A066534 -id:A066534 - cp's OEIS frontend

A007526 a(n) = n*(a(n-1) + 1), a(0) = 0.

0, 1, 4, 15, 64, 325, 1956, 13699, 109600, 986409, 9864100, 108505111, 1302061344, 16926797485, 236975164804, 3554627472075, 56874039553216, 966858672404689, 17403456103284420, 330665665962403999, 6613313319248080000, 138879579704209680021, 3055350753492612960484

Offset: 0

N. J. A. Sloane and Robert G. Wilson v

Eighteenth- and nineteenth-century combinatorialists call this the number of (nonnull) "variations" of n distinct objects, namely the number of permutations of nonempty subsets of {1,...,n}. Some early references to this sequence are Izquierdo (1659), Caramuel de Lobkowitz (1670), Prestet (1675) and Bernoulli (1713). - Don Knuth, Oct 16 2001, Aug 16 2004
Stirling transform of A006252(n-1) = [0,1,1,2,4,14,38,...] is a(n-1) = [0,1,4,15,64,...]. - Michael Somos, Mar 04 2004
In particular, for n >= 1 a(n) is the number of nonempty sequences with n or fewer terms, each a distinct element of {1,...,n}. - Rick L. Shepherd, Jun 08 2005
a(n) = VarScheme(1,n). See A128195 for the definition of VarScheme(k,n). - Peter Luschny, Feb 26 2007
if s(n) is a sequence of the form s(0)=x, s(n)= n(s(n-1)+k), then s(n)= n!*x + a(n)*k. - Gary Detlefs, Jun 06 2010
Exponential convolution of factorials (A000142) and nonnegative integers (A001477). - Vladimir Reshetnikov, Oct 07 2016
For n > 0, a(n) is the number of maps f: {1,...,n} -> {1,...,n} satisfying equal(x,y) <= equal(f(x),f(y)) for all x,y, where equal(x,y) is n if x and y are equal and min(x,y) if not. Here equal(x,y) is the equality predicate in the n-valued Gödel logic, see e. g. the Wikipedia chapter on many-valued logics. - Mamuka Jibladze, Mar 12 2025

			G.f. = x + 4*x^2 + 15*x^3 + 64*x^4 + 325*x^5 + 1956*x^6 + 13699*x^7 + ...
Consider the nonempty subsets of the set {1,2,3,...,n} formed by the first n integers. E.g., for n = 3 we have {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. For each subset S we determine its number of parts, that is nprts(S). The sum over all subsets is written as sum_{S=subsets}. Then we have A007526 = Sum_{S=subsets} nprts(S)!. E.g., for n = 3 we have 1!+1!+1!+2!+2!+2!+3! = 15. - _Thomas Wieder_, Jun 17 2006
a(3)=15: Let the objects be a, b, and c. The fifteen nonempty ordered subsets are {a}, {b}, {c}, {ab}, {ba}, {ac}, {ca}, {bc}, {cb}, {abc}, {acb}, {bac}, {bca}, {cab} and {cba}.

Jacob Bernoulli, Ars Conjectandi (1713), page 127.
Johannes Caramuel de Lobkowitz, Mathesis Biceps Vetus et Nova (Campania: 1670), volume 2, 942-943.
J. K. Horn, personal communication to Robert G. Wilson v.
Sebastian Izquierdo, Pharus Scientiarum (Lyon: 1659), 327-328.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)
J. L. Adams, Conceptual Blockbusting: A Guide to Better Ideas, Freeman, San Francisco, 1974. [Annotated scans of pages 69 and 70 only]
J. Bernoulli, Wahrscheinlichkeitsrechnung (Ars conjectandi) von Jakob Bernoulli (1713) Uebers. und hrsg. von R. Haussner, Leipzig, W. Engelmann, (1899), [124] Kapitel VII. Variationen ohne Wiederholung. (Page 121).
Oscar Cabrera, Introducing loop compression for encoding de Bruijn sequences, engrXiv (2025) Art. No. 4431. See p. 16.
Peter J. Freyd, Core algebra revisited, Theoretical Computer Science, 375 (2007), Issues 1-3, 193-200.
Z. Kasa and Z. Katai, Scattered subwords and composition of natural numbers, Acta Univ. Sapientiae, Informatica, 4, 2 (2012) 225-236. - From _N. J. A. Sloane_, Feb 21 2013
Jean Prestet, Elemens des Mathematiques, (1675), page 341.
Joe Sawada and A. Williams, Successor rules for flipping pancakes and burnt pancakes, Preprint 2015.
Elmar Teufl and Stephan Wagner, Enumeration problems for classes of self-similar graphs, Journal of Combinatorial Theory, Series A, Volume 114, Issue 7, October 2007, Pages 1254-1277.

