A072034 -id:A072034 - cp's OEIS frontend

A336214 a(n) = Sum_{k=0..n} k^n * binomial(n,k)^n, with a(0)=1.

1, 1, 8, 270, 41984, 30706250, 94770093312, 1336016204844832, 76829717664330940416, 19838680914222199482800274, 20521247958509575370600000000000, 94285013320530947020636486516362047300, 1715947732437668013396578734960052732361179136

Offset: 0

Vaclav Kotesovec, Jul 12 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..56

Cf. A072034, A086331, A167008, A167010, A328812, A336188, A336204, A336212, A336213.

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

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

a(n) ~ c * exp(-1/4) * 2^(n^2 - n/2) * n^(n/2) / Pi^(n/2), where c = Sum_{k = -infinity..infinity} exp(-2*k*(k-1)) = exp(1/2) * sqrt(Pi/2) * EllipticTheta(3, -Pi/2, exp(-Pi^2/2)) = 2.036643566277677716389243890291939003151565... if n is even and c = Sum_{k = -infinity..infinity} exp(-2*k^2 + 1/2) = exp(1/2) * EllipticTheta(3, 0, exp(-2)) = 2.096087809957308346119920713317351288828811... if n is odd.

a(n) = n^n * A328812(n-1) for n > 0. - Seiichi Manyama, Jul 15 2020

A336828 a(n) = Sum_{k=0..n} binomial(n,k)^2 * k^n.

Original entry on oeis.org

1, 1, 8, 108, 2144, 56250, 1836792, 71799504, 3269445888, 169974711630, 9934458411800, 644825382429096, 46022332032100800, 3582265183110626740, 302002255041807372080, 27413749834141448520000, 2665789990569658618398720, 276477318687585566522176470

Offset: 0

Ilya Gutkovskiy, Aug 05 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..343

Cf. A000984, A002457, A037966, A037972, A072034, A074334, A187021, A329444, A329913, A336214, A341815.

Join[{1}, Table[Sum[Binomial[n, k]^2 k^n, {k, 0, n}], {n, 1, 17}]]

a(n) = sum(k=0, n, binomial(n, k)^2*k^n); \\ Michel Marcus, Aug 05 2020

a(n) ~ c * d^n * (n-1)!, where d = (1 + 2*LambertW(exp(-1/2)/2)) / (4*LambertW(exp(-1/2)/2)^2) = 6.476217542109791521947605963458797355564... and c = 0.21617818094152997942246965143216887599763501682724844713834495... - Vaclav Kotesovec, Feb 20 2021

A345876 a(n) = Sum_{k=0..n} binomial(2n, n-k) k^n.

Original entry on oeis.org

1, 1, 8, 90, 1408, 28350, 697344, 20264244, 679313408, 25805186550, 1095482736640, 51397070440716, 2640925289349120, 147491783753286700, 8895880971425939456, 576279075821454657000, 39905347440408027725824, 2941534126495441574472870, 229966392623413457628168192

Offset: 0

Vaclav Kotesovec, Oct 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..360

Cf. A072034, A298851.

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

a(n) = sum(k=0, n, binomial(2*n, n-k) * k^n); \\ Michel Marcus, Oct 03 2021

a(n) ~ 2^(2*n + 1/2) * r^(n+1) * n^n / (sqrt(1 + r^2) * exp(n) * (1 - r^2)^n), where r = 0.647918229029602749602061258113970414114660380467168496836586... is the positive root of the equation (1 + r) = (1 - r)*exp(1/r).

A362702 Expansion of e.g.f. 1/(1 + LambertW(-x^2 * exp(x))).

Original entry on oeis.org

1, 0, 2, 6, 60, 500, 6150, 81522, 1300376, 23024808, 459915210, 10104914270, 243652575012, 6378414900156, 180405368976014, 5478759958122570, 177868544365861680, 6146407749811022672, 225262698504062963346, 8727083181657584963766

Offset: 0

Seiichi Manyama, Apr 30 2023

Seiichi Manyama, Table of n, a(n) for n = 0..401
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A072034, A362703.

Cf. A216507, A362347.

