A216507 -id:A216507 - cp's OEIS frontend

A216688 Expansion of e.g.f. exp( x * exp(x^2) ).

1, 1, 1, 7, 25, 121, 841, 4831, 40657, 325585, 2913841, 29910871, 301088041, 3532945417, 41595396025, 531109561711, 7197739614241, 100211165640481, 1507837436365537, 23123578483200295, 376697477235716281, 6348741961892933401, 111057167658053740201, 2032230051717594032767

Offset: 0

Joerg Arndt, Sep 14 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Cf. A216507 (e.g.f. exp(x^2*exp(x))), A216689 (e.g.f. exp(x*exp(x)^2)).

Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).

With[{nn = 25}, CoefficientList[Series[Exp[x Exp[x^2]], {x, 0, nn}], x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *)

x='x+O('x^66);
Vec(serlaplace(exp( x * exp(x^2) )))
/* Joerg Arndt, Sep 14 2012 */

a(n) = n!*sum(k=0, n\2, (n-2*k)^k/(k!*(n-2*k)!)); \\ Seiichi Manyama, Aug 18 2022

a(n) = n!*Sum_{m=floor((n+1)/2)..n} (2*m-n)^(n-m)/((2*m-n)!*(n-m)!). - Vladimir Kruchinin, Mar 09 2013
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (r^n * exp((2*r^2*n)/(1+2*r^2)) * sqrt(3+2*r^2 - 2/(1 + 2*r^2))), where r is the root of the equation r*exp(r^2)*(1+2*r^2) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(2^(1/3)*n^(2/3)/3))) *  sqrt(2/(3*LambertW(2^(1/3)*n^(2/3)/3))).
(End)

A216689 Expansion of e.g.f. exp( x * exp(x)^2 ).

Original entry on oeis.org

1, 1, 5, 25, 153, 1121, 9373, 87417, 898033, 10052353, 121492341, 1573957529, 21729801481, 318121178337, 4917743697805, 79981695655801, 1364227940101857, 24335561350365953, 452874096174214117, 8772713803852981785, 176541611843378273401, 3684142819311127955041, 79596388271096140589949

Offset: 0

Joerg Arndt, Sep 14 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Cf. A216507 (e.g.f. exp(x^2*exp(x))), A216688 (e.g.f. exp(x*exp(x^2))).
Cf. A000248 (e.g.f. exp(x*exp(x))), A003725 (e.g.f. exp(x*exp(-x))).
Cf. A240165 (e.g.f. exp(x*(1+exp(x)^2))).

With[{nn = 25}, CoefficientList[Series[Exp[x Exp[x]^2], {x, 0, nn}], x] Range[0, nn]!] (* Bruno Berselli, Sep 14 2012 *)

x='x+O('x^66);
Vec(serlaplace(exp( x * exp(x)^2 )))
/* Joerg Arndt, Sep 14 2012 */

/* From o.g.f.: */
{a(n)=local(A=1);A=sum(k=0, n, x^k/(1 - 2*k*x +x*O(x^n))^(k+1));polcoeff(A, n)}
for(n=0,25,print1(a(n),", ")) /* Paul D. Hanna, Aug 02 2014 */

/* From binomial sum: */
{a(n)=sum(k=0,n, binomial(n,k)*(2*k)^(n-k))}
for(n=0,30,print1(a(n),", ")) /* Paul D. Hanna, Aug 02 2014 */

O.g.f.: Sum_{n>=0} x^n / (1 - 2*n*x)^(n+1). - Paul D. Hanna, Aug 02 2014
a(n) = Sum_{k=0..n} binomial(n,k) * (2*k)^(n-k) for n>=0. - Paul D. Hanna, Aug 02 2014
From Vaclav Kotesovec, Aug 06 2014: (Start)
a(n) ~ n^n / (exp(2*n*r/(1+2*r)) * r^n * sqrt((1+6*r+4*r^2)/(1+2*r))), where r is the root of the equation r*(1+2*r)*exp(2*r) = n.
(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n/2)))) / LambertW(sqrt(n/2)).
(End)

