A052673 -id:A052673 - cp's OEIS frontend

A083746 a(1) = 1, a(2) = 2; for n>2, a(n) = 3(n-2)(n-2)!.

1, 2, 3, 12, 54, 288, 1800, 12960, 105840, 967680, 9797760, 108864000, 1317254400, 17244057600, 242853811200, 3661488230400, 58845346560000, 1004293914624000, 18140058832896000, 345728180109312000, 6933770723303424000

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003

nonn

a(1) = 1, a(2) = 2, define S(k) = sum of all the terms other than a(k) k < n. a(n) = Sum_{k=1..n-1} S(k).

			a(4) = {a(1) + a(2)} + {a(1) +a(3)} + {a(2) + a(3)} = 12.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

Cf. A001563, A052673, A129379.

[n le 2 select n else 3*(n-2)*Factorial(n-2): n in [1..40]]; // G. C. Greubel, Feb 03 2024

a := proc(n) option remember: if n=1 then RETURN(1) fi: if n=2 then RETURN(2) fi: 3*(n-2)*(n-2)! end: for n from 1 to 40 do printf(`%d,`,a(n)) od: # James Sellers, May 19 2003

Join[{1,2},Table[3n n!,{n,20}]] (* Harvey P. Dale, Feb 27 2012 *)

[1,2]+[3*(n-2)*factorial(n-2) for n in range(3, 41)] # G. C. Greubel, Feb 03 2024

a(n) = (n-2)*Sum_{j=1..n-1} a(j).
E.g.f.: 3*(x-2)*log(1-x) - 5*x + x^2. - Vladeta Jovovic, May 06 2003
From Reinhard Zumkeller, Apr 14 2007: (Start)
Sum_{k=1..n} a(k) = A052560(n-1) for n > 1.
a(n) = A052673(n-2) for n > 2. (End)

Simpler description from Vladeta Jovovic, May 06 2003

More terms from James Sellers, May 19 2003

A122972 a(1) = 1, a(2) = 2; for n>2, a(n+1) = a(n)(n-1) + a(n-1)n.

Original entry on oeis.org

1, 2, 4, 14, 58, 302, 1858, 13262, 107698, 980942, 9905458, 109844942, 1327159858, 17353902542, 244180971058, 3678842132942, 59089527531058, 1007972756756942, 18199148360427058, 346736152866068942

Offset: 1

Reinhard Zumkeller, Sep 21 2006

nonn

a(n+2) - a(n) = 3*(n-1)*(n-1)! = A052673(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

a122972 n = a122972_list !! (n-1)
a122972_list = 1 : 2 : zipWith (+)
   (zipWith (*) [2..] a122972_list) (zipWith (*) [1..] $ tail a122972_list)
-- Reinhard Zumkeller, Nov 15 2011

RecurrenceTable[{a[1]==1,a[2]==2,a[n]==a[n-1](n-2)+a[n-2](n-1)}, a,{n,20}] (* Harvey P. Dale, Nov 02 2011 *)
Table[2*(-1)^n-3*(-1)^n*Sum[(-1)^k*k!,{k,0,n-1}],{n,1,20}] (* Vaclav Kotesovec, Oct 28 2012 *)

a(n) = 2*(-1)^n - 3*(-1)^n*Sum_{k=0..n-1} (-1)^k*k!. - Vaclav Kotesovec, Oct 28 2012

A348555 Numbers k that divide the sum of the digits of 3^k * k!.

Original entry on oeis.org

1, 3, 9, 27, 72, 111, 129, 148, 161, 450, 762, 1233, 1260, 2052, 9153, 15840, 16067, 16302, 16317, 16332, 16435, 74670, 74946, 125046, 208566, 347670, 347685, 583263, 1609667, 1610942, 1616476, 1616532, 1616958, 2683143, 2700261, 4480092, 7469682, 7470432, 7492497

Offset: 1

Kevin P. Thompson, Oct 21 2021

			9 is a term because the sum of the digits of 3^9 * 9! = 7142567040 is 36 which is divisible by 9.

Cf. A032031, A007953, A052673, A108861.

Do[If[Mod[Plus @@ IntegerDigits[3^n * n!], n] == 0, Print[n]], {n, 1, 10000}]

PARI
```
isok(k) = !(sumdigits(3^k * k!) % k);
```

a(36)-a(39) from Martin Ehrenstein, Nov 19 2021

cp's OEIS Frontend

A083746 a(1) = 1, a(2) = 2; for n>2, a(n) = 3(n-2)(n-2)!.

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A122972 a(1) = 1, a(2) = 2; for n>2, a(n+1) = a(n)(n-1) + a(n-1)n.

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A348555 Numbers k that divide the sum of the digits of 3^k * k!.

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Author

Keywords

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Programs

Mathematica

PARI

Extensions

cp's OEIS Frontend

A083746 a(1) = 1, a(2) = 2; for n>2, a(n) = 3*(n-2)*(n-2)!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

Extensions

A122972 a(1) = 1, a(2) = 2; for n>2, a(n+1) = a(n)*(n-1) + a(n-1)*n.

Views

Author

Keywords

Comments

Links

Programs

Haskell

Mathematica

Formula

A348555 Numbers k that divide the sum of the digits of 3^k * k!.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A083746 a(1) = 1, a(2) = 2; for n>2, a(n) = 3(n-2)(n-2)!.

A122972 a(1) = 1, a(2) = 2; for n>2, a(n+1) = a(n)(n-1) + a(n-1)n.