Row sums of A068424.
Partial sums of A001339.
Column k=1 of A326659.
Cf. A000522, A007526, A001339, A128195.

a:=[0];; for n in [2..25] do a[n]:=(n-1)*(a[n-1]+1); od; a; # Muniru A Asiru, Aug 07 2018

a007526 n = a007526_list !! n
a007526_list = 0 : zipWith (*) [1..] (map (+ 1) a007526_list)
-- Reinhard Zumkeller, Aug 27 2013

A007526 := n -> add(n!/k!,k=0..n) - 1;
a := n -> n*hypergeom([1,1-n],[],-1):
seq(simplify(a(n)), n=0..22); # Peter Luschny, May 09 2017
# third Maple program:
a:= proc(n) option remember;
      `if`(n<0, 0, n*(1+a(n-1)))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jan 06 2020

Table[ Sum[n!/(n - r)!, {r, 1, n}], {n, 0, 20}] (* or *) Table[n!*Sum[1/k!, {k, 0, n - 1}], {n, 0, 20}]
a=1;Table[a=(a-1)*(n-1);Abs[a],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 20 2009 *)
FoldList[#1*#2 + #2 &, 0, Range[19]] (* Robert G. Wilson v, Jul 07 2012 *)
f[n_] := Floor[E*n! - 1]; f[0] = 0; Array[f, 20, 0] (* Robert G. Wilson v, Feb 06 2015 *)
a[n_] := n (a[n - 1] +1); a[0] = 0; Array[a, 20, 0] (* Robert G. Wilson v, Feb 06 2015 *)
Round@Table[E n Gamma[n, 1], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Oct 07 2016 *)

{a(n) = if( n<1, 0, n * (a(n-1) + 1))}; /* Michael Somos, Apr 06 2003 */

{a(n) = if( n<0, 0, n! * polcoeff(x * exp(x + x * O(x^n)) / (1 - x), n))}; /* Michael Somos, Mar 04 2004 */

