A111063 -id:A111063 - cp's OEIS frontend

A245334 A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.

1, 2, 1, 3, 4, 2, 4, 9, 12, 6, 5, 16, 36, 48, 24, 6, 25, 80, 180, 240, 120, 7, 36, 150, 480, 1080, 1440, 720, 8, 49, 252, 1050, 3360, 7560, 10080, 5040, 9, 64, 392, 2016, 8400, 26880, 60480, 80640, 40320, 10, 81, 576, 3528, 18144, 75600, 241920, 544320

Offset: 0

Reinhard Zumkeller, Aug 30 2014

row(0) = {1}; row(n+1) = row(n) multiplied by n and prepended with (n+1);
A111063(n+1) = sum of n-th row;
T(2*n,n) = A002690(n), central terms;
T(n,0) = n + 1;
T(n,1) = A000290(n), n > 0;
T(n,2) = A011379(n-1), n > 1;
T(n,3) = A047927(n), n > 2;
T(n,4) = A192849(n-1), n > 3;
T(n,5) = A000142(5) * A027810(n-5), n > 4;
T(n,6) = A000142(6) * A027818(n-6), n > 5;
T(n,7) = A000142(7) * A056001(n-7), n > 6;
T(n,8) = A000142(8) * A056003(n-8), n > 7;
T(n,9) = A000142(9) * A056114(n-9), n > 8;
T(n,n-10) = 11 * A051431(n-10), n > 9;
T(n,n-9) = 10 * A049398(n-9), n > 8;
T(n,n-8) = 9 * A049389(n-8), n > 7;
T(n,n-7) = 8 * A049388(n-7), n > 6;
T(n,n-6) = 7 * A001730(n), n > 5;
T(n,n-5) = 6 * A001725(n), n > 5;
T(n,n-4) = 5 * A001720(n), n > 4;
T(n,n-3) = 4 * A001715(n), n > 2;
T(n,n-2) = A070960(n), n > 1;
T(n,n-1) = A052849(n), n > 0;
T(n,n) = A000142(n);
T(n,k) = A137948(n,k) * A007318(n,k), 0 <= k <= n.

			.  0:   1;
.  1:   2,  1;
.  2:   3,  4,   2;
.  3:   4,  9,  12,    6;
.  4:   5, 16,  36,   48,    24;
.  5:   6, 25,  80,  180,   240,   120;
.  6:   7, 36, 150,  480,  1080,  1440,    720;
.  7:   8, 49, 252, 1050,  3360,  7560,  10080,   5040;
.  8:   9, 64, 392, 2016,  8400, 26880,  60480,  80640,  40320;
.  9:  10, 81, 576, 3528, 18144, 75600, 241920, 544320, 725760, 362880.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A111063 (row sums), A240993 (row products), A002690 (central terms).
Cf. A000290, A011379, A027810, A027818, A047927, A056001, A056003, A056114, A192849.
Cf. A000142, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431, A052849, A070960.
Cf. A007318, A137948.

a245334 n k = a245334_tabl !! n !! k
a245334_row n = a245334_tabl !! n
a245334_tabl = iterate (\row@(h:_) -> (h + 1) : map (* h) row) [1]

Table[(n!)/((n - k)!)*(n + 1 - k), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 10 2017 *)

T(n,k) = n!*(n+1-k)/(n-k)!. - Werner Schulte, Sep 09 2017

A030297 a(n) = n*(n + a(n-1)) with a(0)=0.

Original entry on oeis.org

0, 1, 6, 27, 124, 645, 3906, 27391, 219192, 1972809, 19728190, 217010211, 2604122676, 33853594957, 473950329594, 7109254944135, 113748079106416, 1933717344809361, 34806912206568822, 661331331924807979

Offset: 0

N. J. A. Sloane, "Urkonsaud_admin" (miti(AT)tula.sitek.net)

Exponential convolution of factorials (A000142) and squares (A000290). - Vladimir Reshetnikov, Oct 07 2016

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A019461-A019464, A006183, A054096, A111063.

f := proc(n) options remember; if n <= 1 then n elif n = 2 then 6 else -n*(n-2)*f(n-3)+(n-3)*n*f(n-2)+3*n*f(n-1)/(n-1); fi; end;

a=0;lst={a};Do[a=(a+n)*n;AppendTo[lst, a], {n, 2*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2008 *)
RecurrenceTable[{a[0]==0,a[n]==n(n+a[n-1])},a[n],{n,20}] (* Harvey P. Dale, Oct 22 2011 *)
Round@Table[(2 E Gamma[n, 1] - 1) n, {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Oct 07 2016 *)

a(n) = A019461(2n).
For n>=2, a(n) = floor(2*e*n! - n - 2). - Benoit Cloitre, Feb 16 2003
a(n) = sum_{k=0...n} (n! / k!) * k^2. - Ross La Haye, Sep 21 2004
E.g.f.: x*(1+x)*exp(x)/(1-x). - Vladeta Jovovic, Dec 01 2004

Better description from Henry Bottomley, May 15 2000

A285201 Stage at which Ken Knowlton's elevator (version 1) reaches floor n for the first time.

