A000522 -id:A000522 - cp's OEIS frontend

A186763 Number of increasing odd cycles in all permutations of {1,2,...,n}.

0, 1, 2, 7, 28, 141, 846, 5923, 47384, 426457, 4264570, 46910271, 562923252, 7318002277, 102452031878, 1536780478171, 24588487650736, 418004290062513, 7524077221125234, 142957467201379447, 2859149344027588940, 60042136224579367741

Offset: 0

Emeric Deutsch, Feb 27 2011

nonn

A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)

A cycle is said to be odd if it has an odd number of entries.

			a(3)=7 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), and (132) we have a total of 3+1+1+1+1+0=7 increasing odd cycles.

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

Cf. A000166, A000522, A009628, A186761, A186762, A186764, A186765, A186766, A186768, A186769, A184958.

g := sinh(z)/(1-z): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);
# Alternatively:
A186763 := n -> (exp(1)*GAMMA(1+n,1) - exp(-1)*GAMMA(1+n,-1))/2:
seq(simplify(A186763(n)), n=0..21); # Peter Luschny, Dec 18 2017

a=0;Table[a=n*a+(1/2)(1-(-1)^n),{n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)
CoefficientList[Series[Sinh[x]/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 05 2013 *)

a(n) = Sum_{k>=0} k*A186761(n,k).
E.g.f.: (sinh z)/(1-z).
a(n) ~ n! * (exp(2)-1)*exp(-1)/2. - Vaclav Kotesovec, Oct 05 2013
a(n) = (exp(1)*Gamma(1+n,1) - exp(-1)*Gamma(1+n,-1))/2 = (A000522(n) - A000166(n))/2. - Peter Luschny, Dec 18 2017
a(n) = n! * Sum_{k=0..floor((n-1)/2)} 1 / (2*k+1)!. - Ilya Gutkovskiy, Jul 16 2021
D-finite with recurrence a(n) -n*a(n-1) -a(n-2) +(n-2)*a(n-3)=0. - R. J. Mathar, Jul 26 2022

A006231 a(n) = Sum_{k=2..n} n(n-1)...(n-k+1)/k.

Original entry on oeis.org

0, 1, 5, 20, 84, 409, 2365, 16064, 125664, 1112073, 10976173, 119481284, 1421542628, 18348340113, 255323504917, 3809950976992, 60683990530208, 1027542662934897, 18430998766219317, 349096664728623316, 6962409983976703316, 145841989688186383337

Offset: 1

R. K. Guy

a(n) is also the number of permutations in the symmetric group S_n that are pure cycles, see example. - Avi Peretz (njk(AT)netvision.net.il), Mar 24 2001
Also the number of elementary circuits in a complete directed graph with n nodes [D. B. Johnson, 1975]. - N. J. A. Sloane, Mar 24 2014
If one takes 1,2,3,4, ..., n and starts creating parenthetic products of k-tuples and adding, one gets a(n+1). For 1,2,3,4 one gets (1)+(2)+(3)+(4) = 10; (1*2)+(2*3)+(3*4) = 20; (1*2*3)+(2*3*4) = 30; (1*2*3*4) = 24; and 10+20+30+24 = 84 = a(5). - J. M. Bergot, Apr 24 2014
Let P_n be the set of probability distributions over orderings of n objects that can be obtained by drawing n real numbers from independent probability distributions and sorting. Then a(n) is conjectured to be the dimension of P_n, as a semi-algebraic subset of R^(n!). - Jamie Tucker-Foltz, Jul 29 2024

			a(3) = 5 because the cycles in S_3 are (12), (13), (23), (123), (132).
a(4) = 20 because there are 24 permutations of {1,2,3,4} but we don't count (12)(34), (13)(24), (14)(23) or the identity permutation. - _Geoffrey Critzer_, Nov 03 2012

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 = 1..450 (first 100 terms from T. D. Noe)
Eric Babson, Moon Duchin, Annina Iseli, Pietro Poggi-Corradini, Dylan Thurston, Jamie Tucker-Foltz, Models of Random Spanning Trees, arXiv:2407.20226 [math.CO], 2023.
R. K. Guy, Letter to N. J. A. Sloane, 1977
Donald B. Johnson, Finding all the elementary circuits of a directed graph, SIAM J. Comput. 4 (1975), 77-84. MR0398155 (53 #2010).
Jamie Tucker-Foltz, Code to compute dimension of P_n on GitHub.