PARI
```
a(n)= sum(k=1,n, prod(j=0,k-1,n-j))
```

a(n) = A000522(n) - 1.
a(n) = floor(e*n! - 1). - Joseph K. Horn
a(n) = Sum_{r=1..n} A008279(n, r)= n!*(Sum_{k=0..n-1} 1/k!).
a(n) = n*(a(n-1) + 1).
E.g.f.: x*exp(x)/(1-x). - Vladeta Jovovic, Aug 25 2002
a(n) = Sum_{k=1..n} k!*C(n, k). - Benoit Cloitre, Dec 06 2002
a(n) = Sum_{k=0..n-1} (n! / k!). - Ross La Haye, Sep 22 2004
a(n) = Sum_{k=1..n} (Product_{j=0..k-1} (n-j)). - Joerg Arndt, Apr 24 2011
Binomial transform of n! - !n. - Paul Barry, May 12 2004
Inverse binomial transform of A066534. - Ross La Haye, Sep 16 2004
For n > 0, a(n) = exp(1) * Integral_{x>=0} exp(-exp(x/n)+x) dx. - Gerald McGarvey, Oct 19 2006
a(n) = Integral_{x>=0} (((1+x)^n-1)*exp(-x)). - Paul Barry, Feb 06 2008
a(n) = GAMMA(n+2)*(1+(-GAMMA(n+1)+exp(1)*GAMMA(n+1, 1))/GAMMA(n+1)). - Thomas Wieder, May 02 2009
E.g.f.: -1/G(0) where G(k) = 1 - 1/(x - x^3/(x^2+(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 10 2012
Conjecture : a(n) = (n+2)*a(n-1) - (2*n-1)*a(n-2) + (n-2)*a(n-3). - R. J. Mathar, Dec 04 2012 [Conjecture verified by Robert FERREOL, Aug 04 2018]
G.f.: (Q(0) - 1)/(1-x), where Q(k)= 1 + (2*k + 1)*x/( 1 - x - 2*x*(1-x)*(k+1)/(2*x*(k+1) + (1-x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 09 2013
G.f.: 2/((1-x)*G(0)) - 1/(1-x), where G(k)= 1 + 1/(1 - x*(2*k+2)/(x*(2*k+3) - 1 + x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
a(n) = (...((((((0)+1)*1+1)*2+1)*3+1)*4+1)...*n). - Bob Selcoe, Jul 04 2013
G.f.: Q(0)/(2-2*x) - 1/(1-x), where Q(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) + (1-x)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 09 2013
G.f.: (W(0) - 1)/(1-x), where W(k) = 1 - x*(k+1)/( x*(k+2) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 25 2013
For n > 0: a(n) = n*A000522(n-1). - Reinhard Zumkeller, Aug 27 2013
a(n) = (...(((((0)*1+1)*2+2)*3+3)*4+4)...*n+n). - Bob Selcoe, Apr 30 2014
0 = 1 + a(n)*(+1 + a(n+1) - a(n+2)) + a(n+1)*(+2 +a(n+1)) - a(n+2) for all n >= 0. - Michael Somos, Aug 30 2016
a(n) = n*hypergeom([1, 1-n], [], -1). - Peter Luschny, May 09 2017
Product_{n>=1} (a(n)+1)/a(n) = e, coming from Product_{n=1..N}(a(n)+1)/a(n) = Sum_{n=0..N} 1/n!. - Robert FERREOL, Jul 12 2018
O.g.f.: Sum_{k>=1} k^k*x^k/(1 + (k - 1)*x)^(k+1). - Ilya Gutkovskiy, Oct 09 2018

A010842 Expansion of e.g.f.: exp(2*x)/(1-x).

Original entry on oeis.org

1, 3, 10, 38, 168, 872, 5296, 37200, 297856, 2681216, 26813184, 294947072, 3539368960, 46011804672, 644165281792, 9662479259648, 154599668219904, 2628194359869440, 47307498477912064, 898842471080853504, 17976849421618118656, 377513837853982588928

Offset: 0

N. J. A. Sloane, Simon Plouffe

Incomplete Gamma Function at 2, more precisely: a(n) = exp(2)*Gamma(1+n,2).
Let P(A) be the power set of an n-element set A. Then a(n) = the total number of ways to add 0 or more elements of A to each element x of P(A) where the elements to add are not elements of x and order of addition is important. - Ross La Haye, Nov 19 2007
a(n) is the number of ways to split the set {1,2,...,n} into two disjoint subsets S,T with S union T = {1,2,...,n} and linearly order S and then choose a subset of T. - Geoffrey Critzer, Mar 10 2009

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.1.2.

T. D. Noe, Table of n, a(n) for n = 0..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Cf. A053484, A053485, A053486, A008290, A137346.

