A086331 -id:A086331 - cp's OEIS frontend

A277470 E.g.f.: arcsinh(x)/(1+LambertW(-x)).

0, 1, 2, 11, 104, 1249, 18264, 318163, 6425152, 147344769, 3781848480, 107408279483, 3343875651456, 113227469886881, 4142804357946240, 162871544915116035, 6847004160475236352, 306495323034774157569, 14554502490109085839872, 730777840212988501198059

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

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

Cf. A000312, A086331, A277469.

CoefficientList[Series[ArcSinh[x]/(1+LambertW[-x]), {x, 0, 25}], x] * Range[0, 25]!
Flatten[{0, Table[Sin[Pi*n/2] * (n-2)!!^2 + Sum[Sin[Pi*k/2] * Binomial[n, k] * (k-2)!!^2 * (n-k)^(n-k), {k, 1, n-1}], {n, 1, 25}]}] (* Vaclav Kotesovec, Oct 28 2016 *)

x='x+O('x^50); concat([0], Vec(serlaplace(asinh(x)/(1 + lambertw(-x)) ))) \\ G. C. Greubel, Nov 07 2017

a(n) ~ arcsinh(exp(-1)) * n^n.

a(n) ~ (-1 + log(1 + sqrt(1+exp(2)))) * n^n.

A277485 E.g.f.: -exp(2x)LambertW(-x).

Original entry on oeis.org

0, 1, 6, 33, 216, 1865, 21228, 303765, 5222864, 104540337, 2383558740, 60933722069, 1725392415288, 53590463856345, 1811281159509500, 66172416761172885, 2598298697830360992, 109116931783034360801, 4880122696811960470692, 231565260558289051906965

Offset: 0

Vaclav Kotesovec, Oct 17 2016

nonn

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

Cf. A000169, A069856, A086331, A277457, A277473, A277474.

CoefficientList[Series[-Exp[2*x]*LambertW[-x], {x, 0, 20}], x]*Range[0, 20]!
Table[Sum[Binomial[n, m]*Sum[Binomial[m, k]*k^(k-1), {k, 1, m}], {m, 1, n}], {n, 0, 20}]

x='x+O('x^50); concat([0], Vec(serlaplace(- exp(2*x)*lambertw(-x) ))) \\ G. C. Greubel, Nov 08 2017

a(n) = Sum_{m=1..n} (binomial(n,m) * Sum_{k=1..m} binomial(m,k)*k^(k-1)).

a(n) ~ exp(2*exp(-1)) * n^(n-1).

A290824 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(1 + LambertW(-x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 7, 27, 1, 4, 12, 43, 256, 1, 5, 19, 71, 393, 3125, 1, 6, 28, 117, 616, 4721, 46656, 1, 7, 39, 187, 985, 7197, 69853, 823543, 1, 8, 52, 287, 1584, 11123, 105052, 1225757, 16777216, 1, 9, 67, 423, 2521, 17429, 159093, 1829291, 24866481, 387420489, 1, 10, 84, 601, 3928, 27525, 243256, 2740111, 36922928, 572410513, 10000000000

Offset: 0

Ilya Gutkovskiy, Aug 11 2017

A(n,k) is the k-th binomial transform of A000312 evaluated at n.

			E.g.f. of column k: A_k(x) = 1 + (k + 1)*x/1! + (k^2 + 2*k + 4)*x^2/2! + (k^3 + 3*k^2 + 12*k + 27)*x^3/3! + (k^4 + 4*k^3 + 24*k^2 + 108*k + 256)*x^4/4! + ...
Square array begins:
     1,     1,     1,     1,     1,     1, ...
     1,     2,     3,     4,     5,     6, ...
     4,     7,    12,    19,    28,    39, ...
    27,    43,    71,   117,   187,   287, ...
   256,   393,   616,   985,  1584,  2521, ...
  3125,  4721,  7197, 11123, 17429, 27525, ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
N. J. A. Sloane, Transforms

Columns k=0..2 give A000312, A086331, A277457.
Main diagonal gives A290840.
Cf. A080955, A081576, A271025.

Table[Function[k, n!*SeriesCoefficient[Exp[k x]/(1 + LambertW[-x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten (* G. C. Greubel, Nov 09 2017 *)

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

A(n,k) = Sum_{j=0..n} binomial(n,j)*k^(n-j)*j^j. - Fabian Pereyra, Jul 16 2024

A336213 a(n) = Sum_{k=0..n} k^k * binomial(n,k)^n, with a(0)=1.

Original entry on oeis.org

1, 2, 9, 163, 12609, 3906251, 4835455813, 23882051929709, 470073929716006913, 36867039626275056203923, 11562789460238169439667262501, 14393917436542502296957220221339601, 72060131612303615870363237649174605005057, 1424448870088911493303605765206905153730451241313

Offset: 0

Vaclav Kotesovec, Jul 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..58
Vaclav Kotesovec, Plot of a(n)/f(n) for n = 1..10000000

Cf. A072034, A086331, A167008, A167010, A336188, A336204, A336212, A336214.

Table[1 + Sum[k^k * Binomial[n, k]^n, {k, 1, n}], {n, 0, 15}]

a(n) = if (n==0, 1, sum(k=0, n, k^k * binomial(n,k)^n)); \\ Michel Marcus, Jul 13 2020

Let f(n) = exp(-1/4) * QPochhammer(exp(-4)) * 2^(n^2 - 1/4) * exp((3*log(n)^2 + 3*log(2)^2 + Pi^2 - 1)/24) * n^((1 - log(2))/4) / Pi^(n/2). For sufficiently large n 0.985... < a(n)/f(n) < 1.015...

a(n) ~ exp(-1/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-1)/2, exp(-4)) * 2^(n^2) / Pi^(n/2) if n is even and a(n) ~ exp(-1/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-3)/2, exp(-4)) * sqrt(n) * 2^(n^2 - 1/2) / Pi^(n/2) if n is odd.

