A030297 -id:A030297 - cp's OEIS frontend

A002104 Logarithmic numbers.

0, 1, 3, 8, 24, 89, 415, 2372, 16072, 125673, 1112083, 10976184, 119481296, 1421542641, 18348340127, 255323504932, 3809950977008, 60683990530225, 1027542662934915, 18430998766219336, 349096664728623336, 6962409983976703337, 145841989688186383359, 3201192743180799343844

Offset: 0

N. J. A. Sloane

Prime p divides a(p+1). - Alexander Adamchuk, Jul 05 2006
Also number of lists of elements from {1,..,n} with (1st element) = (smallest element), where a list means an ordered subset (cf. A000262), see also Haskell program. - Reinhard Zumkeller, Oct 26 2010
a(n+1) = p_n(-1)  where p_n(x) is the unique degree-n polynomial such that p_n(k) = A133942(k) for k = 0, 1, ..., n. - Michael Somos, Apr 30 2012
a(n) = A006231(n) + n. - Geoffrey Critzer, Oct 04 2012

			From _Reinhard Zumkeller_, Oct 26 2010: (Start)
a(3) = #{[1], [1,2], [1,2,3], [1,3], [1,3,2], [2], [2,3], [3]} = 8;
a(4) = #{[1], [1,2], [1,2,3], [1,2,3,4], [1,2,4], [1,2,4,3], [1,3], [1,3,2], [1,3,2,4], [1,3,4], [1,3,4,2], [1,4], [1,4,2], [1,4,2,3], [1,4,3], [1,4,3,2], [2], [2,3], [2,3,4], [2,4], [2,4,3], [3], [3,4], [4]} = 24. (End)
G.f. = x + 3*x^2 + 8*x^3 + 24*x^4 + 89*x^5 + 415*x^6 + 2372*x^7 + ...

J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100 (corrected by Michel Marcus, Jan 19 2019)
J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
Mengyao Hu, Eloïc Vallée, Tim Seynnaeve, Patrick Emonts, and Jordi Tura, Characterizing Translation-Invariant Bell Inequalities using Tropical Algebra and Graph Polytopes, arXiv:2407.08783 [quant-ph], 2024. See p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 116
J. C. Tiernan, An efficient search algorithm to find the elementary circuits of a graph, Commun. ACM, 13 (1970), 722-726.
Index entries for sequences related to logarithmic numbers

Cf. A001338, A006231, A007526, A030297, A133942.

import Data.List (subsequences, permutations)
a002104 = length . filter (\xs -> head xs == minimum xs) .
                   tail . choices . enumFromTo 1
   where choices = concat . map permutations . subsequences
-- Reinhard Zumkeller, Feb 21 2012, Oct 25 2010

a := proc(n) option remember; ifelse(n < 2, n, n*a(n-1) - (n-1)*a(n-2) + 1) end:
seq(a(n), n = 0..23); # Peter Luschny, Dec 05 2023

Table[Sum[Sum[m!/k!,{k,0,m}],{m,0,n-1}],{n,1,30}] (* Alexander Adamchuk, Jul 05 2006 *)
a[n_] = n*(HypergeometricPFQ[{1, 1, 1-n}, {2}, -1]); Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Mar 29 2011 *)

x='x+O('x^99); concat([0], Vec(serlaplace(-log(1-x)*exp(x)))) \\ Altug Alkan, Dec 17 2017

{a(n) = sum(k=0, n-1, binomial(n, k) * (n-k-1)!)}; /* Michael Somos, May 08 2019 */