A010843, A000023, A000166, A000142, A000522, A010842, A053486, A053487 have similar properties.

m:=45; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(2*x)/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 16 2018

G(x):=exp(2*x)/(1-x): f[0]:=G(x): for n from 1 to 19 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..19); # Zerinvary Lajos, Apr 03 2009
seq(simplify(exp(1)^2*GAMMA(n+1, 2)), n=0..19); # Peter Luschny, Apr 28 2016
seq(simplify(KummerU(-n, -n, 2)), n=0..21); # Peter Luschny, May 10 2022

With[{r = Round[n! E^2 - 2^(n + 1)/(n + 1)]}, r - Mod[r, 2^(n - Floor[2/n + Log2[n]])]] (* for n>=4; Stan Wagon, Apr 28 2016 *)
a[n_] := n! Sum[2^i/i!, {i, 0, n}]
Table[a[n], {n, 0, 21}] (* Gerry Martens , May 06 2016 *)
With[{nn=30},CoefficientList[Series[Exp[2x]/(1-x),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, May 27 2019 *)

x='x+O('x^44); Vec(serlaplace(exp(2*x)/(1-x))) \\ Joerg Arndt, Apr 29 2016

a(n) = row sums of A090802. - Ross La Haye, Aug 18 2006
a(n) = n*a(n-1) + 2^n = (n+2)*a(n-1) - (2*n-2)*a(n-2) = n!*Sum_{j=0..n} floor(2^j/j!). - Henry Bottomley, Jul 12 2001
a(n) is the permanent of the n X n matrix with 3's on the diagonal and 1's elsewhere. a(n) = Sum_{k=0..n} A008290(n, k)*3^k. - Philippe Deléham, Dec 12 2003
Binomial transform of A000522. - Ross La Haye, Sep 15 2004
a(n) = Sum_{k=0..n} k!*binomial(n, k)*2^(n-k). - Paul Barry, Apr 22 2005
a(n) = A066534(n) + 2^n. - Ross La Haye, Nov 16 2005
G.f.: hypergeom([1,k],[],x/(1-2*x))/(1-2*x) with k=1,2,3 is the generating function for A010842, A081923, and A082031. - Mark van Hoeij, Nov 08 2011
E.g.f.: 1/E(0), where E(k) = 1 - x/(1-2/(2+(k+1)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011
G.f.: 1/Q(0), where Q(k)= 1 - 2*x - x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 18 2013
a(n) ~ n! * exp(2). - Vaclav Kotesovec, Jun 01 2013
From Peter Bala, Sep 25 2013: (Start)
a(n) = n!*e^2 - Sum_{k >= 0} 2^(n + k + 1)/((n + 1)*...*(n + k + 1)).
= n!*e^2 - e^2*( Integral_{t = 0..2} t^n*exp(-t) dt )
= e^2*( Integral_{t >= 2} t^n*exp(-t) dt )
= e^2*( Integral_{t >= 0} t^n*exp(-t)*Heaviside(t-2) dt ),
an integral representation of a(n) as the n-th moment of a nonnegative function on the positive half-axis.
Bottomley's second-order recurrence above a(n) = (n + 2)*a(n-1) - 2*(n - 1)*a(n-2) has n! as a second solution. This yields the finite continued fraction expansion a(n)/n! = 1/(1 - 2/(3 - 2/(4 - 4/(5 - ... - 2*(n - 1)/(n + 2))))) valid for n >= 2. Letting n tend to infinity gives the infinite continued fraction expansion e^2 = 1/(1 - 2/(3 - 2/(4 - 4/(5 - ... - 2*(n - 1)/(n + 2 - ...))))). (End)
a(n) = 2^(n+1)*U(1, n+2, 2), where U is the Bessel U function. - Peter Luschny, Nov 26 2014
For n >= 4, a(n) = r - (r mod 2^(n - floor((2/n) + log_2(n)))) where r = n! * e^2 - 2^(n+1)/(n+1). - Stan Wagon, Apr 28 2016
G.f.: A(x) = 1/(1 - 2*x - x/(1 - x/(1 - 2*x - 2*x/(1 - 2*x/(1 - 2*x - 3*x/(1 - 3*x/(1 - 2*x - 4*x/(1 - 4*x/(1 - 2*x - ... ))))))))). - Peter Bala, May 26 2017
a(n) = Sum_{k=0..n} (-1)^(n-k)*A137346(n, k). - Mélika Tebni, May 10 2022 [This is equivalent to a(n) = KummerU(-n, -n, 2). - Peter Luschny, May 10 2022]
a(n) = F(n), where the function F(x) := 2^(x+1) * Integral_{t >= 0} e^(-2*t)*(1 + t)^x dt smoothly interpolates this sequence to all real values of x. - Peter Bala, Sep 05 2023

