A003319 -id:A003319 - cp's OEIS frontend

A003149 a(n) = Sum_{k=0..n} k!*(n - k)!.

1, 2, 5, 16, 64, 312, 1812, 12288, 95616, 840960, 8254080, 89441280, 1060369920, 13649610240, 189550368000, 2824077312000, 44927447040000, 760034451456000, 13622700994560000, 257872110354432000, 5140559166898176000, 107637093007589376000, 2361827297364885504000

Offset: 0

N. J. A. Sloane, Henry Gould

From Michael Somos, Feb 14 2002: (Start)
The sequence is the resistance between opposite corners of an (n+1)-dimensional hypercube of unit resistors, multiplied by (n+1)!.
The resistances for n+1 = 1,2,3,... are 1, 1, 5/6, 2/3, 8/15, 13/30, 151/420, 32/105, 83/315, 73/315, 1433/6930, ... (see A046878/A046879). (End)
Number of {12,21*,2*1}-avoiding signed permutations in the hyperoctahedral group.
a(n) is the sum of the reciprocals of the binomial coefficients C(n,k), multiplied by n!; example: a(4) = 4!*(1/1 + 1/4 + 1/6 + 1/4 + 1/1) = 64. - Philippe Deléham, May 12 2005
a(n) is the number of permutations on [n+1] that avoid the pattern 13-2|. The absence of a dash between 1 and 3 means the "1" and "3" must be consecutive in the permutation; the vertical bar means the "2" must occur at the end of the permutation. For example, 24153 fails to avoid this pattern: 243 is an offending subpermutation. - David Callan, Nov 02 2005
n!/a(n) is the probability that a random walk on an (n+1)-dimensional hypercube will visit the diagonally opposite vertex before it returns to its starting point. 2^n*a(n)/n! is the expected length of a random walk from one vertex of an (n+1)-dimensional hypercube to the diagonally opposite vertex (a walk which may include one or more passes through the starting point). These "random walk" examples are solutions to IBM's "Ponder This" puzzle for April, 2006. - Graeme McRae, Apr 02 2006
a(n) is the number of strong fixed points in all permutations of {1,2,...,n+1} (a permutation p of {1,2,...,n} is said to have j as a strong fixed point (splitter) if p(k)j for k>j). Example: a(2)=5 because the permutations of {1,2,3}, with marked strong fixed points, are: 1'2'3', 1'32, 312, 213', 231 and 321. - Emeric Deutsch, Oct 28 2008
Coefficients in the asymptotic expansion of exp(-2*x)*Ei(x)^2 for x -> inf, where Ei(x) is the exponential integral. - Vladimir Reshetnikov, Apr 24 2016

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (1.1.11 b, p.342).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Volume 1 (1986), p. 49. [From Emeric Deutsch, Oct 28 2008]

T. D. Noe, Table of n, a(n) for n = 0..100
Joerg Arndt, Generating Random Permutations, PhD thesis, Australian National University, Canberra, Australia, (2010).
Fred Curtis, Resistance-network Problems.
Bakir Farhi, On a curious integer sequence, arXiv:2204.10136 [math.NT], 2022.
Todd Feil, Gary Kennedy and David Callan, Problem E3467, Amer. Math. Monthly, 100 (1993), 800-801. [From _Emeric Deutsch_, Oct 28 2008]
Henry W. Gould, Letter to N. J. A. Sloane, Nov 1973, and various attachments.
IBM, "Ponder This" puzzle for April, 2006
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
F. Nedemeyer and Y. Smorodinsky, Resistance in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.
R. Sprugnoli, Moments of Reciprocals of Binomial Coefficients, Journal of Integer Sequences, 14 (2011), #11.7.8.
V. Strehl, The average number of splitters in a random permutation [Unpublished; included here with the author's permission.]
B. Sury, Sum of the reciprocals of the binomial coefficients, Europ. J. Comb., 14 (1993), 351-353.
Eric Weisstein's World of Mathematics, Incomplete Beta Function.
Eric Weisstein's World of Mathematics, Lerch Transcendent.
Index to sequences related to resistances.

Cf. A046825, A046878, A046879.
Cf. A052186, A006932, A145878. - Emeric Deutsch, Oct 28 2008
Cf. A324495, A324496, A324497 (problem similar to the random walks on the hypercube).

