A217284 -id:A217284 - cp's OEIS frontend

A336247 a(n) = (n!)^n * Sum_{k=0..n} 1 / (k!)^n.

1, 2, 9, 460, 684545, 50547203126, 280807908057046657, 165858480204085842350156792, 13997217669604247492958380810030809089, 218434494471443385260764665498960241287478619115850, 792268399795067334328715213043856435592857850955707257780000000001

Offset: 0

Ilya Gutkovskiy, Jul 14 2020

nonn

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

Cf. A000522, A006040, A036740, A060943, A217284, A336248.

Main diagonal of A343863.

Table[(n!)^n Sum[1/(k!)^n, {k, 0, n}], {n, 0, 10}]

a(n) = (n!)^n * sum(k=0, n, 1/(k!)^n); \\ Michel Marcus, Jul 14 2020

A368769 a(n) = (n!)^3 * Sum_{k=1..n} 1/(k!)^3.

Original entry on oeis.org

0, 1, 9, 244, 15617, 1952126, 421659217, 144629111432, 74050105053185, 53982526583771866, 53982526583771866001, 71850742883000353647332, 124158083701824611102589697, 272775309892908670592389564310, 748495450346141392105516964466641

Offset: 0

Seiichi Manyama, Jan 04 2024

nonn

Cf. A368770, A368771, A368772.
Cf. A002627, A066998.
Cf. A217284, A271574.

Table[(n!)^3 Sum[1/(k!)^3,{k,n}],{n,0,20}] (* Harvey P. Dale, May 11 2025 *)

PARI
```
a(n) = n!^3*sum(k=1, n, 1/k!^3);
```

a(0) = 0; a(n) = n^3 * a(n-1) + 1.
a(n) = A217284(n) - (n!)^3.
a(n) ~ (A271574 - 1) * (n!)^3. - Vaclav Kotesovec, Jan 05 2024

A368772 a(n) = (n!)^3 * Sum_{k=0..n} (k/k!)^3.

Original entry on oeis.org

0, 1, 16, 459, 29440, 3680125, 794907216, 272653175431, 139598425821184, 101767252423643865, 101767252423643866000, 135452212975869985647331, 234061424022303335198589696, 514232948577000427431301564309, 1411055210895289172871491492466640

Offset: 0

Seiichi Manyama, Jan 05 2024

Cf. A368769, A368770, A368771.

Cf. A193563, A217284.

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

a(0) = 0; a(n) = n^3 * a(n-1) + n^3.
a(n) = n^3 * A217284(n-1) for n > 0.
a(n) = Sum_{k=1..n} (k!*binomial(n,k))^3. - Ridouane Oudra, Jun 14 2025

A368775 a(n) = (n+1) * (n!)^3 * Sum_{k=0..n} 1/((k+1) * (k!)^3).

Original entry on oeis.org

1, 3, 37, 1333, 106641, 15996151, 4031030053, 1580163780777, 910174337727553, 737241213559317931, 810965334915249724101, 1177521666296942599394653, 2204320559307876546066790417, 5215422443322435907994026126623

Offset: 0

Seiichi Manyama, Jan 05 2024

nonn

Cf. A217284, A368776.

Cf. A228229, A368771.

a(n) = (n+1)*n!^3*sum(k=0, n, 1/((k+1)*k!^3));

a(n) = (n+1) * n^2 * a(n-1) + 1.

A368776 a(n) = (n+1)^2 * (n!)^3 * Sum_{k=0..n} 1/((k+1)^2 * (k!)^3).

Original entry on oeis.org

1, 5, 91, 4369, 436901, 78642181, 23120801215, 10358118944321, 6712061075920009, 6040854968328008101, 7309434511676889802211, 11578144266496193446702225, 23480476572454280309912112301, 59828254306613506229656062142949

Offset: 0

Seiichi Manyama, Jan 05 2024

nonn

Cf. A217284, A368775.

Cf. A368770.

a(n) = (n+1)^2*n!^3*sum(k=0, n, 1/((k+1)^2*k!^3));

a(n) = (n+1)^2 * n * a(n-1) + 1.

A343863 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = (n!)^k * Sum_{j=1..n} (1/j!)^k.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 9, 16, 5, 1, 2, 17, 82, 65, 6, 1, 2, 33, 460, 1313, 326, 7, 1, 2, 65, 2674, 29441, 32826, 1957, 8, 1, 2, 129, 15796, 684545, 3680126, 1181737, 13700, 9, 1, 2, 257, 94042, 16175105, 427840626, 794907217, 57905114, 109601, 10

Offset: 0

Seiichi Manyama, May 02 2021

			Square array begins:
  1,   1,     1,       1,         1,           1, ...
  2,   2,     2,       2,         2,           2, ...
  3,   5,     9,      17,        33,          65, ...
  4,  16,    82,     460,      2674,       15796, ...
  5,  65,  1313,   29441,    684545,    16175105, ...
  6, 326, 32826, 3680126, 427840626, 50547203126, ...