A240989 Expansion of e.g.f. exp(x^2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 390, 2562, 11816, 105912, 1063530, 8815070, 81342492, 895185876, 9971185406, 112642410090, 1372455608400, 17750397057392, 236950003516626, 3286258330135734, 47688868443593540, 719345273005797900, 11222288509573985382, 181168865439054099266

Offset: 0

Vaclav Kotesovec, Aug 06 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Column k=2 of A292892.
Cf. A000110, A052506, A216507, A060311.
Cf. A216688, A216689, A080108, A000248.

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

x='x+O('x^30); Vec(serlaplace(exp(x^2*(exp(x) - 1)))) \\ G. C. Greubel, Nov 21 2017

a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 2)/(n-2*k)!); \\ Seiichi Manyama, Jul 09 2022

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j/(j-2)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

a(n) ~ exp((n-r^3)/(2+r)-n) * n^(n+1/2) / (r^n * sqrt((2*r^3*(3+r) + n*(1+r)*(4+r))/(2+r))), where r is the root of the equation r^2*((2+r) * exp(r) - 2) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).
From Seiichi Manyama, Jul 09 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,k)/(n-2*k)!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!. (End)

A358080 Expansion of e.g.f. 1/(1 - x^2 * exp(x)).

Original entry on oeis.org

1, 0, 2, 6, 36, 260, 2190, 21882, 248696, 3181320, 45229050, 707208590, 12063902532, 222939837276, 4436813677478, 94605994108290, 2151763873634160, 51999544476324752, 1330540380342907506, 35936656483848501654, 1021700660649312689660

Offset: 0

Seiichi Manyama, Oct 30 2022

Seiichi Manyama, Table of n, a(n) for n = 0..425

Cf. A006153, A358081.

Cf. A216507, A345747, A358064.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^2*exp(x))))

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

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

a(n) ~ n! / ((1 + LambertW(1/2)) * 2^(n+1) * LambertW(1/2)^n). - Vaclav Kotesovec, Oct 30 2022

A292978 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = k! * Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * T(n-1-i,k) for n > 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 3, 5, 1, 0, 2, 10, 15, 1, 0, 0, 6, 41, 52, 1, 0, 0, 6, 24, 196, 203, 1, 0, 0, 0, 24, 140, 1057, 877, 1, 0, 0, 0, 24, 60, 870, 6322, 4140, 1, 0, 0, 0, 0, 120, 480, 5922, 41393, 21147, 1, 0, 0, 0, 0, 120, 360, 5250, 45416, 293608, 115975

Offset: 0

Seiichi Manyama, Sep 27 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   1,  1,  0,  0,  0, ...
   2,  3,  2,  0,  0, ...
   5, 10,  6,  6,  0, ...
  15, 41, 24, 24, 24, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-4 give: A000110, A000248, A216507, A292889, A292979.
Rows n=0 gives A000012.
Main diagonal gives A000142.
Cf. A292973.

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << f(k) * (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
  ary
end
def A292978(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A292978(20)

T(n,k) = n! * Sum_{j=0..floor(n/k)} j^(n-k*j)/(j! * (n-k*j)!) for k > 0. - Seiichi Manyama, Jul 10 2022

A345747 a(n) = n! * Sum_{k=0..floor(n/2)} k^(n - 2*k)/k!.

Original entry on oeis.org

1, 0, 2, 6, 36, 240, 2280, 27720, 425040, 7862400, 171188640, 4319330400, 125199708480, 4142318019840, 155388782989440, 6557345831836800, 308677784640825600, 16079233115648102400, 920518264903690252800, 57603377545940850624000

Offset: 0

Seiichi Manyama, Sep 17 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..315

Cf. A354436, A355575.