F:=Factorial;; List([0..20], n-> Sum([0..n], k-> F(k)*F(n-k)) ); # G. C. Greubel, Dec 29 2019

F:=Factorial; [ (&+[F(k)*F(n-k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 29 2019

seq( add(k!*(n-k)!, k=0..n), n=0..20); # G. C. Greubel, Dec 29 2019
# second Maple program:
a:= proc(n) option remember; `if`(n<2, n+1,
      ((3*n+1)*a(n-1)-n^2*a(n-2))/2)
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Aug 08 2025

Table[Sum[k!(n-k)!,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Mar 28 2012 *)
Table[(n+1)!/2^n*Sum[2^k/(k+1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 27 2012 *)
Round@Table[-2 (n+1)! Re[LerchPhi[2, 1, n+2]], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 12 2015 *)
Table[(n+1)!*Sum[Binomial[n+1, 2*j+1]/(2*j+1), {j, 0, n}]/2^n, {n, 0, 20}] (* Vaclav Kotesovec, Dec 04 2015 *)
Series[Exp[-2x] ExpIntegralEi[x]^2, {x, Infinity, 20}][[3]] (* Vladimir Reshetnikov, Apr 24 2016 *)
Table[2*(-1)^n * Sum[(2^k - 1) * StirlingS1[n, k] * BernoulliB[k], {k, 0, n}], {n, 1, 25}] (* Vaclav Kotesovec, Oct 04 2022 *)

PARI
```
a(n)=sum(k=0,n,k!*(n-k)!)
```

a(n)=if(n<0,0,(n+1)!*polcoeff(log(1-x+x^2*O(x^n))/(x/2-1),n+1))

a(n) = my(A = 1, B = 1); for(k=1, n, B *= k; A = (n-k+1)*A + B); A \\ Mikhail Kurkov, Aug 08 2025

def a(n: int) -> int:
    if n < 2: return n + 1
    app, ap = 1, 2
    for i in range(2, n + 1):
        app, ap = ap, ((3 * i + 1) * ap - (i * i) * app) >> 1
    return ap
print([a(n) for n in range(23)])  # Peter Luschny, Aug 08 2025

f=factorial; [sum(f(k)*f(n-k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Dec 29 2019

a(n) = n! + ((n+1)/2)*a(n-1), n >= 1. - Leroy Quet, Sep 06 2002
a(n) = ((3n+1)*a(n-1) - n^2*a(n-2))/2, n >= 2. - David W. Wilson, Sep 06 2002; corrected by N. Sato, Jan 27 2010
G.f.: (Sum_{k>=0} k!*x^k)^2. - Vladeta Jovovic, Aug 30 2002
E.g.f: log(1-x)/(x/2 - 1) if offset 1.
Convolution of A000142 [factorial numbers] with itself. - Ross La Haye, Oct 29 2004
a(n) = Sum_{k=0..n+1} k*A145878(n+1,k). - Emeric Deutsch, Oct 28 2008
a(n) = A084938(n+2,2). - Philippe Deléham, Dec 17 2008
a(n) = 2*Integral_{t=0..oo} Ei(t)*exp(-2*t)*t^(n+1) where Ei is the exponential integral function. - Groux Roland, Dec 09 2010
Empirical: a(n-1) = 2^(-n)*(A103213(n) + n!*H(n)) with H(n) harmonic number of order n. - Groux Roland, Dec 18 2010; offset fixed by Vladimir Reshetnikov, Apr 24 2016
O.g.f.: 1/(1-I(x))^2 where I(x) is o.g.f. for A003319. - Geoffrey Critzer, Apr 27 2012
a(n) ~ 2*n!. - Vaclav Kotesovec, Oct 04 2012
a(n) = (n+1)!/2^n * Sum_{k=0..n} 2^k/(k+1). - Vaclav Kotesovec, Oct 27 2012
E.g.f.: 2/((x-1)*(x-2)) + 2*x/(x-2)^2*G(0) where G(k) = 1 + x*(2*k+1)/(2*(k+1) - 4*x*(k+1)^2/(2*x*(k+1) + (2*k+3)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 14 2012
a(n) = 2 * n! * (1 + Sum_{k>=1} A005649(k-1)/n^k). - Vaclav Kotesovec, Aug 01 2015
From Vladimir Reshetnikov, Nov 12 2015: (Start)
a(n) = -(n+1)!*Re(Beta(2; n+2, 0))/2^(n+1), where Beta(z; a, b) is the incomplete Beta function.
a(n) = -2*(n+1)!*Re(LerchPhi(2, 1, n+2)), where LerchPhi(z, s, a) is the Lerch transcendent. (End)
a(n) = (n+1)!*(H(n+1) + (n+1)*hypergeom([1, 1, -n], [2, 2], -1))/2^(n+1), where H(n) is the harmonic number. - Vladimir Reshetnikov, Apr 24 2016
Expansion of square of continued fraction 1/(1 - x/(1 - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - ...))))))). - Ilya Gutkovskiy, Apr 19 2017
a(n) = Sum_{k=0..n+1} (-1)^(n-k)*A226158(k)*Stirling1(n+1, k). - Mélika Tebni, Feb 22 2022
E.g.f.: x/((1-x)*(2-x))-(2*log(1-x))/(2-x)^2+1/(1-x). - Vladimir Kruchinin, Dec 17 2022