E.g.f.: -log(1 - x) * exp(x).
a(n) = Sum_{k=1..n} Sum_{i=0..n-k} (n-k)!/i!.
a(n) = Sum_{k=1..n} n(n-1)...(n-k+1)/k = A006231(n) + n - Avi Peretz (njk(AT)netvision.net.il), Mar 24 2001
a(n+1) - a(n) = A000522(n).
a(n) = sum{k=0..n-1, binomial(n, k)*(n-k-1)!}, row sums of A111492. - Paul Barry, Aug 26 2004
a(n) = Sum[Sum[m!/k!,{k,0,m}],{m,0,n-1}]. a(n) = Sum[A000522(m),{m,0,n-1}]. - Alexander Adamchuk, Jul 05 2006
For n > 1, the arithmetic mean of the first n terms is a(n-1) + 1. - Franklin T. Adams-Watters, May 20 2010
a(n) = n * 3F1((1,1,1-n); (2); -1). - Jean-François Alcover, Mar 29 2011
Conjecture: a(n) +(-n-1)*a(n-1) +2*(n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Dec 02 2012
From Emanuele Munarini, Dec 16 2017: (Start)
The generating series A(x) = -exp(x)*log(1-x) satisfies the differential equations:
  (1-x)*A'(x) - (1-x)*A(x) = exp(x)
  (1-x)*A''(x) - (3-2*x)*A'(x) + (2-x)*A(x) = 0.
  From the first one, we have the recurrence reported below by R. R. Forberg. From the second one, we have the recurrence conjectured above. (End)
G.f.: conjecture: T(0)*x/(1-2*x)/(1-x), where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 - (1 - 2*x*(k+1))*(1 - 2*x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
a(n) ~ exp(1)*(n-1)!. - Vaclav Kotesovec, Mar 10 2014
a(n) = n*a(n-1) - (n-1)*a(n-2) + 1, a(0) = 0, a(1) = 1. - Richard R. Forberg, Dec 15 2014
a(n) = A007526(n) + A006231(n+1) - A030297(n). - Anton Zakharov, Sep 05 2016
0 =  +a(n)*(+a(n+1) -4*a(n+2) +4*a(n+3) -a(n+4)) +a(n+1)*(+2*a(n+2) -5*a(n+3) +2*a(n+4)) +a(n+2)*(+2*a(n+2) -a(n+3) -a(n+4)) +a(n+3)*(+a(n+3)) for all n>=0. - Michael Somos, May 08 2019
From Peter Bala, Sep 12 2022: (Start)
For n, m >= 0, a(n) - a(n + m) == ( a(1) - a(m) ) (mod m). The sequence {mod(a(1) - a(m+1), m): m >= 1} begins [0, 1, 1, 0, 1, 5, 1, 0, 3, 7, 1, 4, 1, 9, 8, 0, 1, 15, 1, 4, ...].
Conjectures:
1) for n, m >= 0, k >= 2, a(n + m*2^k) - a(n) is divisible by 2^k.
2) for n >= 0, a(n + m*p^k) - a(n) + m*p^(k-1) is divisible by p^k for all positive integers m and k, and for all odd primes p. The particular case n = m = k = 1 is stated in the Comments section by Adamchuk. (End)
a(n) = Integral_{t=0..oo} ((t + 1)^n - 1)/(t*e^t) dt. - Velin Yanev, Apr 13 2024
a(n) = Gamma(n)*(e - ((-1)^n)*Gamma(1 - n, -1)) + hypergeom([1, 1], [2, n + 2], 1)/(n + 1) - polygamma(n) - 1/n + i*Pi for n > 0, where polygamma is the digamma function and the bivariate gamma function is the upper incomplete gamma function. - Velin Yanev, Apr 13 2024

More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001

A337001 a(n) = n! * Sum_{k=0..n} k^3 / k!.

Original entry on oeis.org

0, 1, 10, 57, 292, 1585, 9726, 68425, 547912, 4931937, 49320370, 542525401, 6510306540, 84633987217, 1184875823782, 17773137360105, 284370197765776, 4834293362023105, 87017280516421722, 1653328329812019577, 33066566596240399540, 694397898521048399601

Offset: 0

Ilya Gutkovskiy, Aug 10 2020

nonn

Exponential convolution of cubes (A000578) and factorial numbers (A000142).

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

Cf. A000142, A000522, A000578, A007526, A030297, A256016, A337002.

Row sums of A121682.

Table[n! Sum[k^3/k!, {k, 0, n}], {n, 0, 21}]
nmax = 21; CoefficientList[Series[x (1 + 3 x + x^2) Exp[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = n (n^2 + a[n - 1]); Table[a[n], {n, 0, 21}]

a(n) = n! * sum(k=0, n, k^3/k!); \\ Michel Marcus, Aug 12 2020

E.g.f.: x * (1 + 3*x + x^2) * exp(x) / (1 - x).
a(0) = 0; a(n) = n * (n^2 + a(n-1)).
a(n) ~ 5*exp(1)*n!. - Vaclav Kotesovec, Jan 13 2024

A337002 a(n) = n! * Sum_{k=0..n} k^4 / k!.

Original entry on oeis.org

0, 1, 18, 135, 796, 4605, 28926, 204883, 1643160, 14795001, 147960010, 1627574751, 19530917748, 253901959285, 3554627468406, 53319412076715, 853110593292976, 14502880086064113, 261051841549259010, 4959984989436051511, 99199699788721190220

Offset: 0

Ilya Gutkovskiy, Aug 10 2020

nonn

Exponential convolution of fourth powers (A000583) and factorial numbers (A000142).