A090802 Triangle read by rows: a(n,k) = number of k-length walks in the Hasse diagram of a Boolean algebra of order n.

Original entry on oeis.org

1, 2, 1, 4, 4, 2, 8, 12, 12, 6, 16, 32, 48, 48, 24, 32, 80, 160, 240, 240, 120, 64, 192, 480, 960, 1440, 1440, 720, 128, 448, 1344, 3360, 6720, 10080, 10080, 5040, 256, 1024, 3584, 10752, 26880, 53760, 80640, 80640, 40320

Offset: 0

Ross La Haye, Feb 10 2004

Row sums = A010842(n); Row sums from column 1 on = A066534(n) = n*A010842(n-1) = A010842(n) - 2^n.
a(n,k) = n! = k! = A000142(n) for n = k; a(n,n-1) = 2*n! = A052849(n) for n > 1; a(n,n-2) = 2*n! = A052849(n) for n > 2; a(n,n-3) = (4/3)*n! = A082569(n) for n > 3; a(n,n-1)/a(2,1) = n!/2! = A001710(n) for n > 1; a(n,n-2)/ a(3,1) = n!/3! = A001715(n) for n > 2; a(n,n-3)/a(4,1) = n!/4! = A001720(n) for n > 3.
a(2k, k) = A052714(k+1). a(2k-1, k) = A034910(k).
a(n,0) = A000079(n); a(n,1) = A001787(n) = row sums of A003506; a(n,2) = A001815(n) = 2!*A001788(n-1); a(n,3) = A052771(n) = 3!*A001789(n); a(n,4) = A052796(n) = 4!*A003472(n); ceiling[a(n,1) / 2] = A057711(n); a(n,5) = 5!*A054849(n).
In a class of n students, the number of committees (of any size) that contain an ordered k-sized subcommittee is a(n,k). - Ross La Haye, Apr 17 2006
Antidiagonal sums [1,2,5,12,30,76,198,528,1448,4080,...] appear to be binomial transform of A000522 interleaved with itself, i.e., 1,1,2,2,5,5,16,16,65,65,... - Ross La Haye, Sep 09 2006
Let P(A) be the power set of an n-element set A. Then a(n,k) = the number of ways to add k elements of A to each element x of P(A) where the k elements are not elements of x and order of addition is important. - Ross La Haye, Nov 19 2007
The derivatives of x^n evaluated at x=2. - T. D. Noe, Apr 21 2011

			{1};
{2, 1};
{4, 4, 2};
{8, 12, 12, 6};
{16, 32, 48, 48, 24};
{32, 80, 160, 240, 240, 120};
{64, 192, 480, 960, 1440, 1440, 720};
{128, 448, 1344, 3360, 6720, 10080, 10080, 5040};
{256, 1024, 3584, 10752, 26880, 53760, 80640, 80640, 40320}
a(5,3) = 240 because P(5,3) = 60, 2^(5-3) = 4 and 60 * 4 = 240.

T. D. Noe, Rows n = 0..100, flattened
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Eric Weisstein, Walk
Eric Weisstein, Boolean Algebra
Eric Weisstein, Hasse Diagram

Cf. A000142, A001710, A001715, A001720, A001787, A001788, A001789, A001815, A003472, A010842, A052771, A052796, A052849, A054849, A057711, A066534, A082569.

Cf. A038207, A007318.

