A036740 -id:A036740 - cp's OEIS frontend

A122222 Difference between (n!)^n and the next smaller factorial.

2, 96, 291456, 18656179200, 17668969095168000, 67095201210572537856000000, 6721833410207820593461922365440000000, 75658161802407509372174837302453333917696000000000, 365526772920711815200262962616603688918661180831039488000000000000

Offset: 2

Hugo Pfoertner, Sep 25 2006

nonn

			a(3)=96 because the difference between (3!)^3=216 and the next smaller factorial 5!=120 is 96.

Cf. A036740, A121348, A122221.

s={};m=1;Do[Until[m!>(n!)^n,m++];AppendTo[s,(n!)^n-(m-1)!],{n,2,10}];s (* James C. McMahon, Oct 26 2024 *)

a(10) from James C. McMahon, Oct 26 2024

A214651 Count down from n to 1, n times.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 5

Offset: 1

Arkadiusz Wesolowski, Jul 24 2012

This sequence contains every positive integer infinitely often.

This is a fractal sequence. Striking out the first instance of every term produces 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, ..., which is the same as the original sequence, as far as it goes.

			1;
2, 1, 2, 1;
3, 2, 1, 3, 2, 1, 3, 2, 1;
...

Arkadiusz Wesolowski, Rows n = 1..14 of triangle, flattened
Eric Weisstein's World of Mathematics, Positive Integer
Wikipedia, Fractal sequence

Cf. A056520 (locations of new values), A060432 (locations of 1's).

Cf. A000290 (row lengths), A002411 (row sums), A036740 (row products).

f[n_] := Table[Range[n, 1, -1], {n}]; Flatten@Array[f, 6] (* Wesolowski *)
Flatten[Table[Table[Range[n, 1, -1], {n}], {n, 6}]] (* Alonso del Arte, Jul 24 2012 *)

A291547 a(n) = ((2*n-1)!!)^n.

Original entry on oeis.org

1, 9, 3375, 121550625, 753631499840625, 1261673443947253805015625, 822952789790387281855874669859609375, 285018362247755338974104595257347347998199462890625, 68512882179510153729154120317673085873841328059500855014801025390625

Offset: 1

Vaclav Kotesovec, Aug 26 2017

nonn

Cf. A001147, A036740, A254866, A291332.

Table[Product[2*k - 1, {k, 1, n}]^n, {n, 1, 10}]
Table[((2*n - 1)!!)^n, {n, 1, 10}]

a(n) = ((2*n)!/n!)^n / 2^(n^2).

a(n) ~ 2^(n^2 + n/2) * n^(n^2) / exp(n^2 + 1/24).

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

A336248 a(n) = (n!)^n * Sum_{k=0..n} (-1)^k / (k!)^n.

Original entry on oeis.org

1, 0, 1, 26, 20481, 774403124, 2173797080953345, 645067515585218711490294, 27280857986486289638369834192338945, 213095986405176211170558965907644717041658073416, 386654453940903446694477049963665295677203885863801760000000001

Offset: 0

Ilya Gutkovskiy, Jul 14 2020

nonn

Cf. A000166, A036740, A073701, A336247.

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

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

A351580 a(n) is the number of multisets of size n-1 consisting of permutations of n elements.

Original entry on oeis.org

1, 2, 21, 2600, 9078630, 1634935320144, 22831938997720867560, 34390564970975286088924022400, 7457911916650283082000186530740981347120, 300682790088737748950725540713718365319268411170195200, 2830053444386286847574443631356044745870287426798365860653876609636480