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

Cf. A000142, A000522, A000583, A007526, A030297, A256016, A337001.

Table[n! Sum[k^4/k!, {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[x (1 + 7 x + 6 x^2 + x^3) Exp[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = n (n^3 + a[n - 1]); Table[a[n], {n, 0, 20}]

a(n) = n! * sum(k=0, n, k^4/k!); \\ Michel Marcus, Aug 12 2020

E.g.f.: x * (1 + 7*x + 6*x^2 + x^3) * exp(x) / (1 - x).
a(0) = 0; a(n) = n * (n^3 + a(n-1)).
a(n) ~ 15*exp(1)*n!. - Vaclav Kotesovec, Jan 13 2024

A111063 a(0) = 1; a(n) = (n-1)*a(n-1) + n.

Original entry on oeis.org

1, 1, 3, 9, 31, 129, 651, 3913, 27399, 219201, 1972819, 19728201, 217010223, 2604122689, 33853594971, 473950329609, 7109254944151, 113748079106433, 1933717344809379, 34806912206568841, 661331331924807999, 13226626638496160001, 277759159408419360043

Offset: 0

N. J. A. Sloane, Oct 08 2005

From Frank Ruskey, Nov 24 2009: (Start)
If the initial 1 were deleted, the recurrence relation becomes a(n) = n+1+n*a(n-1) with a(0) = 1. Furthermore, it can then be shown that a(n) is the number of nonempty subsets of binary strings with n 1's and 2 0's that are closed under the operation of replacing the leftmost 01 with 10. Taking the maximal elements under this relation,
a(2) = 9 = |{0011},{0101},{1001},{1010},{1100},{0110}, {0110,1001},{0101,0110},{0011,0110}|.
We also have the e.g.f. (1+x)/(1-x) e^x and the formula a(n) = -1 + 2*n!*sum_{j=0..n} 1/j!. (End)
a(n+1) = sum of n-th row in triangle A245334. - Reinhard Zumkeller, Aug 30 2014 [A-number corrected by N. J. A. Sloane, May 03 2017]
Eigensequence of triangle with (1, 2, 3, ...) as the right and left borders and the rest zeros. - Gary W. Adamson, Aug 01 2016
The following remarks apply to the sequence without the initial term a(0) = 1: For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k. It follows that for each k the sequence a(n) taken modulo k is periodic with period dividing k. For example, modulo 10 the sequence becomes 1, 3, 9, 1, 9, 1, 3, 9, 1, 9, ... with period 5. Cf. A000522. - Peter Bala, Nov 19 2017

F. Drewes et al., Tight Bounds for Cut-Operations on Deterministic Finite Automata, in Lecture Notes in Computer Science, Volume 9288 2015, Machines, Computations, and Universality, 7th International Conference, MCU 2015, Famagusta, North Cyprus, September 9-11, 2015, Editors: Jerome Durand-Lose, Benedek Nagy, ISBN: 978-3-319-23110-5 (Print) 978-3-319-23111-2 (Online). ["In the On-Line Encyclopedia of Integer Sequences (OEIS) this matches the sequence A111063."]

Reinhard Zumkeller, Table of n, a(n) for n = 0..400

Cf. A000522, A007526, A030297, A245334, A240993.

a111063 n = a111063_list !! n
a111063_list = 1 : zipWith (+) [1..] (zipWith (*) [0..] a111063_list)
-- Reinhard Zumkeller, Aug 30 2014

a:=proc(n) option remember; if n=0 then RETURN(1); fi; (n-1)*a(n-1)+n; end;
# Alternatively:
a := n -> `if`(n=0, 1, 2*exp(1)*GAMMA(n, 1) - 1):
seq(simplify(a(n)), n=0..22); # Peter Luschny, Nov 21 2017

FoldList[#1*#2 + #2 + 1 &, 1, Range[21]] (* Robert G. Wilson v, Jul 07 2012 *)

a(n+1) = Sum_{k=0..2*n} C(n,floor(k/2))(n-floor(k/2))!. - Paul Barry, May 04 2007
a(n) = A030297(n)/n, n>0.
a(n) = A007526(n) + A000522(n). - Gary Detlefs, Jun 10 2010
a(n) = 2*floor(e*n!) - 1, n>1. - Gary Detlefs, Jun 10 2010
E.g.f.: exp(x)*(1+x)/(1-x), - N. J. A. Sloane, May 03 2017
a(n) ~ 2*sqrt(2*Pi)*exp(1)*n^n*sqrt(n)/exp(n). - Ilya Gutkovskiy, Aug 02 2016
a(n) = 2*exp(1)*GAMMA(n, 1) - 1 for n>=1. - Peter Luschny, Nov 21 2017

A006183 a(n) = (n+1)a(n-1) + (2-n)a(n-2).