Seiichi Manyama, Antidiagonals n = 0..59, flattened

Columns 0..3 give A000027(n+1), A000522, A006040, A217284.
Main diagonal gives A336247.
Cf. A291556.

T[n_, k_] := Sum[(n!/j!)^k, {j, 0, n}]; Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, May 03 2021 *)

PARI
```
T(n, k) = sum(j=0, n, (n!/j!)^k);
```

T(0,k) = 1 and T(n,k) = n^k * T(n-1,k) + 1 for n > 0.

A228513 a(n) = Sum_{k=0..n} 2^k*(n!/k!)^2.

Original entry on oeis.org

1, 3, 16, 152, 2448, 61232, 2204416, 108016512, 6913057024, 559957619456, 55995761946624, 6775487195543552, 975670156158275584, 164888256390748581888, 32318098252586722066432, 7271572106832012464979968, 1861522459348995191034937344, 537979990751859610209097023488

Offset: 0

Vaclav Kotesovec, Aug 24 2013

Generally, Sum_{k=0..n} x^k*(n!/k!)^2 is asymptotic to BesselI(0,2*sqrt(x))*(n!)^2

Cf. A000522, A006040, A217284.

Table[(n!)^2*Sum[2^j/(j!)^2, {j, 0, n}], {n, 0, 20}]
Total/@Table[2^k (n!/k!)^2,{n,0,20},{k,0,n}] (* Harvey P. Dale, Jun 10 2018 *)

a(n) = (n^2+2)*a(n-1) - 2*(n-1)^2*a(n-2).

a(n) ~ 2*Pi*BesselI(0,2*sqrt(2)) * n^(2*n+1)/exp(2*n).

A295610 a(n) = Sum_{k=0..n} (n!/(n - k)!)^k.

Original entry on oeis.org

1, 2, 7, 256, 345749, 25090776406, 139507578065088907, 82622801516492599819822772, 6985137485409222182920705065038896201, 109110989095384931538566720095053550173384985449034, 395940975233113726268241745444050219538058574725338743701918216111

Offset: 0

Ilya Gutkovskiy, Nov 24 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..30
Eric Weisstein's World of Mathematics, Binomial Sums

Cf. A000522, A006040, A036740, A217284, A219206.

Table[Sum[(n!/(n - k)!)^k, {k, 0, n}], {n, 0, 10}]
Table[Sum[(Gamma[n + 1]/Gamma[k + 1])^(n - k), {k, 0, n}], {n, 0, 10}]
Table[Sum[(Binomial[n, k] k!)^k, {k, 0, n}], {n, 0, 10}]

a(n) = sum(k=0, n, (n!/(n - k)!)^k); \\ Michel Marcus, Nov 25 2017

a(n) = Sum_{k=0..n} A219206(n,k)*A036740(k).

a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n^2 + n/2) / exp(n^2 - 1/12). - Vaclav Kotesovec, Nov 25 2017

A368852 a(n) = (n!)^3 * Sum_{k=0..n} (-1)^k/(k!)^3.

Original entry on oeis.org

1, 0, 1, 26, 1665, 208124, 44954785, 15419491254, 7894779522049, 5755294271573720, 5755294271573720001, 7660296675464621321330, 13236992655202865643258241, 29081672863480695818238355476, 79800110337391029325246047426145

Offset: 0

Seiichi Manyama, Jan 07 2024

Cf. A000166, A073701.

Cf. A217284.

PARI
```
a(n) = n!^3*sum(k=0, n, (-1)^k/k!^3);
```

a(n) = n^3 * a(n-1) + (-1)^n.

cp's OEIS Frontend

A336247 a(n) = (n!)^n * Sum_{k=0..n} 1 / (k!)^n.

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A368769 a(n) = (n!)^3 * Sum_{k=1..n} 1/(k!)^3.

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A368772 a(n) = (n!)^3 * Sum_{k=0..n} (k/k!)^3.

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A368775 a(n) = (n+1) * (n!)^3 * Sum_{k=0..n} 1/((k+1) * (k!)^3).

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A368776 a(n) = (n+1)^2 * (n!)^3 * Sum_{k=0..n} 1/((k+1)^2 * (k!)^3).

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A343863 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = (n!)^k * Sum_{j=1..n} (1/j!)^k.

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Formula

A228513 a(n) = Sum_{k=0..n} 2^k*(n!/k!)^2.

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A295610 a(n) = Sum_{k=0..n} (n!/(n - k)!)^k.

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Mathematica

PARI

Formula

A368852 a(n) = (n!)^3 * Sum_{k=0..n} (-1)^k/(k!)^3.

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