A355464 Expansion of Sum_{k>=0} x^k/(1 - k^k * x)^(k+1).

Original entry on oeis.org

1, 2, 4, 17, 210, 9217, 1399298, 811229225, 2071392232962, 20710319937493889, 1137259214532706572162, 255141201504146525745627265, 348787971214016591166179037803522, 2262996819897931095524655885144485185409

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Cf. A086331, A320287, A349893, A355440, A355463.

Cf. A000248, A135746, A355473.

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

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

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

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

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

A355493 Expansion of Sum_{k>=0} (k^3 * x)^k/(1 - x)^(k+1).

Original entry on oeis.org

1, 2, 67, 19879, 16856337, 30601661681, 101743314190033, 559257425236996361, 4726837695171258085569, 58192258417571877186113281, 1000581709943568968705788233921, 23236157618902718144948494353385025, 709080642850925779233576351761544968833

Offset: 0

Seiichi Manyama, Jul 04 2022

nonn

Winston de Greef, Table of n, a(n) for n = 0..152

Cf. A086331, A323280.

Cf. A355470, A355473, A355496.

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

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

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

E.g.f.: exp(x) * Sum_{k>=0} (k^3 * x)^k/k!.

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

A135749 a(n) = Sum_{k=0..n} binomial(n,k)(n-k)^kk^k.

Original entry on oeis.org

1, 1, 3, 19, 217, 3821, 95761, 3214975, 137501505, 7226764921, 455941716481, 33983083953611, 2954163633223969, 296027886705639973, 33823026186790043841, 4363561123325076879991, 630392564294402819207041

Offset: 0

Paul D. Hanna, Nov 27 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000248, A086331, A349965.

Table[Sum[Binomial[n,k](n-k)^k k^k,{k,n}],{n,0,20}]+1 (* Harvey P. Dale, Oct 08 2012 *)

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

a(n)=n!*polcoeff(sum(k=0,n,exp((n-k)*k*x +x*O(x^n))*x^k/k!),n)

a(n) = n!*[x^n] Sum_{k=0..n} exp((n-k)*x)^k * x^k/k!.

A277453 a(n) = Sum_{k=0..n} binomial(n,k) * 2^k * n^k * k!.

Original entry on oeis.org

1, 3, 41, 1531, 111393, 13262051, 2336744233, 570621092091, 184341785557121, 76092709735150723, 39064090158380196201, 24408768326642565035963, 18237590837527919131499041, 16056004231253610384348995811, 16448689708899063469247204152553

Offset: 0

Vaclav Kotesovec, Oct 16 2016

nonn

Cf. A086331, A277373, A277423, A277452.

Flatten[{1, Table[Sum[Binomial[n, k]*2^k*n^k*k!, {k, 0, n}], {n, 1, 20}]}]

a(n) = exp(1/(2*n)) * 2^n * n^n * Gamma(n+1, 1/(2*n)).

a(n) ~ 2^n * n^n * n!.

A308863 Expansion of e.g.f. (1 + LambertW(-x))/(1 + 2*LambertW(-x)).

Original entry on oeis.org

1, 1, 6, 57, 736, 11985, 235296, 5403937, 142073856, 4206560769, 138483596800, 5017244970441, 198363105460224, 8498001799768273, 392127481640165376, 19388814120804416625, 1022681739669784231936, 57317273018414456262273, 3401527253966521309200384

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Cf. A000312, A002866, A086331, A218688, A277610, A308861, A308862.

nmax = 18; CoefficientList[Series[(1 + LambertW[-x])/(1 + 2 LambertW[-x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

E.g.f.: 1 / (1 - Sum_{k>=1} k^k*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k^k * a(n-k).
a(n) ~ sqrt(Pi) * 2^(n - 3/2) * n^(n + 1/2) / exp(n/2). - Vaclav Kotesovec, Jun 29 2019

A110140 Binomial transform of n^n (with interpolated zeros).

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 103, 351, 1321, 5113, 21201, 90465, 406621, 1879021, 9051309, 44754061, 229059633, 1201271409, 6488957521, 35853950609, 203303964341, 1177299817093, 6976709899853, 42161309544541, 260154190137865

Offset: 0

Paul Barry, Jul 13 2005

Binomial transform of 1,0,1,0,4,0,27,....

Cf. A000312, A086331, A294353.

Table[1 + Sum[Binomial[n, k]*(k/2)^(k/2)*(1 + (-1)^k)/2, {k, 1, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 30 2017 *)

a(n)=sum{k=0..n, C(n, k)(k/2)^(k/2)*(1+(-1)^k)/2}

cp's OEIS Frontend

A277470 E.g.f.: arcsinh(x)/(1+LambertW(-x)).

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A277485 E.g.f.: -exp(2*x)*LambertW(-x).

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A290824 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(k*x)/(1 + LambertW(-x)).

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A336213 a(n) = Sum_{k=0..n} k^k * binomial(n,k)^n, with a(0)=1.

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A355464 Expansion of Sum_{k>=0} x^k/(1 - k^k * x)^(k+1).

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A355493 Expansion of Sum_{k>=0} (k^3 * x)^k/(1 - x)^(k+1).

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A135749 a(n) = Sum_{k=0..n} binomial(n,k)*(n-k)^k*k^k.

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A277453 a(n) = Sum_{k=0..n} binomial(n,k) * 2^k * n^k * k!.

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A277485 E.g.f.: -exp(2x)LambertW(-x).

A135749 a(n) = Sum_{k=0..n} binomial(n,k)(n-k)^kk^k.