Original entry on oeis.org

1, 2, 6, 22, 98, 522, 3262, 23486, 191802, 1753618, 17755382, 197282022, 2387112466, 31249472282, 440096734638, 6635304614542, 106638824162282, 1819969265702946, 32873194861759462, 626524419718239158, 12565295306571352002, 264532532769923200042

Offset: 1

N. J. A. Sloane, Simon Plouffe

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. C. Holroyd and W. J. G. Wingate, Cycles in the complement of a tree or other graph, Discrete Math., 55 (1985), 267-282.

Equals A030297(n-1) - A030297(n-2) + 1. Cf. A054096.

Equals 2 * A001339(n+2).

[n le 2 select n else n*Self(n-1)+(3-n)*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 06 2016

RecurrenceTable[{a[n] == (n + 1) a[n - 1] + (2 - n) a[n - 2], a[0] == 1, a[1] == 2}, a, {n, 20}] (* Robert G. Wilson v, Jun 15 2013 *)

G.f.: 2*Sum_{k>=0} k!*(x/(1-x))^k - 1 = Q(0) -1, where Q(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) + (1-x)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 08 2013

More terms from James Sellers, Aug 21 2000
a(1) from Robert G. Wilson v, Jun 15 2013
a(21)-a(22) from Vincenzo Librandi, Mar 06 2016

A121662 Triangle read by rows: T(i,j) for the recurrence T(i,j) = (T(i-1,j) + 1)*i.

Original entry on oeis.org

1, 4, 2, 15, 9, 3, 64, 40, 16, 4, 325, 205, 85, 25, 5, 1956, 1236, 516, 156, 36, 6, 13699, 8659, 3619, 1099, 259, 49, 7, 109600, 69280, 28960, 8800, 2080, 400, 64, 8, 986409, 623529, 260649, 79209, 18729, 3609, 585, 81, 9, 9864100, 6235300, 2606500, 792100, 187300, 36100, 5860, 820, 100, 10

Offset: 1

Thomas Wieder, Aug 15 2006

The first column is A007526 = "the number of (nonnull) "variations" of n distinct objects, namely the number of permutations of nonempty subsets of {1,...,n}." E.g. for n=3 there are 15 subsets: {a}, {b}, {c}, {ab}, {ba}, {ac}, {ca}, {bc}, {cb}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}. These are subsets with a number of elements l=1,...,n. The second column excludes all subsets with l=n elements. For n=3 one has therefore only the 9 subsets {a}, {b}, {c}, {ab}, {ba}, {ac}, {ca}, {bc}, {cb}. The third column excludes all subsets with l>=n-1 elements. For n=3 one has therefore only the 3 subsets {a}, {b},{c}. See also A121684. The second column is A038156 = n!*Sum(1/k!, k=1..n-1). The first lower diagonal are the squares A000290 = n^2. The second lower diagonal (15, 40, 85...) is A053698 = n^3 + n^2 + n + 1. The row sum is A030297 = a(n) = n*(n+a(n-1)).

T(i, j) is the total number of ordered sets of size 1 to i-j+1 that can be created from i distinct items. - Manfred Boergens, Jun 22 2022

			Triangle T(i,j) begins:
       1
       4     2
      15     9     3
      64    40    16     4
     325   205    85    25    5
    1956  1236   516   156   36   6
   13699  8659  3619  1099  259  49  7
   ...

Cf. A007526, A030297, A053698, A038156, A000290, A121682.

Mirror of triangle A285268.

T:= proc(i, j) option remember;
      `if`(j<1 or j>i, 0, T(i-1, j)*i+i)
    end:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Jun 22 2022

Table[Sum[m!/(m - i)!, {i, n}], {m, 9}, {n, m, 1, -1}] // Flatten (* Michael De Vlieger, Apr 22 2017 *)
(* Sum-free code *)
b[j_] = If[j == 0, 0, Floor[j! E - 1]];
T[i_, j_] = b[i] - i! b[j - 1]/(j - 1)!;
Table[T[i, j], {i, 24}, {j, i}] // Flatten
(* Manfred Boergens, Jun 22 2022 *)

From Manfred Boergens, Jun 22 2022: (Start)
T(i, j) = Sum_{k=1..i-j+1} i!/(i-k)! = Sum_{k=j-1..i-1} i!/k!.
Sum-free formula: T(i, j) = b(i) - i!*b(j-1)/(j-1)! where b(0)=0, b(j)=floor(j!*e-1) for j>0.
(End)

Edited by N. J. A. Sloane, Sep 15 2006

Formula in name corrected by Alois P. Heinz, Jun 22 2022

A337085 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} j^k/j!.

