A354436 -id:A354436 - cp's OEIS frontend

A256016 a(n) = n! * Sum_{k=0..n} k^n/k!.

1, 1, 6, 57, 796, 15145, 374526, 11669665, 447595800, 20733553809, 1141067915290, 73552752257281, 5484203261135028, 467864288815609465, 45236104846954021014, 4915818294874879570305, 596044703812665607374256, 80118478395137652912476449, 11870487496575403846760198322

Offset: 0

Vaclav Kotesovec, Jun 01 2015

Seiichi Manyama, Table of n, a(n) for n = 0..283
Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials

Cf. A000110, A031971, A072034, A242446, A337001, A337002, A354436.

Main diagonal of A337085.

Join[{1}, Table[n!*Sum[k^n/k!,{k,0,n}],{n,1,20}]]

a(n) = n!*sum(k=0, n, k^n/k!); \\ Michel Marcus, Aug 15 2020

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*x)^k/(k!*(1-k*x))))) \\ Seiichi Manyama, Aug 23 2022

a(n) ~ e*Bell(n)*n!, for the Bell numbers see A000110.
a(n) ~ sqrt(2*Pi) * n^(2*n+1/2) * exp(n/LambertW(n)-2*n) / (sqrt(1+LambertW(n)) * LambertW(n)^n).
E.g.f.: Sum_{k>=0} (k * x)^k / (k! * (1 - k * x)). - Seiichi Manyama, Aug 23 2022
a(n) = n! * [x^n] B_n(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials. - Seiichi Manyama, Jan 04 2024
a(n) = Sum_{k=0..n} k^n*(n-k)!*binomial(n,k). - Ridouane Oudra, Jun 16 2025

a(0)=1 prepended by Seiichi Manyama, Aug 14 2020

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

A356629 a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3k)^k/(n - 3k)!.

Original entry on oeis.org

1, 1, 1, 1, 25, 121, 361, 5881, 82321, 547345, 6053041, 167991121, 2179469161, 22892967241, 788375451865, 18046198202761, 245523704069281, 7548055281543841, 270833271588545761, 5369819950838359585, 141456920470310708281, 6760255576117937586841

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..393

Cf. A354436, A356628, A356630.

Cf. A354553, A356633, A358065.

a[n_] := n! * Sum[(n - 3*k)^k/(n - 3*k)!, {k, 0, Floor[n/3]}]; a[0] = 1; Array[a, 22, 0] (* Amiram Eldar, Aug 19 2022 *)

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

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

E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x^3)).

a(n) ~ sqrt(Pi/3) * exp((2*n - 3)/(6*LambertW(exp(1/4)*(2*n - 3)/8)) - 4*n/3) * n^(4*n/3 + 1/2) / (sqrt(1 + LambertW(exp(1/4)*(2*n - 3)/8)) * 2^(2*n/3 + 1/2) * LambertW(exp(1/4)*(2*n - 3)/8)^(n/3)). - Vaclav Kotesovec, Nov 01 2022

A356673 a(n) = n! * Sum_{k=0..n} k^(3*(n-k))/k!.

Original entry on oeis.org

1, 1, 3, 31, 901, 45741, 3960871, 584698843, 130554106761, 40790044059481, 17681098707667531, 10491554658622447191, 8198225417359164798733, 8172446419302496167191941, 10264848632098736708582150511

Offset: 0

Seiichi Manyama, Aug 22 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..185

Cf. A354436, A356672.

Cf. A349880, A356629, A358687.

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

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

E.g.f: Sum_{k>=0} x^k / (k! * (1 - k^3 * x)).

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

A355575 a(n) = n! * Sum_{k=0..floor(n/3)} k^(n - 3*k)/k!.

Original entry on oeis.org

1, 0, 0, 6, 24, 120, 1080, 10080, 120960, 1874880, 34473600, 738460800, 17982518400, 489858969600, 14834839219200, 498452777222400, 18583796335104000, 768773914900992000, 35220800475250790400, 1779227869201400217600, 98469904378626772992000

Offset: 0

Seiichi Manyama, Sep 17 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..323

Cf. A345747, A354436.

Cf. A292889, A352945.

