A358064 -id:A358064 - cp's OEIS frontend

A358065 Expansion of e.g.f. 1/(1 - x * exp(x^3)).

1, 1, 2, 6, 48, 360, 2880, 27720, 322560, 4173120, 58665600, 911433600, 15567552000, 287740252800, 5710178073600, 121450256928000, 2758495490150400, 66563938106265600, 1699990278213427200, 45828946821385728000, 1300703752243703808000

Offset: 0

Seiichi Manyama, Oct 29 2022

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

Cf. A354553, A358064.

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

PARI
```
a(n) = n!*sum(k=0, n\3, (n-3*k)^k/k!);
```

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

a(n) ~ n! * 3^(n/3) / ((1 + LambertW(3)) * LambertW(3)^(n/3)). - Vaclav Kotesovec, Nov 01 2022

A356628 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2k)^k/(n - 2k)!.

Original entry on oeis.org

1, 1, 1, 7, 25, 181, 1561, 12811, 188497, 2071945, 38889361, 620762671, 12917838121, 291278938237, 6667342764265, 194869722610291, 5137978752994081, 177509783765281681, 5610285632192738977, 215195998789004395735, 8228064506323330305721

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..366

Cf. A354436, A356629, A356630.

Cf. A216688, A356632, A358064.

a[n_] := n! * Sum[(n - 2*k)^k/(n - 2*k)!, {k, 0, Floor[n/2]}]; a[0] = 1; Array[a, 21, 0] (* Amiram Eldar, Aug 19 2022 *)

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

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

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

a(n) ~ sqrt(Pi) * exp((n-1)/(2*LambertW(exp(1/3)*(n-1)/3)) - 3*n/2) * n^((3*n + 1)/2) / (sqrt(1 + LambertW(exp(1/3)*(n - 1)/3)) * 3^((n+1)/2) * LambertW(exp(1/3)*(n-1)/3)^(n/2)). - Vaclav Kotesovec, Nov 01 2022

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

A365283 E.g.f. satisfies A(x) = 1 + xA(x)exp(x^2*A(x)^2).

Original entry on oeis.org

1, 1, 2, 12, 120, 1380, 19440, 341040, 7029120, 164762640, 4355769600, 128527439040, 4181332700160, 148633442717760, 5734427199621120, 238676208285715200, 10659325532663808000, 508452777299622355200, 25800664274991135129600

Offset: 0

Seiichi Manyama, Aug 31 2023

nonn

Cf. A358064, A365282, A365284.

Cf. A161633, A365287.

Join[{1},Table[n!/(n+1) * Sum[(n-2*k)^k * Binomial[n+1,n-2*k]/k!, {k, 0, Floor[n/2]}], {n,1,20}]] (* Vaclav Kotesovec, Nov 08 2023 *)

a(n) = n!*sum(k=0, n\2, (n-2*k)^k*binomial(n+1, n-2*k)/k!)/(n+1);

a(n) = (n!/(n+1)) * Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(n+1,n-2*k)/k!.

a(n) ~ 2^(n/2) * (1 + 3*LambertW(2^(1/3)/3))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(2^(1/3)/3)) * 3^(3*n/2 + 2) * exp(n) * LambertW(2^(1/3)/3)^(3*(n+1)/2)). - Vaclav Kotesovec, Nov 08 2023

A365282 E.g.f. satisfies A(x) = 1 + xA(x)exp(x^2*A(x)).

Original entry on oeis.org

1, 1, 2, 12, 96, 900, 10800, 157080, 2634240, 50455440, 1089849600, 26157479040, 690848040960, 19924295751360, 623024501299200, 20996216063222400, 758724126031872000, 29267547577396128000, 1200407895406514995200

Offset: 0

Seiichi Manyama, Aug 31 2023

nonn

Cf. A358064, A365283, A365284.

Cf. A365285.

Join[{1}, Table[n! * Sum[(n-2*k)^k * Binomial[n-k+1,n-2*k] / ((n-k+1)*k!), {k,0,Floor[n/2]}], {n,1,20}]] (* Vaclav Kotesovec, Aug 31 2023 *)

a(n) = n!*sum(k=0, n\2, (n-2*k)^k*binomial(n-k+1, n-2*k)/((n-k+1)*k!));

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(n-k+1,n-2*k)/( (n-k+1)*k! ).

a(n) ~ sqrt((s+1)/(2*s-1)) * (s-1)^((n+1)/2) * s^(n/2 + 1) * n^(n-1) / exp(n), where s = 3.011547791499065828694160466323712196300874261862... is the root of the equation (s-1)*LambertW(2*(s-1)^2/s) = 2. - Vaclav Kotesovec, Aug 31 2023

A375627 Expansion of e.g.f. exp(2x^2) / (1 - x exp(x^2)).

Original entry on oeis.org

1, 1, 6, 24, 168, 1260, 11760, 126000, 1545600, 21304080, 326350080, 5497873920, 101048048640, 2011924474560, 43139969832960, 991088998099200, 24286975237324800, 632358338278867200, 17433184834127462400, 507307459608530380800, 15539683835941532467200

Offset: 0

Seiichi Manyama, Aug 21 2024

nonn

Cf. A368265, A375628.