Original entry on oeis.org

1, 0, 2, 0, 1, 5, 0, 1, 4, 16, 0, 1, 6, 15, 65, 0, 1, 10, 27, 64, 326, 0, 1, 18, 57, 124, 325, 1957, 0, 1, 34, 135, 292, 645, 1956, 13700, 0, 1, 66, 345, 796, 1585, 3906, 13699, 109601, 0, 1, 130, 927, 2404, 4605, 9726, 27391, 109600, 986410, 0, 1, 258, 2577, 7804, 15145, 28926, 68425, 219192, 986409, 9864101

Offset: 0

Seiichi Manyama, Aug 14 2020

			Square array begins:
     1,    0,    0,    0,     0,     0,      0, ...
     2,    1,    1,    1,     1,     1,      1, ...
     5,    4,    6,   10,    18,    34,     66, ...
    16,   15,   27,   57,   135,   345,    927, ...
    65,   64,  124,  292,   796,  2404,   7804, ...
   326,  325,  645, 1585,  4605, 15145,  54645, ...
  1957, 1956, 3906, 9726, 28926, 98646, 374526, ...

Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials

Columns k=0..5 give A000522, A007526, A030297, A337001, A337002, A368719.
Main diagonal gives A256016.
Cf. A368724.

T[n_, k_] := n! * Sum[If[j == k == 0, 1, j^k]/j!, {j, 0, n}]; Table[T[k, n-k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 29 2021 *)

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

E.g.f. of column k: B_k(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials. - Seiichi Manyama, Jan 04 2024

A053817 a(0)=1, a(n) = n*(a(n-1) + n).

Original entry on oeis.org

1, 2, 8, 33, 148, 765, 4626, 32431, 259512, 2335689, 23356990, 256927011, 3083124276, 40080615757, 561128620794, 8416929312135, 134670868994416, 2289404772905361, 41209285912296822, 782976432333639979, 15659528646672799980, 328850101580128800021

Offset: 0

mindy and steve schwartz (minsteve(AT)netvision.net.il), Apr 07 2000

			Start with 1, add 1, multiply by 1 (getting 2), add 2, multiply by 2 (getting 8).

Cf. A019460, A030297.

a:= proc(n) a(n):= n*(a(n-1) + n) end: a(0):= 1:
seq(a(n), n=0..23);  # Alois P. Heinz, May 12 2021

a=1;lst={a};Do[a=(a+n)*n;AppendTo[lst, a], {n, 2*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2008 *)
RecurrenceTable[{a[0]==1,a[n]==n(a[n-1]+n)},a,{n,20}] (* Harvey P. Dale, Feb 14 2015 *)

a(n) = A030297(n) + n! = A019460(2n+1) - n. - Henry Bottomley, Nov 10 2000
a(n)= n! + floor(2*e*n!-n) - 2, n>1. - Gary Detlefs, Jun 22 2010
E.g.f.: (1 + x * (1 + x) * exp(x)) / (1 - x). - Ilya Gutkovskiy, Oct 11 2021

More terms from James Sellers, Apr 08 2000

A285268 Triangle read by rows: T(m,n) = Sum_{i=1..n} P(m,i) where P(m,n) = m!/(m-n)! is the number of permutations of m items taken n at a time, for 1 <= n <= m.

Original entry on oeis.org

1, 2, 4, 3, 9, 15, 4, 16, 40, 64, 5, 25, 85, 205, 325, 6, 36, 156, 516, 1236, 1956, 7, 49, 259, 1099, 3619, 8659, 13699, 8, 64, 400, 2080, 8800, 28960, 69280, 109600, 9, 81, 585, 3609, 18729, 79209, 260649, 623529, 986409, 10, 100, 820, 5860, 36100, 187300, 792100, 2606500, 6235300, 9864100

