1/(1 + LambertW[Log[1 - x]]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[Abs[StirlingS1[n, k]] k^k, {k, n}], {n, 20}]]

a(n) = sum(k=0, n, (-1)^(n-k)*k^k*stirling(n, k, 1)); \\ Seiichi Manyama, Feb 05 2022

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(log(1-x))))) \\ Seiichi Manyama, Feb 05 2022

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

a(n) ~ n^n / ((exp(exp(-1)) - 1)^(n + 1/2) * exp(n*(1 - exp(-1)) + 1/2)). - Vaclav Kotesovec, Aug 18 2018

A308491 a(0) = 1, a(n) = Sum_{k=1..n} stirling2(n,k) * k^(3*k).

Original entry on oeis.org

1, 1, 65, 19876, 16895763, 30685843321, 102018812632786, 560682901512212459, 4738032814084465062121, 58320000513552476843995786, 1002620283226568243192938115197, 23280221638971518379191182864465213, 710336441472841166799952152725333251616

Offset: 0

Vaclav Kotesovec, May 31 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..152

Cf. A282190, A308490, A316748.

Join[{1}, Table[Sum[k^(3*k)*StirlingS2[n, k], {k, 1, n}], {n, 1, 15}]]

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k^3*(exp(x)-1))^k/k!))) \\ Seiichi Manyama, Feb 04 2022

a(n) ~ n^(3*n).

E.g.f.: Sum_{k>=0} (k^3 * (exp(x) - 1))^k / k!. - Seiichi Manyama, Feb 04 2022

A351218 a(n) = Sum_{k=0..n} (-k)^k * Stirling2(n,k).

Original entry on oeis.org

1, -1, 3, -16, 121, -1181, 14114, -199543, 3257139, -60279214, 1247164055, -28525394481, 714681439212, -19465007759913, 572609747089735, -18093710202583480, 611202186074834221, -21979340746682042249, 838330656532184312218, -33803668628843391999843

Offset: 0

Seiichi Manyama, Feb 05 2022

sign

Alois P. Heinz, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A282190, A305981.

b:= proc(n, m) option remember; `if`(n=0,
      (-m)^m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 17 2022

Table[Sum[(-1)^k * k^k * StirlingS2[n,k], {k,1,n}], {n,0,20}] (* Vaclav Kotesovec, Feb 05 2022 *)

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

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(exp(x)-1))))

E.g.f.: 1/(1 + LambertW(exp(x) - 1)), where LambertW() is the Lambert W-function.

a(n) ~ (-1)^n * n^n / (sqrt(exp(1)-1) * (1 - log(exp(1)-1))^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Feb 05 2022

A351276 a(n) = Sum_{k=0..n} (2k)^k Stirling2(n,k).

Original entry on oeis.org

1, 2, 18, 266, 5506, 146602, 4772162, 183618794, 8152995138, 410307648938, 23079780216386, 1434953808618090, 97716253164212034, 7233006174407149866, 578233606405444793410, 49651123488091636885994, 4557474786380802233761090, 445324385454834015896585386

Offset: 0

Seiichi Manyama, Feb 05 2022

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A282190, A308490, A351274, A351277.

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

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

E.g.f.: 1/(1 + LambertW( 2 * (1 - exp(x)) )), where LambertW() is the Lambert W-function.

a(n) ~ n^n / (sqrt(1 + 2*exp(1)) * (log(exp(1) + 1/2) - 1)^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Feb 06 2022

A316145 a(n) = Sum_{k=0..n} Stirling2(n,k) * A000041(k) * k^k.

Original entry on oeis.org

1, 9, 106, 1823, 36821, 932080, 26666067, 876727561, 32137538059, 1305168046976, 57774609056649, 2783202675369037, 144453227105110782, 8035192765567735275, 476686201707606976317, 30053582893540865299197, 2005019178999976881804130, 141111387620531900621281975

Offset: 1

Vaclav Kotesovec, Jun 25 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..370

Cf. A167137, A282190, A306022, A316146.

Table[Sum[StirlingS2[n, k] * PartitionsP[k] * k^k, {k, 1, n}], {n, 1, 20}]

Limit_{n -> infinity} (a(n)/n!)^(1/n) = 1/(log(1+ exp(1)) - 1) = 3.1922192845297391106277924019427161296056687330974482534324... - Vaclav Kotesovec, Nov 21 2021

log(a(n) / A316146(n)) ~ (sqrt(2) - 1) * Pi * sqrt(n) / sqrt(3*(1 + exp(1)) * log(1 + exp(-1))). - Vaclav Kotesovec, Nov 22 2021

A316146 a(n) = Sum_{k=0..n} Stirling2(n,k) * A000009(k) * k^k.