Cf. A059760, A000295.

Column k=1 of A136394.

a006231 n = numerator $
   sum $ tail $ zipWith (%) (scanl1 (*) [n,(n-1)..1]) [1..n]
-- Reinhard Zumkeller, Dec 27 2011

A006231 := proc(n)
    n*( hypergeom([1,1,1-n],[2],-1)-1) ;
    simplify(%) ;
end proc: # R. J. Mathar, Aug 06 2013

a[n_] = n*(HypergeometricPFQ[{1,1,1-n}, {2}, -1] - 1); Table[a[n], {n, 1, 20}] (* Jean-François Alcover,  Mar 29 2011 *)
Table[Sum[Times@@Range[n-k+1,n]/k,{k,2,n}],{n,20}] (* Harvey P. Dale, Sep 23 2011 *)

a(n) = n--; sum(ip=1, n, sum(j=1, n-ip+1, prod(k=j, j+ip-1, k))); \\ Michel Marcus, May 07 2014 after comment by J. M. Bergot

a(n+1) - a(n) = A000522(n) - 1.
a(n) = n*( 3F1(1,1,1-n; 2;-1) -1). - Jean-François Alcover,  Mar 29 2011
E.g.f.: exp(x)*(log(1/(1-x))-x). - Geoffrey Critzer, Sep 12 2012
G.f.: (Q(0) - 1)/(1-x)^2, 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
Conjecture: a(n) + (-n-2)*a(n-1) + (3*n-2)*a(n-2) + 3*(-n+2)*a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Aug 06 2013

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

A009179 E.g.f. cosh(x)/(1+x).

Original entry on oeis.org

1, -1, 3, -9, 37, -185, 1111, -7777, 62217, -559953, 5599531, -61594841, 739138093, -9608795209, 134523132927, -2017846993905, 32285551902481, -548854382342177, 9879378882159187, -187708198761024553, 3754163975220491061

Offset: 0

R. H. Hardin

Unsigned sequence satisfies a(n)=n*a(n-1)+a(n-2)-(n-2)*a(n-3), a(0)=1,a(1)=1,a(2)=3 with e.g.f. cosh(z)/(1-z). - Mario Catalani (mario.catalani(AT)unito.it), Feb 07 2003
(-1)^n*(A000166(n) + A000522(n))/2 = this_sequence, (-1)^n*(A000166(n) - A000522(n))/2 = A009628(n).
The positive sequence has e.g.f. cosh(x)/(1-x), with a(n)=sum{k=0..floor(n/2), binomial(n,2k)(n-2k)!}. It is the mean of the binomial and inverse binomial transforms of n!. - Paul Barry, May 01 2005

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

Cf. A000166, A000522, A001540, A009628.

restart: G(x):= cosh(x)/(1+x): f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..20); # Zerinvary Lajos, Apr 03 2009

a[n_] := (-1)^n (Exp[1] Gamma[1 + n, 1] + Exp[-1] Gamma[1 + n, -1])/2;
Table[a[n], {n, 0, 20}] (* Peter Luschny, Dec 18 2017 *)

x='x+O('x^99); Vec(serlaplace(cosh(x)/(1+x))) \\ Altug Alkan, Dec 18 2017

a(n) = (-1)^n*floor(n!*cosh(1)). - Vladeta Jovovic, Aug 10 2002
a(n) = (1+(-1)^n)/2-n*a(n-1). - Vladeta Jovovic, Apr 19 2003
a(n) = (-1)^n * n! * sum{k=0, [n/2], 1/(2k)!}.
E.g.f.: U(0)/(1+x) where U(k)= 1 + x^2/((4*k+1)*(4*k+2) - x^2*(4*k+1)*(4*k+2)/(x^2 + (4*k+3)*(4*k+4)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2012
a(n) = (-1)^n*(exp(1)*Gamma(1+n,1) + exp(-1)*Gamma(1+n,-1))/2 - Peter Luschny, Dec 18 2017

Extended with signs by Olivier Gérard, Mar 15 1997

A010845 a(n) = 3na(n-1) + 1, a(0) = 1.