Cf. A104872, A216507, A356628.

Join[{1}, Table[n!*Sum[k^(n - 2*k)/k!, {k, 0, n/2}], {n, 1, 20}]] (* Vaclav Kotesovec, Oct 30 2022 *)

PARI
```
a(n) = n!*sum(k=0, n\2, k^(n-2*k)/k!);
```

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

E.g.f.: Sum_{k>=0} x^(2*k) / (k! * (1 - k * x)).

a(n) ~ sqrt(2*Pi) * exp((n - 1/2)/LambertW(exp(2/3)*(2*n - 1)/6) - 2*n) * n^(2*n + 1/2) / (3^(n + 1/2) * sqrt(1 + LambertW(exp(2/3)*(2*n - 1)/6)) * LambertW(exp(2/3)*(2*n - 1)/6)^n). - Vaclav Kotesovec, Oct 30 2022

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)!).

A292907 E.g.f.: exp(x^2 * exp(-x)).

Original entry on oeis.org

1, 0, 2, -6, 24, -140, 870, -5922, 45416, -381096, 3442410, -33382910, 345803172, -3801763836, 44156760830, -539962736250, 6929042527920, -93032248209872, 1303556965679826, -19018807375195638, 288341417011487420, -4534168069704168420

Offset: 0

Seiichi Manyama, Sep 26 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..516

Column k=2 of A292973.

Cf. A216507.

x='x+O('x^66); Vec(serlaplace(exp(x^2*exp(-x))))

a(n) = (-1)^n * A216507(n).

A366459 Expansion of e.g.f. -log(1 - x^2 * exp(x)).

Original entry on oeis.org

0, 0, 2, 6, 24, 140, 990, 8442, 84056, 955656, 12227130, 173812430, 2717859012, 46362339036, 856770362630, 17050946225250, 363576478312560, 8269357341437072, 199837364514425586, 5113346326011170838, 138106722548779770620, 3926456810081828991780

Offset: 0

Seiichi Manyama, Dec 14 2023

nonn

Cf. A009444, A366546.

Cf. A216507, A346753, A358080.

a(n) = n!*sum(k=1, n\2, k^(n-2*k-1)/(n-2*k)!);

a(n) = n! * Sum_{k=1..floor(n/2)} k^(n-2*k-1)/(n-2*k)!.

a(n) ~ (n-1)! / (2^n *LambertW(1/2)^n). - Vaclav Kotesovec, Dec 29 2023

A336961 Expansion of e.g.f. exp(x * (2 + x) * exp(x)).

Original entry on oeis.org

1, 2, 10, 56, 384, 3022, 26626, 258624, 2734360, 31168682, 380196414, 4932536908, 67717987948, 979613124414, 14877703575130, 236469561581768, 3922587278751504, 67743812585483218, 1215417753459838198, 22609895367286957572, 435341977596130683316

Offset: 0

Ilya Gutkovskiy, Aug 09 2020

nonn

Exponential transform of the oblong numbers (A002378).

Cf. A000248, A002378, A033462, A209801, A216507, A279361, A336960.

nmax = 20; CoefficientList[Series[Exp[x (2 + x) Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] k (k + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

E.g.f.: exp(x * (2 + x) * exp(x)).

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * k * (k + 1) * a(n-k).

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A216688 Expansion of e.g.f. exp( x * exp(x^2) ).

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A216689 Expansion of e.g.f. exp( x * exp(x)^2 ).

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A240989 Expansion of e.g.f. exp(x^2 * (exp(x) - 1)).

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A358080 Expansion of e.g.f. 1/(1 - x^2 * exp(x)).

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A292978 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = k! * Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * T(n-1-i,k) for n > 0.

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A345747 a(n) = n! * Sum_{k=0..floor(n/2)} k^(n - 2*k)/k!.

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

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A292907 E.g.f.: exp(x^2 * exp(-x)).

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