Offset: 1

Rick Nungester, Apr 15 2017

T(m, n) is the total number of ordered sets of size 1 to n that can be created from m distinct items. For example, for 4 items taken 1 to 3 at a time, there are P(4, 1) + P(4, 2) + P(4, 3) = 4 + 12 + 24 = 40 total sets: 1, 12, 123, 124, 13, 132, 134, 14, 142, 143, 2, 21, 213, 214, 23, 231, 234, 24, 241, 243, 3, 31, 312, 314, 32, 321, 324, 34, 341, 342, 4, 41, 412, 413, 42, 421, 423, 43, 431, 432.
T(m, n) is the total number of tests in a software program that generates all P(m, n) possible solutions to a problem, allowing early-termination testing on each partial permutation, and not doing any such early termination. For example, from a deck of 52 cards, evaluate all possible 5-card deals as each card is dealt. T(52, 5) = 318507904 total evaluations.
T(m, n) counts the numbers with <= n distinct nonzero digits in base m+1. - M. F. Hasler, Oct 10 2019

			Triangle begins:
  1;
  2,  4;
  3,  9,  15;
  4, 16,  40,   64;
  5, 25,  85,  205,   325;
  6, 36, 156,  516,  1236,  1956;
  7, 49, 259, 1099,  3619,  8659,  13699;
  8, 64, 400, 2080,  8800, 28960,  69280, 109600;
  9, 81, 585, 3609, 18729, 79209, 260649, 623529, 986409;
  ...

Rick Nungester, Table of n, a(n) for n = 1..20100

Mirror of triangle A121662.
Row sums give A030297.
Diagonals (1..4): A007526 (less the initial 0), A038156 (less the initial 0, 0), A224869 (less the initial -1, 0), A079750 (less the initial 0).
Columns (1..3): A000027, A000290 (less the initial 0, 1), A053698 (less the initial 1, 4).

SumPermuteTriangle := proc(M)
  local m;
  for m from 1 to M do print(seq(add(m!/(m-k)!, k=1..n), n=1..m)) od;
end:
SumPermuteTriangle(10);
# second Maple program:
T:= proc(n, k) option remember;
     `if`(k<1, 0, T(n-1, k-1)*n+n)
    end:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Jun 26 2022

Table[Sum[m!/(m - i)!, {i, n}], {m, 9}, {n, m}] // Flatten (* Michael De Vlieger, Apr 22 2017 *)
(* Sum-free code *)
b[j_] = If[j==0, 0, Floor[j! E - 1]]; T[m_,n_] = b[m] - m! b[m-n]/(m-n)!; Table[T[m, n],{m, 24},{n, m}]//Flatten
(* Manfred Boergens, Jun 22 2022 *)

A285268(m,n,s=m-n+1)={for(k=m-n+2,m,s=(s+1)*k);s} \\ Much faster than sum(k=1,n,m!\(m-k)!), e.g., factor 6 for m=1..99, factor 57 for m=1..199.
apply( A285268_row(m)=vector(m,n,A285268(m,n)), [1..9]) \\ M. F. Hasler, Oct 10 2019

T(n, k) = {exp(1)*(incgam(n+1, 1) - incgam(n-k, 1)*(n-k)*n!/(n-k)!) - 1;}
  apply(Trow(n) = vector(n, k, round(T(n, k))), [1..10]) \\ Adjust the realprecision if needed. Peter Luschny, Oct 10 2019

T(m, n) = Sum_{k=1..n} m!/(m-k)! = m*(1 + (m-1)*(1 + (m-2)*(1 + ... + m-n+1)...)), cf. PARI code. - M. F. Hasler, Oct 10 2019
T(n, k) = exp(1)*(Gamma(n+1, 1) - Gamma(n-k, 1)*(n-k)*n!/(n-k)!) - 1. - Peter Luschny, Oct 10 2019
Sum-free and Gamma-free formula: T(m, n) = b(m) - m!*b(m-n)/(m-n)! where b(0)=0, b(j)=floor(j!*e-1) for j>0. - Manfred Boergens, Jun 22 2022

cp's OEIS Frontend

A002104 Logarithmic numbers.

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A337001 a(n) = n! * Sum_{k=0..n} k^3 / k!.

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A337002 a(n) = n! * Sum_{k=0..n} k^4 / k!.

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A111063 a(0) = 1; a(n) = (n-1)*a(n-1) + n.

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A006183 a(n) = (n+1)*a(n-1) + (2-n)*a(n-2).

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A121662 Triangle read by rows: T(i,j) for the recurrence T(i,j) = (T(i-1,j) + 1)*i.

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A337085 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} j^k/j!.

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A006183 a(n) = (n+1)a(n-1) + (2-n)a(n-2).