A072034 -id:A072034 - cp's OEIS frontend

A355466 Expansion of Sum_{k>=0} (k^k * x)^k/(1 - k^k * x)^(k+1).

1, 2, 19, 19879, 4297094601, 298028721578591321, 10314430386430205371442173873, 256923580889667562995278943476559835493321, 6277101737079381674883855772624745947410338680458857322625

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Cf. A072034, A242446, A355470.

Cf. A349886.

my(N=10, x='x+O('x^N)); Vec(sum(k=0, N, (k^k*x)^k/(1-k^k*x)^(k+1)))

my(N=10, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, exp(k^k*x)*(k^k*x)^k/k!)))

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

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

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

A360816 Expansion of Sum_{k>=0} (kx)^(2k) / (1 - k*x)^(k+1).

Original entry on oeis.org

1, 0, 1, 2, 19, 100, 1118, 10034, 134993, 1715140, 27589661, 449763360, 8522965956, 168431719308, 3698624353289, 85523954588806, 2142927489388319, 56618555339223572, 1596938935380604858, 47399670488829289678, 1487559109670284821841

Offset: 0

Seiichi Manyama, Feb 21 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..418

Cf. A072034, A360817.

Cf. A360814.

Join[{1}, Table[Sum[k^n * Binomial[n-k,k], {k,0,n/2}], {n,1,20}]] (* Vaclav Kotesovec, Aug 04 2025 *)

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^(2*k)/(1-k*x)^(k+1)))

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

a(n) = Sum_{k=0..floor(n/2)} k^n * binomial(n-k,k).

a(n) ~ (1-r)^(1 + n*(1-r)) * r^(1/2 + n*(1-r)) * n^n / (sqrt(1 - 2*r + 2*r^2) * (1-2*r)^(n*(1-2*r))), where r = 0.42401262950134202779147542659633991972637211375... is the root of the equation log(r*(1-r)) - 2*log(1-2*r) = 1/r. - Vaclav Kotesovec, Aug 04 2025

A360817 Expansion of Sum_{k>=0} (kx)^(3k) / (1 - k*x)^(k+1).

Original entry on oeis.org

1, 0, 0, 1, 2, 3, 68, 389, 1542, 24810, 251564, 1814487, 27520734, 391640548, 4295115396, 69305652406, 1221344986380, 18207710383335, 329699350020676, 6759819628538561, 126950556666301050, 2624697847966227077, 60825028694289947940, 1365568620213461601924

Offset: 0

Seiichi Manyama, Feb 21 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A072034, A360816.

Cf. A360815.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^(3*k)/(1-k*x)^(k+1)))

a(n) = sum(k=0, n\3, k^n*binomial(n-2*k, k));

a(n) = Sum_{k=0..floor(n/3)} k^n * binomial(n-2*k,k).

A360834 Expansion of Sum_{k>=0} (k * x)^k / (1 - (k * x)^2)^(k+1).

Original entry on oeis.org

1, 1, 4, 29, 304, 4100, 67520, 1314167, 29520128, 751658635, 21393444864, 673046604600, 23192501108736, 868730852002205, 35145114836811776, 1527192185786650417, 70941146068492943360, 3508043437942077557884, 183989995827118805352448

Offset: 0

Seiichi Manyama, Feb 22 2023

nonn

Cf. A072034, A360835.

Cf. A000045, A360782, A360787.

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-(k*x)^2)^(k+1)))

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

a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^n * binomial(n-k,k).

A360835 Expansion of Sum_{k>=0} (k * x)^k / (1 - (k * x)^3)^(k+1).

Original entry on oeis.org

1, 1, 4, 27, 258, 3221, 49572, 905466, 19122502, 458161191, 12275530636, 363646493044, 11801356347294, 416365459777150, 15867258718677348, 649548679156603983, 28426564854590132236, 1324406974148881529057, 65448443631801436742052

Offset: 0

Seiichi Manyama, Feb 22 2023

nonn

Cf. A072034, A360834.

Cf. A000930, A360783, A360788.

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-(k*x)^3)^(k+1)))

a(n) = sum(k=0, n\3, (n-3*k)^n*binomial(n-2*k, k));

a(n) = Sum_{k=0..floor(n/3)} (n-3*k)^n * binomial(n-2*k,k).

A362699 Expansion of e.g.f. 1/(1 + LambertW(-x * exp(x^3))).

Original entry on oeis.org

1, 1, 4, 27, 280, 3605, 56376, 1041103, 22188496, 535856553, 14460919120, 431287416131, 14087063106216, 500112706900573, 19174548699128200, 789598137339356535, 34757031591555021856, 1628640121039415039057, 80938770039259919191584

Offset: 0

Seiichi Manyama, Apr 30 2023

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

Cf. A072034, A362604.