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

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

A074932 Row sums of unsigned triangle A075513.

Original entry on oeis.org

1, 3, 18, 170, 2200, 36232, 725200, 17095248, 463936896, 14246942336, 488428297984, 18491942300416, 766293946203136, 34498781924766720, 1676731077272217600, 87501958444207351808, 4880017252828686155776

Offset: 1

Wolfdieter Lang, Oct 02 2002

			E.g.f.: A(x) = x + 3*x^2/2! + 18*x^3/3! + 170*x^4/4! + 2200*x^5/5! +...
where exp(A(x)) = 1 + x + 4*x^2/2! + 28*x^3/3! + 288*x^4/4! + 3936*x^5/5! + 67328*x^6/6! +...+ A201595(n)*x^n/n! +...

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Victor J. W. Guo, Yiting Yang, Proof of a conjecture of Kløve on permutation codes under the Chebychev distance, arXiv:1704.01295 [cs.IT], 2017. Also in Designs, Codes and Cryptography, June 2017, Volume 83, Issue 3, pp 685-690.
Torleiv Kløve, Spheres of Permutations under the Infinity Norm - Permutations with limited displacement, Reports in Informatics, Department of Informatics, University of Bergen, Norway, no. 376, November 2008.

Cf. A201595.

Rest[CoefficientList[Series[Log[x-LambertW[-x*Exp[x]]]-Log[2*x], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Dec 04 2012 *)
a[n_] := Sum[Binomial[n-1, k]*(k+1)^(n-1), {k, 0, n-1}]; Table[a[n], {n, 1, 17}] (* Jean-François Alcover, Jul 09 2013, after Paul D. Hanna *)

{a(n)=sum(k=0,n-1,binomial(n-1,k)*(k+1)^(n-1))} \\ Paul D. Hanna, Aug 02 2012

{a(n)=local(A201595=serreverse(x-x*tanh(x+x^2*O(x^n)))/x);n!*polcoeff(log(A201595), n)} \\ Paul D. Hanna, Aug 02 2012

{a(n) = my(A); if( n<0, 0, A = O(x); for(k=1, n, A = log( (1 + exp( 2*x * exp(A))) / 2 )); n! * polcoeff(A, n))}; /* Michael Somos, Apr 10 2018 */

a(n) = sum(|A075513(n, m)|, m=0..n-1) = sum(binomial(n-1, m)*(m+1)^(n-1), m=0..n-1), n>=1.
E.g.f.: log(G(x)) where G(x) = (1 + exp(2*x*G(x)))/2 is the e.g.f. of A201595. - Paul D. Hanna, Aug 02 2012
E.g.f: log(x-LambertW(-x*exp(x)))-log(2*x). - Vaclav Kotesovec, Dec 04 2012
a(n) ~ n!/(sqrt(2*Pi*(1+LambertW(exp(-1))))*n^(3/2)*LambertW(exp(-1))^n). - Vaclav Kotesovec, Dec 04 2012
a(n) = A072034(n)/n. - Vladimir Reshetnikov, Nov 09 2016
O.g.f.: Sum_{k>=1} k^(k-1)*x^k/(1 - k*x)^k. - Ilya Gutkovskiy, Oct 09 2018

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.

A343928 a(n) = Sum_{k=0..n} (k!)^n * binomial(n,k).

Original entry on oeis.org

1, 2, 7, 244, 337061, 24923091206, 139331988275478727, 82607113404338664216300296, 6984967577834038055008791270166057993, 109110690950275218023122492287310115968068596613130, 395940866518366059877297056617763923418318903997411043997258716171

Offset: 0

Seiichi Manyama, May 04 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..30

Cf. A000522, A046662, A072034, A336247, A339311, A343898, A343899, A343929.

a[n_] := Sum[(k!)^n * Binomial[n, k], {k, 0, n} ]; Array[a, 11, 0] (* Amiram Eldar, May 04 2021 *)

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

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

a(n) = n! * [x^n] exp(x) * Sum_{k>=0} (k!)^(n-1) * x^k.

A355470 Expansion of Sum_{k>=0} (k^3 * x)^k/(1 - k^3 * x)^(k+1).