Flatten[Table[n!/(n-k)! * 2^(n-k), {n, 0, 8}, {k, 0, n}]] (* Ross La Haye, Feb 10 2004 *)

a(n, k) = 0 for n < k. a(n, k) = k!*C(n, k)*2^(n-k) = P(n, k)*2^(n-k) = (2n)!!/((n-k)!*2^k) = k!*A038207(n, k) = A068424*2^(n-k) = Sum[C(n, m)*P(n-m, k), {m, 0, n-k}] = Sum[C(n, n-m)*P(n-m, k), {m, 0, n-k}] = n!*Sum[1/(m!*(n-m-k)!), {m, 0, n-k}] = k!*Sum[C(n, m)*C(n-m, k), {m, 0, n-k}] = k!*Sum[C(n, n-m)*C(n-m, k), {m, 0, n-k}] = k!*C(n, k)*Sum[C(n-k, n-m-k), {m, 0, n-k}] = k!*C(n, k)*Sum[C(n-k, m), {m, 0, n-k}] for n >= k.
a(n, k) = 0 for n < k. a(n, k) = n*a(n-1, k-1) for n >= k >= 1.
E.g.f. (by columns): exp(2x)*x^k.

More terms from Ray Chandler, Feb 26 2004

Entry revised by Ross La Haye, Aug 18 2006

A346394 Expansion of e.g.f. -log(1 - x) * exp(2*x).

Original entry on oeis.org

0, 1, 5, 20, 78, 324, 1520, 8336, 53872, 405600, 3492416, 33798016, 362543104, 4264455168, 54540715008, 753246711808, 11168972683264, 176937613586432, 2982069587042304, 53271637651996672, 1005385746384846848, 19987620914387812352, 417489079682758213632

Offset: 0

Ilya Gutkovskiy, Jul 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A002104, A010842, A066534, A346395, A346396.

nmax = 22; CoefficientList[Series[-Log[1 - x] Exp[2 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[2^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 22}]

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(i+1)*v[i]-2*(i-1)*v[i-1]+2^(i-1)); v; \\ Seiichi Manyama, May 27 2022

a(n) = n! * Sum_{k=0..n-1} 2^k / ((n-k) * k!).
a(n) = Sum_{k=0..n} binomial(n,k) * A002104(k).
a(n) ~ exp(2) * (n-1)!. - Vaclav Kotesovec, Aug 09 2021
a(0) = 0, a(1) = 1, a(n) = (n+1) * a(n-1) - 2 * (n-1) * a(n-2) + 2^(n-1). - Seiichi Manyama, May 27 2022

A335111 a(n) = n! * Sum_{k=0..n-1} (-2)^k / k!.

Original entry on oeis.org

0, 1, -2, 6, -8, 40, 48, 784, 5248, 49536, 490240, 5403904, 64822272, 842742784, 11798284288, 176974510080, 2831591636992, 48137058942976, 866467058614272, 16462874118651904, 329257482362552320, 6914407129635618816, 152116956851937476608, 3498690007594658430976

Offset: 0

Ilya Gutkovskiy, May 23 2020

sign

Inverse binomial transform of A000240.

Cf. A000023, A000240, A008290, A066534, A087981, A122803.

Table[n! Sum[(-2)^k/k!, {k, 0, n - 1}], {n, 0, 23}]
nmax = 23; CoefficientList[Series[Sum[k! x^k/(1 + 2 x)^(k + 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[x Exp[-2 x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = n! * sum(k=0, n-1, (-2)^k / k!); \\ Michel Marcus, May 23 2020

G.f.: Sum_{k>=1} k! * x^k / (1 + 2*x)^(k + 1).
E.g.f.: x*exp(-2*x) / (1 - x).
a(n) = A000023(n) - A122803(n).
a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Jun 08 2022
a(n) = Sum_{k=0..n} (-1)^k * k * A008290(n,k). - Alois P. Heinz, May 20 2023

A348312 a(n) = n! * Sum_{k=0..n-1} 3^k / k!.