Original entry on oeis.org

1, 4, 25, 226, 2713, 40696, 732529, 15383110, 369194641, 9968255308, 299047659241, 9868572754954, 355268619178345, 13855476147955456, 581929998214129153, 26186849919635811886, 1256968796142518970529

Offset: 0

Simon Plouffe

a(n)/(A000142*A000244) is an increasingly good approximation to cube root of e.
Related to Incomplete Gamma Function at 1/3. - Michael Somos, Mar 26 1999
For positive n, a(n) equals 3^n times the permanent of the n X n matrix with (4/3)'s along the main diagonal, and 1's everywhere else. - John M. Campbell, Jul 10 2011

			1 + 4*x + 25*x^2 + 226*x^3 + 2713*x^4 + 40696*x^5 + 732529*x^6 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
M. Z. Spivey and L. L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.

Cf. A000522, A010844, A056545, A056546, A056547 for analogs.

Table[ Gamma[ n, 1/3 ]*Exp[ 1/3 ]*3^(n-1), {n, 1, 24} ]
a[ n_] := If[ n<0, 0, Floor[ n! E^(1/3) 3^n ]] (* Michael Somos, Sep 04 2013 *)
Range[0, 20]! CoefficientList[Series[Exp[x]/(1 - 3 x), {x, 0, 20}], x] (* Vincenzo Librandi, Feb 17 2014 *)

{a(n) = if( n<0, 0, n! * sum(k=0, n, 3^(n-k) / k!))} /* Michael Somos, Sep 04 2013 */

E.g.f.: exp(x)/(1-3*x).
a(n) = floor( n!*e^(1/3)*3^n ) = n! * (Sum_{k=0..n} 3^(n-k) / k!) = n! * (e^(1/3) * 3^n - Sum_{k>n} 3^(n-k) / k!). - Michael Somos, Mar 26 1999
a(n) = Sum_{k=0..n} P(n, k)*3^k. - Ross La Haye, Aug 29 2005
Binomial transform of A032031. - Carl Najafi, Sep 11 2011
Conjecture: a(n) +(-3*n-1)*a(n-1) +3*(n-1)*a(n-2)=0. - R. J. Mathar, Feb 16 2014
a(n) = hypergeometric_U(1,n+2,1/3)/3. - Peter Luschny, Nov 26 2014
From Peter Bala, Mar 01 2017: (Start)
a(n) = Integral_{x >= 0} (3*x + 1)^n*exp(-x) dx.
The e.g.f. y = exp(x)/(1 - 3*x) satisfies the differential equation (1 - 3*x)*y' = (4 - 3*x)*y. Mathar's recurrence above follows easily from this.
The sequence b(n) := (3^n)*n! also satisfies Mathar's recurrence with b(0) = 1, b(1) = 3. This leads to the continued fraction representation a(n) = (3^n)*n!*( 1 + 1/(3 - 3/(7 - 6/(10 - ... - (3*n - 3)/(3*n + 1) )))) for n >= 2. Taking the limit as n -> oo gives the continued fraction representation exp(1/3) = 1 + 1/(3 - 3/(7 - 6/(10 - ... - (3*n - 3)/((3*n + 1) - ... )))). Cf. A010844. (End)

Better description and formulas from Michael Somos

More terms from James Sellers, Jul 04 2000

A019460 Add 1, multiply by 1, add 2, multiply by 2, etc., start with 2.

Original entry on oeis.org

2, 3, 3, 5, 10, 13, 39, 43, 172, 177, 885, 891, 5346, 5353, 37471, 37479, 299832, 299841, 2698569, 2698579, 26985790, 26985801, 296843811, 296843823, 3562125876, 3562125889, 46307636557, 46307636571, 648306911994, 648306912009, 9724603680135, 9724603680151, 155593658882416

Offset: 0

N. J. A. Sloane

After a(7) = 43, the next prime in the sequence is a(649) with 676 digits. - M. F. Hasler, Jan 12 2011

New York Times, Oct 13, 1996.

Ivan Panchenko, Table of n, a(n) for n = 0..200
Nick Hobson, Python program for this sequence

Cf. A019461 (same, but start with 0), A019463 (start with 1), A019462 (start with 3), A082448 (start with 4).