Cf. A358064, A375604.

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

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

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

A358264 Expansion of e.g.f. 1/(1 - x * exp(x^2/2)).

Original entry on oeis.org

1, 1, 2, 9, 48, 315, 2520, 23415, 248640, 2972025, 39463200, 576413145, 9184855680, 158550787395, 2947473809280, 58707685211175, 1247293022976000, 28156003910859825, 672972205556851200, 16978695795089253225, 450907982644863744000, 12573634144960773960075

Offset: 0

Seiichi Manyama, Nov 06 2022

Cf. A006153, A358265.

Cf. A354550, A358064.

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

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

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

a(n) ~ n! / ((1 + LambertW(1)) * LambertW(1)^(n/2)). - Vaclav Kotesovec, Nov 13 2022

A365284 E.g.f. satisfies A(x) = 1 + xA(x)exp(x^2*A(x)^3).

Original entry on oeis.org

1, 1, 2, 12, 144, 1980, 31680, 630840, 15093120, 411883920, 12607660800, 430740858240, 16265744732160, 671629503504960, 30093198326231040, 1454898560062147200, 75503612563771392000, 4186035286381024876800, 246916968958719605145600

Offset: 0

Seiichi Manyama, Aug 31 2023

nonn

Cf. A358064, A365282, A365283.

Join[{1}, Table[n! * Sum[(n - 2*k)^k*Binomial[n + k + 1, n - 2*k]/((n + k + 1)*k!), {k, 0, Floor[n/2]}], {n, 1, 20}]] (* Vaclav Kotesovec, Mar 10 2024 *)

a(n) = n!*sum(k=0, n\2, (n-2*k)^k*binomial(n+k+1, n-2*k)/((n+k+1)*k!));

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(n+k+1,n-2*k)/( (n+k+1)*k! ).

a(n) ~ sqrt((1 + 2*r^2*s^3) / (12*r^2*s + 9*r^4*s^4)) * n^(n-1) / (exp(n) * r^n), where s = 1.766482823850997284176450269002863328615073785089684545740773169... is the root of the equation 3*(s-1)*LambertW(2*s*(s-1)^2) = 2 and r = 1/sqrt(3*s^3*(s-1)) = 0.280882078734447087396397749882018030987007964077248... - Vaclav Kotesovec, Mar 10 2024

A377534 Expansion of e.g.f. 1/(1 - x * exp(x^2))^3.

Original entry on oeis.org

1, 3, 12, 78, 648, 6300, 72000, 939960, 13749120, 223035120, 3969907200, 76890733920, 1609732776960, 36214043785920, 871131980759040, 22310233170825600, 606026217929932800, 17401756135956192000, 526641334386809241600, 16753142420507766873600

Offset: 0

Seiichi Manyama, Oct 31 2024

Cf. A358064, A377533.

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

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

a(n) ~ n! * n^2 * 2^(n/2 - 1) / ((1 + LambertW(2))^3 * LambertW(2)^(n/2)). - Vaclav Kotesovec, Oct 31 2024

A377533 Expansion of e.g.f. 1/(1 - x * exp(x^2))^2.

Original entry on oeis.org

1, 2, 6, 36, 264, 2280, 23760, 283920, 3830400, 57728160, 959212800, 17424348480, 343508014080, 7302340805760, 166504724305920, 4053311579116800, 104916366780825600, 2877212787562713600, 83332056329006284800, 2541707625791324390400, 81432631127484628992000

Offset: 0

Seiichi Manyama, Oct 31 2024

Cf. A358064, A377534.

a(n) = n!*sum(k=0, n\2, (n-2*k+1)*(n-2*k)^k/k!);

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k+1) * (n-2*k)^k/k!.

a(n) ~ n! * n * 2^(n/2) / ((1+LambertW(2))^2 * LambertW(2)^(n/2)). - Vaclav Kotesovec, Oct 31 2024

cp's OEIS Frontend

A358065 Expansion of e.g.f. 1/(1 - x * exp(x^3)).

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A356628 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^k/(n - 2*k)!.

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

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A365283 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^2*A(x)^2).

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A365282 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^2*A(x)).

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A375627 Expansion of e.g.f. exp(2*x^2) / (1 - x * exp(x^2)).

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A358264 Expansion of e.g.f. 1/(1 - x * exp(x^2/2)).

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A365284 E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^2*A(x)^3).

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A377534 Expansion of e.g.f. 1/(1 - x * exp(x^2))^3.

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A356628 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2k)^k/(n - 2k)!.

A365283 E.g.f. satisfies A(x) = 1 + xA(x)exp(x^2*A(x)^2).

A365282 E.g.f. satisfies A(x) = 1 + xA(x)exp(x^2*A(x)).

A375627 Expansion of e.g.f. exp(2x^2) / (1 - x exp(x^2)).

A365284 E.g.f. satisfies A(x) = 1 + xA(x)exp(x^2*A(x)^3).