Original entry on oeis.org

0, 1, 8, 51, 312, 1965, 13248, 97839, 800208, 7260921, 72806040, 801515979, 9620317512, 125071036389, 1751016829968, 26265324194055, 420245416687392, 7144172815479921, 128595113003161512, 2443307154421058019, 48866143111666389720, 1026189005418216656541, 22576158119430894214368

Offset: 0

Ilya Gutkovskiy, Oct 11 2021

nonn

Cf. A007526, A053486, A066534, A348314.

Table[n! Sum[3^k/k!, {k, 0, n - 1}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[x Exp[3 x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = n!*sum(k=0, n-1, 3^k/k!); \\ Michel Marcus, Oct 11 2021

E.g.f.: x * exp(3*x) / (1 - x).
a(0) = 0; a(n) = n * (a(n-1) + 3^(n-1)).
a(n) ~ exp(3)*n!. - Stefano Spezia, Oct 11 2021

A348314 a(n) = n! * Sum_{k=0..n-1} 4^k / k!.

Original entry on oeis.org

0, 1, 10, 78, 568, 4120, 30864, 244720, 2088832, 19389312, 196514560, 2173194496, 26128665600, 339890756608, 4759410116608, 71395178280960, 1142340032364544, 19419853564641280, 349557673401188352, 6641597100292636672, 132831947503410872320, 2789470920661372502016

Offset: 0

Ilya Gutkovskiy, Oct 11 2021

nonn

Cf. A007526, A053487, A066534, A348312.

Table[n! Sum[4^k/k!, {k, 0, n - 1}], {n, 0, 21}]
nmax = 21; CoefficientList[Series[x Exp[4 x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = n!*sum(k=0, n-1, 4^k/k!); \\ Michel Marcus, Oct 11 2021

E.g.f.: x * exp(4*x) / (1 - x).
a(0) = 0; a(n) = n * (a(n-1) + 4^(n-1)).
a(n) ~ exp(4)*n!. - Stefano Spezia, Oct 11 2021

A137387 Triangular sequence from coefficients of the expansion of p(x,t)=Exp[2xt]*t/(1 - t).

Original entry on oeis.org

0, 1, 2, 4, 6, 12, 12, 24, 48, 48, 32, 120, 240, 240, 160, 80, 720, 1440, 1440, 960, 480, 192, 5040, 10080, 10080, 6720, 3360, 1344, 448, 40320, 80640, 80640, 53760, 26880, 10752, 3584, 1024, 362880, 725760, 725760, 483840, 241920, 96768, 32256, 9216

Offset: 1

Roger L. Bagula, Apr 26 2008

Row sums = A066534.

			{0},
{1},
{2, 4},
{6, 12, 12},
{24, 48, 48, 32},
{120, 240, 240, 160, 80},
{720, 1440, 1440, 960, 480, 192},
{5040, 10080, 10080, 6720, 3360, 1344, 448},
{40320, 80640, 80640, 53760, 26880, 10752, 3584, 1024},
{362880, 725760, 725760, 483840, 241920, 96768, 32256, 9216, 2304},
{3628800, 7257600, 7257600, 4838400, 2419200, 967680, 322560, 92160, 23040,5120}

Terrell Hill, Statistical Mechanics, Dover, 1987, page 417

Cf. A066534.

p[t_] = Exp[2*x*t]*t/(1 - t); Table[ ExpandAll[n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]

p(x,t)=Exp[2*x*t]*t/(1 - t)=Sum[P(x,n)*t6n/n!,{n,1,Infinity}]; out_n,m=n!*Coefficients(P(x,n)).

A359110 Number of Boolean monoids of order 2^n up to isomorphism.

Original entry on oeis.org

1, 5, 83, 242547

Offset: 1

Choiwah Chow, Dec 18 2022

MathStructures, Boolean Monoids.

Cf. A090802, A066534, A198085, A342908, A058129, A058129.

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