Cf. A082458, A019464, A019465, A019466 (similar, but first multiply, then add; starting with 0,1,2,3).

a[n_] := If[ OddQ@n, a[n - 1] + (n + 1)/2, a[n - 1]*n/2]; a[0] = 2; Table[ a@n, {n, 0, 28}] (* Robert G. Wilson v, Jul 21 2009 *)

A019460(n)=2*(A000522(n\2)+(n\2)!)-if(bittest(n,0),1,n\2+2)
/* For producing the terms in increasing order, the following 'hack' can be used M. F. Hasler, Jan 12 2011 */
lastn=0; an1=1; A000522(n)={ an1=if(n, n==lastn && return(an1); n==lastn+1||error(); an1*lastn=n)+1 }

l=[2]
for n in range(1, 101):
    l.append(l[n - 1] + ((n + 1)//2) if n%2 else l[n - 1]*(n//2))
print(l) # Indranil Ghosh, Jul 05 2017

a(2n) = 2*(A000522(n) + n!) - n - 2.
a(2n+1) = 2*(A000522(n) + n!) - 1.
Recursive: a(0) = 2, a(n) = (1 + floor((n-1)/2) - ceiling((n-1)/2))*(a(n-1) + (n+2)/2) + (ceiling((n-1)/2) - floor((n-1)/2))*(n/2)*a(n-1). - Wesley Ivan Hurt, Jan 12 2013

One more term from Robert G. Wilson v, Jul 21 2009
Formula provided by Nathaniel Johnston, Nov 11 2010
Formula double-checked and PARI code added by M. F. Hasler, Nov 12 2010
Edited by M. F. Hasler, Feb 25 2018

A056545 a(n) = 4na(n-1) + 1 with a(0)=1.

Original entry on oeis.org

1, 5, 41, 493, 7889, 157781, 3786745, 106028861, 3392923553, 122145247909, 4885809916361, 214975636319885, 10318830543354481, 536579188254433013, 30048434542248248729, 1802906072534894923741, 115385988642233275119425

Offset: 0

Henry Bottomley, Jun 20 2000

For positive n, a(n) equals 4^n times the permanent of the n X n matrix with (5/4)'s along the main diagonal and 1's everywhere else. - John M. Campbell, Jul 10 2011

			a(2) = 4*2*a(1) + 1 = 8*5 + 1 = 41.

Harvey P. Dale, Table of n, a(n) for n = 0..365
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A000522, A010844, A010845, A056546, A056547, A001907 for analogs. A056545/(A000142*A000302) is an increasingly good approximation to 4th root of e.

Round@Table[Exp[1/4] 4^n Gamma[n + 1, 1/4], {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster; Vladimir Reshetnikov, Oct 14 2016 *)
nxt[{n_,a_}]:={n+1,4a(n+1)+1}; NestList[nxt,{0,1},20][[All,2]] (* Harvey P. Dale, Mar 19 2019 *)

a(n) = floor(e^(1/4)*4^n*n!).
From Philippe Deléham, Mar 14 2004: (Start)
a(n) = n!*Sum_{k=0..n} 4^(n-k)/k!.
E.g.f.: exp(x)/(1 - 4*x). (End)
a(n) = Sum_{k=0..n} P(n, k)*4^k. - Ross La Haye, Aug 29 2005
a(n) = hypergeometric_U(1, n+2 , 1/4)/4. - Peter Luschny, Nov 26 2014
a(n) = exp(1/4)*4^n*Gamma(n+1, 1/4). a(n) ~ sqrt(2*Pi)*4^n*n^(n+1/2)*exp(1/4-n). - Vladimir Reshetnikov, Oct 14 2016
From Peter Bala, Mar 01 2017: (Start)
a(n) = Integral_{x = 0..inf} (4*x + 1)^n*exp(-x) dx.
The e.g.f. y = exp(x)/(1 - 4*x) satisfies the differential equation (1 - 4*x)*y' = (5 - 4*x)*y.
a(n) = (4*n + 1)*a(n-1) - 4*(n - 1)*a(n-2).
The sequence b(n) := 4^n*n! also satisfies the same recurrence with b(0) = 1, b(1) = 4. This leads to the continued fraction representation a(n) = 4^n*n!*( 1 + 1/(4 - 4/(9 - 8/(13 - ... - (4*n - 4)/(4*n + 1) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(1/4) = 1 + 1/(4 - 4/(9 - 8/(13 - ... - (4*n - 4)/((4*n + 1) - ... )))). Cf. A010844. (End)

More terms from James Sellers, Jul 04 2000

A009628 Expansion of e.g.f.: sinh(x)/(1+x).