Offset: 1

Dan Eilers, Feb 13 2022

nonn

a(n) is the number of reduced men's ranking tables in the stable marriage problem of order n. In the SMP (as noted in A351409), relabeling men or women has no effect on the number of stable matchings. So the women can be relabeled to normalize the order of man #1's rankings (with woman #1 as his first choice and woman n as his last choice), and then the men except man #1 can be relabeled to normalize the lexicographic order of those men's rankings. Since man #1's rankings end up fixed in natural order, they do not contribute to the number of possibilities, leaving n! multichoose (n-1) ways to arrange the rankings of the other n-1 men.
The number of unreduced men's ranking tables is given by A036740. Relabeling just the women reduces this to A134366. Alternately, relabeling just the men reduces A036740 to A344690. Relabeling both men and women reduces the men's relabeling reduction, A344690, by a factor of (n!+n-1)/n to a(n).
It might be tempting to try to reduce A344690 by a factor of n!, but that doesn't work because not all of man #1's rankings are equally likely after relabeling all the men to give man #1 the lexicographically least rankings.
There is room for further relabeling reduction from a(n), given by A263921. The reduction from a(n) to A263921 is analogous to the reduction from reduced latin squares, A000315, to A123234.
Each of the a(n) reduced men's ranking tables can be combined with the A036740 possible unreduced women's ranking tables to form complete instances, but these instances have more possibilities than A351409. For example, a(3)*A036740(3)=21*216=4536 > A351409(3)=3888. However, fewer possibilities result from using A263921 in place of a(n), although the men's ranking tables of A263921 may not be as straightforward to generate. With A263921(3)=10, 10*216=2160 < 3888.

			Starting with the following men's ranking table of order 3, where row k represents man k's rankings, the 1 in the 2nd position of row 3 means that man #3 ranks woman #2 as his 1st choice.
  213
  321
  213
Step 1: reorder columns so row 1 is in natural order:
  123
  231
  123
Step 2: reorder rows 2 to n so rows are in lexical order:
  123
  123
  231
a(3)=21 because there are 1+2+3+4+5+6 = 21 possibilities for the last two rows in lexical order, with 3!=6 possible permutations for each row.
The 21 tables for a(3) are the following:
  123   123   123   123   123   123   123
  123   123   123   123   123   123   132
  123   132   213   231   312   321   132
.
  123   123   123   123   123   123   123
  132   132   132   132   213   213   213
  213   231   312   321   213   231   312
.
  123   123   123   123   123   123   123
  213   231   231   231   312   312   321
  321   231   312   321   312   321   321

Alois P. Heinz, Table of n, a(n) for n = 1..31
Wikipedia, Counting multisets

Cf. A036740, A071919, A134366, A344690, A263921, A351409, A000315, A185141, A030495.

Table[Binomial[n!+n-2,n-1],{n,15}] (* Harvey P. Dale, Jun 02 2023 *)

a(n) = binomial(n! + n - 2, n - 1) \\ Andrew Howroyd, Feb 13 2022

a(n) = binomial(n! + n - 2, n - 1).
a(n) = n*A344690(n)/A030495(n-1).
a(n) = A344690*n/(n! + n - 1).
a(n) = A071919(n-1,n!). - Alois P. Heinz, Feb 16 2022

Erroneous Mathematica program deleted by N. J. A. Sloane, Jun 02 2023

A366306 a(n) = Product_{k=1..n} (k^n - (k-1)^n).

Original entry on oeis.org

1, 3, 133, 170625, 10733002621, 50465283999665535, 25145494699347449245677097, 1787473773567267792523164108726890625, 23480751910878672340765325385856840967957995534681, 71672834655019406921956925590632596034005848922160549420728589375

Offset: 1

Vaclav Kotesovec, Oct 06 2023

nonn

Cf. A036740, A323575, A323588, A323589, A366305.

Table[Product[k^n - (k-1)^n, {k, 1, n}], {n, 1, 10}]

a(n) = (n!)^n * Product_{k=1..n} (1 - (1 - 1/k)^n).
a(n) ~ n!^n * d^n, where d = exp(Integral_{x=0..1} log(1 - exp(-1/x)) dx) = 0.84207793096051704199642805288991601369639823969574423397520945175552718...
a(n) ~ (2*Pi)^(n/2) * d^n * n^(n*(2*n+1)/2) / exp(n^2 - 1/12).