Original entry on oeis.org

1, 1, 66, 21222, 18927560, 36030104000, 125486684755152, 722272396672485568, 6391048590559497227904, 82362961035803105954736768, 1482370265813455598541301007360, 36031982428595760278113744699088384, 1150873035676373345725887922070318410752

Offset: 0

Seiichi Manyama, Jul 03 2022

nonn

Cf. A355469, A355473.

Cf. A072034, A242446, A355466.

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

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

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

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

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

A215080 T(n,k) = Sum_{j=0..k} (k-j)^n * binomial(n,j).

Original entry on oeis.org

1, 0, 1, 0, 1, 6, 0, 1, 11, 54, 0, 1, 20, 151, 680, 0, 1, 37, 413, 2569, 11000, 0, 1, 70, 1128, 9450, 52431, 217392, 0, 1, 135, 3104, 34416, 243255, 1251921, 5076400, 0, 1, 264, 8637, 125248, 1113027, 7025016, 34282879, 136761984, 0, 1, 521, 24327, 457807, 5064143, 38811015, 225930121, 1059812993, 4175432064, 0, 1, 1034, 69334, 1685266, 23031680, 212609518, 1465077802, 8026643702, 36519075583, 142469423360

Offset: 0

Olivier Gérard, Aug 02 2012

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   6;
  0, 1,  11,   54;
  0, 1,  20,  151,    680;
  0, 1,  37,  413,   2569,   11000;
  0, 1,  70, 1128,   9450,   52431,  217392;
  0, 1, 135, 3104,  34416,  243255, 1251921,  5076400;
  0, 1, 264, 8637, 125248, 1113027, 7025016, 34282879, 136761984;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Row sums give 215077 (binomial convolution of descending powers).

Main diagonal gives A072034.

Flatten[Table[Table[Sum[(k - j)^n*Binomial[n, j], {j, 0, k}], {k, 0, n}], {n, 0, 10}], 1]

T(n,k) = sum( (k-j)^n * binomial(n,j), j=0..k).

A341815 a(n) = Sum_{k=0..n} binomial(n,k)^3 * k^n.

Original entry on oeis.org

1, 1, 12, 270, 8960, 406250, 23293872, 1617774592, 132075970560, 12397121784954, 1315528361642000, 155743010418063860, 20351866171905066240, 2909818652684404979440, 451849287590990124662400, 75730203998219999637000000, 13625593688459657260608782336, 2619521322904712777031960349850

Offset: 0

Vaclav Kotesovec, Feb 20 2021

nonn

For m>0, Sum_{k=0..n} binomial(n,k)^m * k^n ~ c(m) * d^n * n! / n^(m/2), where d = (1 + m*LambertW(exp(-1/m)/m))^(m-1) / (m^m * LambertW(exp(-1/m)/m)^m) and c(m) is a constant independent of n.

Cf. A072034, A336828.

Join[{1}, Table[Sum[k^n * Binomial[n, k]^3, {k, 0, n}], {n, 1, 20}]]

a(n) ~ c * d^n * n! / n^(3/2), where d = (1 + 3*LambertW(exp(-1/3)/3))^2 / (27 * LambertW(exp(-1/3)/3)^3) = 12.3645613141726293982008517178673172577947617775... and c = 0.143687082995832067469009730530027989920523409582173778129054767279...

cp's OEIS Frontend

A336214 a(n) = Sum_{k=0..n} k^n * binomial(n,k)^n, with a(0)=1.

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A336828 a(n) = Sum_{k=0..n} binomial(n,k)^2 * k^n.

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A345876 a(n) = Sum_{k=0..n} binomial(2*n, n-k) * k^n.

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A362702 Expansion of e.g.f. 1/(1 + LambertW(-x^2 * exp(x))).

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A074932 Row sums of unsigned triangle A075513.

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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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A343928 a(n) = Sum_{k=0..n} (k!)^n * binomial(n,k).

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A355470 Expansion of Sum_{k>=0} (k^3 * x)^k/(1 - k^3 * x)^(k+1).

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A345876 a(n) = Sum_{k=0..n} binomial(2n, n-k) k^n.