Original entry on oeis.org

0, 1, -2, 7, -28, 141, -846, 5923, -47384, 426457, -4264570, 46910271, -562923252, 7318002277, -102452031878, 1536780478171, -24588487650736, 418004290062513, -7524077221125234, 142957467201379447, -2859149344027588940, 60042136224579367741

Offset: 0

R. H. Hardin

(-1)^n*(A000166 + A000522)/2 = A009179, (-1)^n*(A000166-A000522)/2 = this_sequence.

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

Cf. A000166, A000522, A009179, A051397, A087208, A186763.

G(x):= sinh(x)/(1+x): f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..20); # Zerinvary Lajos, Apr 03 2009

a[n_] := (-1)^n (Exp[-1] Gamma[1 + n, -1] - Exp[1] Gamma[1 + n, 1])/2;
Table[a[n], {n, 0, 20}] (* Peter Luschny, Dec 18 2017 *)
With[{nn=30},CoefficientList[Series[Sinh[x]/(1+x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 19 2023 *)

a(n) = n!*polcoeff((sinh(x)/(1+x) + x * O(x^n)), n) \\ Charles R Greathouse IV, Sep 09 2016

x='x+O('x^99); concat([0], Vec(serlaplace(sinh(x)/(1+x)))) \\ Altug Alkan, Dec 18 2017

def A009628(n)
  a = 0
  (0..n).map{|i| a = -i * a + i % 2}
end # Seiichi Manyama, Sep 09 2016

a(n) = (-1)^(n+1)*floor(n!*sinh(1)), n>=1. - Vladeta Jovovic, Aug 10 2002
Let u(1) = 1, u(n) = n*u(n-1) + n (mod 2); then for n>0, a(n) = (-1)^(n+1)*u(n). - Benoit Cloitre, Jan 12 2003
Unsigned sequence satisfies a(n) = n*a(n-1)+a(n-2)-(n-2)*a(n-3), with E.g.f. sinh(z)/(1-z). - Mario Catalani (mario.catalani(AT)unito.it), Feb 08 2003
a(n) = (-1)^(n+1) * n! * Sum_{k=1..floor((n+1)/2)} 1/(2*k-1)!.
a(n) = -n*a(n-1) + n (mod 2). - Seiichi Manyama, Sep 09 2016
a(n) = (-1)^n*(exp(-1)*Gamma(1+n,-1) - exp(1)*Gamma(1+n,1))/2. - Peter Luschny, Dec 18 2017

Extended with signs by Olivier Gérard, Mar 15 1997

Definition clarified by Harvey P. Dale, Mar 19 2023

A090210 Triangle of certain generalized Bell numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 7, 1, 1, 15, 87, 34, 1, 1, 52, 1657, 2971, 209, 1, 1, 203, 43833, 513559, 163121, 1546, 1, 1, 877, 1515903, 149670844, 326922081, 12962661, 13327, 1, 1, 4140, 65766991, 66653198353, 1346634725665, 363303011071, 1395857215, 130922, 1, 1

Offset: 1

Wolfdieter Lang, Dec 01 2003

Let B_{n}(x) = sum_{j>=0}(exp(j!/(j-n)!*x-1)/j!) and
S(n,k) = k! [x^k] taylor(B_{n}(x)), where [x^k] denotes the
coefficient of x^k in the Taylor series for B_{n}(x).
Then S(n,k) (n>0, k>=0) is the square array representation of the triangle.
To illustrate the cross-references of T(n,k) when written as a square array.
n\k         A000012,A000012,A002720,A069948,A182925,A182924,...
0: A000012: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1: A000110: 1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
2: A020556: 1, 1, 7, 87, 1657, 43833, 1515903, ...
3: A069223: 1, 1, 34, 2971, 513559, 149670844, ...
4: A071379: 1, 1, 209, 163121, 326922081, ...
5: A090209: 1, 1, 1546, 12962661, 363303011071,...
6:  ...     1, 1, 13327, 1395857215, 637056434385865,...
Note that the sequence T(0,k) is not included in the data.
- Peter Luschny, Mar 27 2011

			Triangle begins:
1;
1, 1;
2, 1, 1;
5, 7, 1, 1;
15, 87, 34, 1, 1;
52, 1657, 2971, 209, 1, 1;
203, 43833, 513559, 163121, 1546, 1, 1;

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
W. Lang, First 8 rows.
K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp,Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. 50, 083512 (2009).

Cf. A000110, A020556, A069223, A071379, A090209, A002720, A069948, A182924, A182925, A182933.