Join[{1}, Table[n!*Sum[k^(n - 3*k)/k!, {k, 0, n/3}], {n, 1, 20}]] (* Vaclav Kotesovec, Oct 30 2022 *)

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

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

E.g.f.: Sum_{k>=0} x^(3*k) / (k! * (1 - k * x)).

a(n) ~ sqrt(Pi) * exp((n - 1/2)/LambertW(exp(3/4)*(2*n - 1)/8) - 2*n) * n^(2*n + 1/2) / (sqrt(1 + LambertW(exp(3/4)*(2*n - 1)/8)) * 2^(2*n + 1/2) * LambertW(exp(3/4)*(2*n - 1)/8)^n). - Vaclav Kotesovec, Oct 30 2022

A356608 a(n) = n! * Sum_{k=0..floor(n/4)} (n - 4k)^k/(24^k (n - 4*k)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 31, 106, 281, 1261, 13861, 106261, 558361, 2709136, 32802771, 447762316, 4093711441, 28011714641, 293624974441, 5549250905281, 80454378591121, 815886496908946, 8379058314620071, 168672787637953446, 3514729162490432041, 51656083670790267901

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..455

Cf. A354436, A356029, A356328.

Cf. A354552, A356630, A356634.

a[n_] := n! * Sum[(n - 4*k)^k/(24^k*(n - 4*k)!), {k, 0, Floor[n/4]}]; a[0] = 1; Array[a, 26, 0] (* Amiram Eldar, Aug 19 2022 *)

a(n) = n!*sum(k=0, n\4, (n-4*k)^k/(24^k*(n-4*k)!));

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x^4/24)))))

E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x^4/24)).

A356029 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2k)^k/(2^k (n - 2*k)!).

Original entry on oeis.org

1, 1, 1, 4, 13, 61, 421, 2626, 27049, 245953, 3069721, 40222216, 576988501, 10058716669, 169773404893, 3596206855606, 73450508303761, 1775382487932001, 43993288886533489, 1183551336464017708, 34806599282992709341, 1043452963148195577181

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Cf. A354436, A356328, A356608.

Cf. A354550, A356628, A356632.

a[n_] := n! * Sum[(n - 2*k)^k/(2^k*(n - 2*k)!), {k, 0, Floor[n/2]}]; a[0] = 1; Array[a, 22, 0] (* Amiram Eldar, Aug 19 2022 *)

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

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

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

A356328 a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3k)^k/(6^k (n - 3*k)!).

Original entry on oeis.org

1, 1, 1, 1, 5, 21, 61, 281, 2521, 15625, 84841, 971521, 10646461, 83366141, 962405445, 15445935961, 181502928881, 2182235585041, 42297481449361, 714940186390465, 10007476059187381, 204722588272279141, 4600003555996715021, 80767827313930590681

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Cf. A354436, A356029, A356608.

Cf. A354551, A356629, A356633.

a[n_] := n! * Sum[(n - 3*k)^k/(6^k*(n - 3*k)!), {k, 0, Floor[n/3]}]; a[0] = 1; Array[a, 24, 0] (* Amiram Eldar, Aug 19 2022 *)

a(n) = n!*sum(k=0, n\3, (n-3*k)^k/(6^k*(n-3*k)!));

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x^3/6)))))

E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x^3/6)).

A356630 a(n) = n! * Sum_{k=0..floor(n/4)} (n - 4k)^k/(n - 4k)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 378001, 7287841, 59930641, 319429441, 7524471241, 353072319601, 5897248517161, 55827317669761, 726274560953761, 53139878190826561, 1650487849152976801, 25981849479032542081, 317292238756098973081

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Cf. A354436, A356628, A356629.

Cf. A354554, A356634.

a[n_] := n! * Sum[(n - 4*k)^k/(n - 4*k)!, {k, 0, Floor[n/4]}]; a[0] = 1; Array[a, 22, 0] (* Amiram Eldar, Aug 19 2022 *)

a(n) = n!*sum(k=0, n\4, (n-4*k)^k/(n-4*k)!);

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x^4)))))

E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x^4)).

cp's OEIS Frontend

A256016 a(n) = n! * Sum_{k=0..n} k^n/k!.

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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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A356629 a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3*k)^k/(n - 3*k)!.

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A356673 a(n) = n! * Sum_{k=0..n} k^(3*(n-k))/k!.

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

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A355575 a(n) = n! * Sum_{k=0..floor(n/3)} k^(n - 3*k)/k!.

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A356608 a(n) = n! * Sum_{k=0..floor(n/4)} (n - 4*k)^k/(24^k * (n - 4*k)!).

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A356029 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^k/(2^k * (n - 2*k)!).

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

A356629 a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3k)^k/(n - 3k)!.

A356608 a(n) = n! * Sum_{k=0..floor(n/4)} (n - 4k)^k/(24^k (n - 4*k)!).

A356029 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2k)^k/(2^k (n - 2*k)!).

A356328 a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3k)^k/(6^k (n - 3*k)!).

A356630 a(n) = n! * Sum_{k=0..floor(n/4)} (n - 4k)^k/(n - 4k)!.