Original entry on oeis.org

1, 5, 67, 865, 15906, 365514, 9545026, 276368635, 9188742238, 343857717788, 13998751394662, 618098575755637, 29469995998980356, 1510585321262760900, 83100039017148288635, 4873627957977247842223, 302388593396139280682588, 19804146883678522219587314

Offset: 1

Vaclav Kotesovec, Jun 25 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..370

Cf. A282190, A305550, A306023, A316145.

Table[Sum[StirlingS2[n, k] * PartitionsQ[k] * k^k, {k, 1, n}], {n, 1, 20}]

Limit_{n -> infinity} (a(n)/n!)^(1/n) = 1/(log(1+ exp(1)) - 1) = 3.1922192845297391106277924019427161296056687330974482534324... - Vaclav Kotesovec, Nov 21 2021

log(A316145(n) / a(n)) ~ (sqrt(2) - 1) * Pi * sqrt(n) / sqrt(3*(1 + exp(1)) * log(1 + exp(-1))). - Vaclav Kotesovec, Nov 22 2021

A351181 a(n) = Sum_{k=0..n} k^(k+n) * Stirling2(n,k).

Original entry on oeis.org

1, 1, 17, 826, 79107, 12553011, 2979141058, 988163147091, 436562014218313, 247800100563125728, 175732698005376526429, 152264214647249387402567, 158273183995563848011907696, 194391589002961482387840145341

Offset: 0

Seiichi Manyama, Feb 04 2022

nonn

Cf. A108459, A229233, A229261, A282190, A308490.

Cf. A350722, A351180.

a[0] = 1; a[n_] := Sum[k^(k + n) * StirlingS2[n, k], {k, 1, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 04 2022 *)

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*(exp(k*x)-1))^k/k!)))

E.g.f.: Sum_{k>=0} (k * (exp(k*x) - 1))^k / k!.

a(n) ~ c * r^(2*n) * (1 + exp(1 + 1/r))^n * n^(2*n) / exp(2*n), where r = 0.942405403803582963024019065398882138211529545249588032669864757847... is the root of the equation r*(1 + exp(-1 - 1/r)) * LambertW(-exp(-1/r)/r) = -1 and c = 0.94346979328254581112250921799629823027437848684764713214690470878402... - Vaclav Kotesovec, Feb 18 2022

A351281 a(n) = Sum_{k=0..n} k! * k^k * Stirling2(n,k).

Original entry on oeis.org

1, 1, 9, 187, 7173, 440611, 39631509, 4910795107, 802015652853, 166948755155971, 43146953460348309, 13555255072473665827, 5087595330217093070133, 2248298922174973220446531, 1155512971750307157457879509, 683392198848998191062416885347

Offset: 0

Seiichi Manyama, Feb 06 2022

nonn

Cf. A000670, A122399, A229234, A282190, A350722, A351280.

a[0] = 1; a[n_] := Sum[k! * k^k * StirlingS2[n, k], {k, 1, n}]; Array[a, 16, 0] (* Amiram Eldar, Feb 06 2022 *)

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*(exp(x)-1))^k)))

E.g.f.: Sum_{k>=0} (k * (exp(x) - 1))^k.

a(n) ~ exp(exp(-1)/2) * n! * n^n. - Vaclav Kotesovec, Feb 06 2022

cp's OEIS Frontend

A305819 Expansion of e.g.f. 1/(1 + LambertW(-log(1 + x))).

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A308490 a(0) = 1, a(n) = Sum_{k=1..n} stirling2(n,k) * k^(2*k).

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A305981 Expansion of e.g.f. 1/(1 + LambertW(log(1 - x))).

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A308491 a(0) = 1, a(n) = Sum_{k=1..n} stirling2(n,k) * k^(3*k).

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A351218 a(n) = Sum_{k=0..n} (-k)^k * Stirling2(n,k).

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A351276 a(n) = Sum_{k=0..n} (2*k)^k * Stirling2(n,k).

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A316145 a(n) = Sum_{k=0..n} Stirling2(n,k) * A000041(k) * k^k.

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A316146 a(n) = Sum_{k=0..n} Stirling2(n,k) * A000009(k) * k^k.

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A351276 a(n) = Sum_{k=0..n} (2k)^k Stirling2(n,k).