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

a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k)^(n-2*k) / (k! * (n-3*k)!).

A362700 Expansion of e.g.f. 1/(1 + LambertW(-x * exp(x^2/2))).

Original entry on oeis.org

1, 1, 4, 30, 304, 3950, 62736, 1177288, 25489024, 625404060, 17149867840, 519779815016, 17253698140416, 622521514670536, 24257636227003648, 1015256151136695840, 45421939756667293696, 2163253599528596482832, 109270502354027312243712

Offset: 0

Seiichi Manyama, Apr 30 2023

nonn

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

Cf. A072034, A362701.

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

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)^(n-k) / (2^k * k! * (n-2*k)!).

A362703 Expansion of e.g.f. 1/(1 + LambertW(-x^3 * exp(x))).

Original entry on oeis.org

1, 0, 0, 6, 24, 60, 1560, 20370, 161616, 2601144, 53827920, 829605150, 14894289960, 360575394036, 8234733389064, 188800085076330, 5145737430116640, 148419618327231600, 4278452209330445856, 134018446273097264694, 4529883358179857555640

Offset: 0

Seiichi Manyama, Apr 30 2023

Seiichi Manyama, Table of n, a(n) for n = 0..414
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A072034, A362702.

Cf. A292889, A362348, A362699.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-x^3*exp(x)))))

a(n) = n! * Sum_{k=0..floor(n/3)} k^(n-2*k) / (k! * (n-3*k)!).

A362705 Expansion of e.g.f. 1/(1 + LambertW(-x^3/6 * exp(x))).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 60, 595, 4536, 34524, 361320, 4333725, 51214460, 651628406, 9448719644, 146868322055, 2376666773040, 41077757951000, 762599081332176, 14918668387075449, 305774990501285940, 6602482711971622210, 149921553418087172260, 3557552268845721893131

Offset: 0

Seiichi Manyama, Apr 30 2023

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

Cf. A072034, A362704.

Cf. A362351.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(-x^3/6*exp(x)))))

a(n) = n! * Sum_{k=0..floor(n/3)} k^(n-2*k) / (6^k * k! * (n-3*k)!).

A216858 Number of connected functions from {1,2,...,n} into a subset of {1,2,...,n} summed over all subsets.

Original entry on oeis.org

0, 1, 5, 38, 422, 6184, 112632, 2453296, 62202800, 1799623296, 58507176320, 2111633645824, 83777729991936, 3624054557443072, 169759643117603840, 8560585769442662400, 462387289560368764928, 26633435981686107701248, 1629609677806398679646208, 105555926477075661655441408, 7215930505311133152120995840

Offset: 0

Geoffrey Critzer, Sep 17 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 0..370

Cf. A216857, A000248, A048802, A072034.

nn=20; a=-ProductLog[-x Exp[x] ]; Range[0,nn]! CoefficientList[Series[Log[1/(1-a)], {x,0,nn}], x]

x='x+O('x^30); concat([0], Vec(serlaplace(log(1/(1+ lambertw(-x*exp(x))))))) \\ G. C. Greubel, Nov 16 2017

E.g.f.: log(1/(1-T(x*exp(x)))) where T(x) is the e.g.f. for A000169.

a(n) ~ n!/(2*n*LambertW(exp(-1))^n) * (1 - sqrt(2*(1 + LambertW(exp(-1))) /(Pi*n))/3). - Vaclav Kotesovec, Sep 24 2013

cp's OEIS Frontend

A355466 Expansion of Sum_{k>=0} (k^k * x)^k/(1 - k^k * x)^(k+1).

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A360816 Expansion of Sum_{k>=0} (k*x)^(2*k) / (1 - k*x)^(k+1).

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A360817 Expansion of Sum_{k>=0} (k*x)^(3*k) / (1 - k*x)^(k+1).

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A360834 Expansion of Sum_{k>=0} (k * x)^k / (1 - (k * x)^2)^(k+1).

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A360835 Expansion of Sum_{k>=0} (k * x)^k / (1 - (k * x)^3)^(k+1).

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A362699 Expansion of e.g.f. 1/(1 + LambertW(-x * exp(x^3))).

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PARI

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A362700 Expansion of e.g.f. 1/(1 + LambertW(-x * exp(x^2/2))).

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PARI

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A362703 Expansion of e.g.f. 1/(1 + LambertW(-x^3 * exp(x))).

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A362705 Expansion of e.g.f. 1/(1 + LambertW(-x^3/6 * exp(x))).

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A360816 Expansion of Sum_{k>=0} (kx)^(2k) / (1 - k*x)^(k+1).

A360817 Expansion of Sum_{k>=0} (kx)^(3k) / (1 - k*x)^(k+1).