T(n,n) gives A070227.

A090210_AsSquareArray := proc(n,k) local r,s,i;
if k=0 then 1 else r := [seq(n+1,i=1..k-1)]; s := [seq(1,i=1..k-1)];
exp(-x)*n!^(k-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
seq(lprint(seq(A090210_AsSquareArray(n,k),k=0..6)),n=0..6);
# Peter Luschny, Mar 30 2011

t[n_, k_] := t[n, k] = Sum[(n+j)!^(k-1)/(j!^k*E), {j, 0, Infinity}]; t[_, 0] = 1;
Flatten[ Table[ t[n-k+1, k], {n, 0, 8}, {k, n, 0, -1}]][[1 ;; 43]] (* Jean-François Alcover, Jun 17 2011 *)

a(n, m) = Bell(m;n-(m-1)), n>= m-1 >=0, with Bell(m;k) := Sum_{p=m..m*k} S2(m;k, p), where S2(m;k, p) := (((-1)^p)/p!) * Sum_{r=m..p} ((-1)^r)*binomial(p, r)*fallfac(r, m)^k; with fallfac(n, m) := A008279(n, m) (falling factorials) and m<=p<=k*m, k>=1, m=1, 2, ..., else 0. From eqs.(6) with r=s->m and eq.(19) with S_{r, r}(n, k)-> S2(r;n, k) of the Blasiak et al. reference. [Corrected by Sean A. Irvine, Jun 03 2024]
a(n, m) = (Sum_{k>=m} fallfac(k, m)^(n-(m-1)))/exp(1), n>=m-1>=0, else 0. From eq.(26) with r->m of the Schork reference which is rewritten eq.(11) of the original Blasiak et al. reference.
E.g.f. m-th column (no leading zeros): (Sum_{k>=m} exp(fallfac(k, m)*x)/k!) + A000522(m)/m!)/exp(1). Rewritten from the top of p. 4656 of the Schork reference.

A096307 E.g.f.: exp(x)/(1-x)^6.

Original entry on oeis.org

1, 7, 55, 481, 4645, 49171, 566827, 7073725, 95064361, 1369375615, 21054430591, 344231563897, 5964569413645, 109196040092491, 2106381399472435, 42705264827626261, 907920105215691217, 20198878182718877815

Offset: 0

Philippe Deléham, Jun 26 2004

Sum_{k = 0..n} A094816(n,k)*x^k give A000522(n), A001339(n), A082030(n), A095000(n), A095177(n) for x = 1, 2, 3, 4, 5 respectively.

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000522, A001339, A082030, A095000, A095177, A001689, A094794.

Table[HypergeometricPFQ[{6, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)
With[{nn = 250}, CoefficientList[Series[Exp[x]/(1 - x)^6, {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, May 27 2016 *)

a(n) = Sum_{k = 0..n} A094916(n, k)*6^k.
a(n) = Sum_{k = 0..n} binomial(n, k)*(k+5)!/5!.
a(n) = 2F0(6,-n;;-1). - Benedict W. J. Irwin, May 27 2016
From Peter Bala, Jul 25 2021: (Start)
a(n) = (n+6)*a(n-1) - (n-1)*a(n-2) with a(0) = 1 and a(1) = 7. Cf. A001689.
First-order recurrence: P(n-1)*a(n) = n*P(n)*a(n-1) - 1 with a(0) = 1, where P(n) = n^5 + 10*n^4 + 45*n^3 + 100*n^2 + 109*n + 44 = A094794(n).
(End)

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A142995 a(0) = 0, a(1) = 1, a(n+1) = (2*n^2 + 2*n + 3)*a(n) - n^4*a(n-1), n >= 1.

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A186763 Number of increasing odd cycles in all permutations of {1,2,...,n}.

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A006231 a(n) = Sum_{k=2..n} n(n-1)...(n-k+1)/k.

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A009179 E.g.f. cosh(x)/(1+x).

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A010845 a(n) = 3*n*a(n-1) + 1, a(0) = 1.

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A019460 Add 1, multiply by 1, add 2, multiply by 2, etc., start with 2.

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A142995 a(0) = 0, a(1) = 1, a(n+1) = (2n^2 + 2n + 3)a(n) - n^4a(n-1), n >= 1.

A010845 a(n) = 3na(n-1) + 1, a(0) = 1.

A056545 a(n) = 4na(n-1) + 1 with a(0)=1.