Original entry on oeis.org

1, 2, 5, 14, 45, 174, 825, 4738, 32137, 251338, 2224157, 21952358, 238962581, 2843085270, 36696680241, 510647009850, 7619901954001, 121367981060434, 2055085325869813, 36861997532438654, 698193329457246653, 13924819967953406654, 291683979376372766697, 6402385486361598687666, 146948520147021794869977

Offset: 1

R. L. Graham, May 02 2017

nonn

Indices of records in A285200.

When prefixed by a(0)=0, the first differences give A111063. - N. J. A. Sloane, May 03 2017

N. J. A. Sloane, Table of n, a(n) for n = 1..400
Ken Knowlton, Illustration showing what floor the elevator is on for the first 49 stages

Cf. A285200, A285203, A111063, A098558, A052849, A286281, A286282.

a:= proc(n) option remember; `if`(n<3, n, ((n-1)^2*a(n-1)
      -(n-2)*(2*n-3)*a(n-2)+(n-1)*(n-3)*a(n-3))/(n-2))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 11 2018

a[n_] := 2 - n + 2 Sum[k!/j!, {k, 0, n-2}, {j, 0, k}];
Array[a, 25] (* Jean-François Alcover, Nov 01 2020 *)

a(n) = 2 - n + 2 * Sum_{k=0..n-2} Sum_{j=0..k} k!/j!.
For n >= 2, a(n) = 1+n+2*Sum_{k=2..n} C(n,k)*(k-1)! = 1+n+2*n!*Sum_{k=2..n} 1/(k*(n-k)!). - N. J. A. Sloane, May 03 2017
E.g.f.: exp(x)*(1-x-2*log(1-x)). Omitting the factor exp(x), this gives (essentially) the e.g.f. for A098558 (or A052849). - N. J. A. Sloane, May 03 2017

A240993 A000142 (n+1) * A002109(n), a product of factorials and hyperfactorials.

Original entry on oeis.org

1, 2, 24, 2592, 3317760, 62208000000, 20316635136000000, 133852981198454784000000, 20211123400293732996612096000000, 78302033109811407811828935756349440000000, 8613223642079254859301182933198438400000000000000000

Offset: 0

Reinhard Zumkeller, Aug 31 2014

nonn

a(n+1) / a(n) = A055897(n+2);

row products of the triangle A245334.

Reinhard Zumkeller, Table of n, a(n) for n = 0..36

Cf. A000142, A002109, A245334, A055897, A245334, A111063, A074962.

Haskell
```
a240993 n = a000142 (n + 1) * a002109 n
```

Table[(n+1)!*Hyperfactorial[n], {n, 0, 10}] (* Vaclav Kotesovec, Nov 14 2014 *)
Table[(n+1)*(n!)^(n+1)/BarnesG[n+1], {n, 0, 10}] (* Vaclav Kotesovec, Nov 14 2014 *)

a(n) ~ A * sqrt(2*Pi) * n^(n^2/2+3*n/2+19/12) / exp(n*(n+4)/4), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 14 2014

A347667 Triangle read by rows: T(n,k) = Sum_{j=0..k} binomial(n,j) * j! (0 <= k <= n).

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 4, 10, 16, 1, 5, 17, 41, 65, 1, 6, 26, 86, 206, 326, 1, 7, 37, 157, 517, 1237, 1957, 1, 8, 50, 260, 1100, 3620, 8660, 13700, 1, 9, 65, 401, 2081, 8801, 28961, 69281, 109601, 1, 10, 82, 586, 3610, 18730, 79210, 260650, 623530, 986410

Offset: 0

Ilya Gutkovskiy, Sep 10 2021

			Triangle begins:
  1;
  1, 2;
  1, 3,  5;
  1, 4, 10,  16;
  1, 5, 17,  41,   65;
  1, 6, 26,  86,  206,  326;
  1, 7, 37, 157,  517, 1237, 1957;
  1, 8, 50, 260, 1100, 3620, 8660, 13700;
  ...

T(n,n) = A000522, T(2*n,n) = A066211.

Cf. A008279, A008949, A111063, A121662, A285268.

T[n_, k_] := Sum[Binomial[n, j] j!, {j, 0, k}]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

cp's OEIS Frontend

A245334 A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.

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A030297 a(n) = n*(n + a(n-1)) with a(0)=0.

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A285201 Stage at which Ken Knowlton's elevator (version 1) reaches floor n for the first time.

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Maple

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A240993 A000142 (n+1) * A002109(n), a product of factorials and hyperfactorials.

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Haskell

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A347667 Triangle read by rows: T(n,k) = Sum_{j=0..k} binomial(n,j) * j! (0 <= k <= n).

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