A366329 a(n) = Product_{k=1..n} Sum_{j=1..k} j^n.

Original entry on oeis.org

1, 5, 324, 589764, 52393770000, 347773153451938500, 244632735619259069507040000, 24547871392966749661547369532868031040, 455140097017244017295446005144727669016636127744000, 1960564895414510364772369567330640938816177001699555385515625000000

Offset: 1

Vaclav Kotesovec, Oct 07 2023

nonn

Cf. A031971, A036740, A366305, A366342.

Table[Product[Sum[j^n, {j, 1, k}], {k, 1, n}], {n, 1, 12}]
Table[Product[HarmonicNumber[k, -n], {k, 1, n}], {n, 1, 12}] // FunctionExpand

a(n) = A036740(n) * Product_{k=1..n} Sum_{j=1..k} (j/k)^n.
a(n) ~ n!^n * c * d^n, where d = exp(-Integral_{x=0..1} log(1 - exp(-1/x)) dx) = 1.187538543919977798892363400109897833660222697152558038684860736484... and c = exp(1 - 1/(exp(1) - 1)) / (exp(1) - 1) = 0.88399704290317414073109479991305699114875723090346..., updated Apr 19 2024
a(n) ~ c * d^n * (2*Pi)^(n/2) * n^(n*(2*n+1)/2) / exp(n^2 - 1/12).

A368754 a(n) = (n!)^n * [x^n] * 1/(1 - polylog(n,x)).

Original entry on oeis.org

1, 1, 5, 278, 404768, 28436662624, 151309093659896512, 86745908552613198656020224, 7184659625769578063908866060107907072, 110866279942987479997999976181870531647691458347008, 399488258540989429698770032526869852804662313023226648081962369024

Offset: 0

Alois P. Heinz, Jan 04 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..30

Cf. A000051, A000142, A007840, A011782, A036740, A323339, A323340, A336258, A336259, A336260, A336261.

a:= n-> n!^n*coeff(series(1/(1-polylog(n, x)), x, n+1), x, n):
seq(a(n), n=0..10);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)/j^k, j=1..n))
    end:
a:= n-> n!^n*b(n$2):
seq(a(n), n=0..10);

a(n) = (n!)^n*b(n,n) with b(n,k) = Sum_{j=1..n} b(n-j,k)/j^k for n>0, b(0,k) = 1.

A078669 Number of times n appears among the decimal digits of (n!)^n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 4, 0, 0, 1, 2, 2, 2, 1, 2, 4, 3, 1, 5, 3, 2, 11, 5, 1, 2, 7, 8, 15, 8, 5, 11, 11, 10, 8, 7, 10, 16, 17, 15, 19, 12, 17, 18, 23, 27, 19, 24, 30, 18, 25, 40, 30, 23, 27, 29, 31, 33, 48, 48, 50, 30, 49, 51, 51, 58, 55, 67, 52, 59, 50, 52, 63, 78, 67, 92, 107, 94, 74

Offset: 1

Jason Earls, Dec 16 2002

			a(6)=1 because 6 appears one time in 6!^6 = 139314069504000000.

Cf. A036740.

{mcf(d, n)=my(c=0, m=10^#digits(d)); while(n>0, if(n%m==d, c++); n\=10; ); c }
a(n) = {mcf(n, (n!)^n)}

cp's OEIS Frontend

A122222 Difference between (n!)^n and the next smaller factorial.

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A214651 Count down from n to 1, n times.

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A291547 a(n) = ((2*n-1)!!)^n.

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Mathematica

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

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A336248 a(n) = (n!)^n * Sum_{k=0..n} (-1)^k / (k!)^n.

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A351580 a(n) is the number of multisets of size n-1 consisting of permutations of n elements.

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A366306 a(n) = Product_{k=1..n} (k^n - (k-1)^n).

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A366329 a(n) = Product_{k=1..n} Sum_{j=1..k} j^n.

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A368754 a(n) = (n!)^n * [x^n] * 1/(1 - polylog(n,x)).

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