More terms from Michel ten Voorde, Apr 11 2001

A075834 Coefficients of power series A(x) such that n-th term of A(x)^n = n! x^(n-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 2, 7, 34, 206, 1476, 12123, 111866, 1143554, 12816572, 156217782, 2057246164, 29111150620, 440565923336, 7101696260883, 121489909224618, 2198572792193786, 41966290373704332, 842706170872913634, 17759399688526009020, 391929722837419044420

Offset: 0

Paul D. Hanna, Oct 14 2002, Jul 30 2008

nonn

Also, number of stablized-interval-free permutations on [n] (see Callan link).

Coefficients in the series reversal of the asymptotic expansion of exp(-x)*Ei(x) for x -> inf, where Ei(x) is the exponential integral. - Vladimir Reshetnikov, Apr 24 2016

			At n=7, the 7th term of A(x)^7 is 7! x^6, as demonstrated by A(x)^7 = 1 + 7 x + 28 x^2 + 91 x^3 + 294 x^4 + 1092 x^5 + 5040 x^6 + 29093 x^7 + 203651 x^8 + ... .
A(x) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 206*x^6 + ... = x/series_reversion(x + x^2 + 2*x^3 + 6*x^4 + 24*x^5 + 120*x^6 + ...).
Related expansions:
log(A(x)) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 136*x^5/5 + 1030*x^6/6 + ...;
1 - x/(A(x) - 1) = x + x^2 + 4*x^3 + 21*x^4 + 136*x^5 + 1030*x^6 +...;
(d/dx)((A(x) - 1)/x) = 1 + 4*x + 21*x^2 + 136*x^3 + 1030*x^4 + ... .

Vincenzo Librandi, Table of n, a(n) for n = 0..100
F. Ardila, F. Rincón and L. Williams, Positroids and non-crossing partitions, arXiv preprint arXiv:1308.2698 [math.CO], 2013.
Daniel Birmajer, Juan B. Gil and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
David Callan, Counting stabilized-interval-free permutations, arXiv:math/0310157 [math.CO], 2003.
David Callan, Counting Stabilized-Interval-Free Permutations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.
Colin Defant and Nathan Williams, Coxeter Pop-Tsack Torsing, arXiv:2106.05471 [math.CO], 2021.
Jesse Elliott, Asymptotic expansions of the prime counting function, arXiv:1809.06633 [math.NT], 2018.
Hyungju Park, An Asymptotic Formula for the Number of Stabilized-Interval-Free Permutations, J. Int. Seq. (2023) Vol. 26, Art. 23.9.3.

Cf. A209881, A091063, A084938.

Cf. A003319.

a = ConstantArray[0,20]; a[[1]]=1; a[[2]]=1; a[[3]]=2; Do[a[[n]] = (n-1)*a[[n-1]] + Sum[(j-1)*a[[j]]*a[[n-j]],{j,2,n-2}],{n,4,20}]; Flatten[{1,a}] (* Vaclav Kotesovec after David Callan, Feb 22 2014 *)
InverseSeries[Series[Exp[-x] ExpIntegralEi[x], {x, Infinity, 20}]][[3]] (* Vladimir Reshetnikov, Apr 24 2016 *)

a(n)=if(n<0,0,if(n<=1,1,(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j));))

a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,(k-1)!))))[n+1] \\ Paul D. Hanna, Jul 09 2006

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/(1-x*deriv(A)/A));polcoeff(A,n)}

{a(n)=local(F=1+x*O(x^n)); for(i=0,n,F=1+x*F+x^2*F*deriv(F)+x*O(x^n));polcoeff(1+x*F,n)} \\ Paul D. Hanna, Sep 02 2008

a(0)=a(1)=1, a(n) = (n-1)*a(n-1) + Sum_{j=2..n-2}(j-1)*a(j)*a(n-j), n >= 2. - David Callan
G.f.: A(x) = x/series_reversion(x*G(x)); G(x) = A(x*G(x)); A(x) = G(x/A(x)); where G(x) is the g.f. of the factorials (A000142). - Paul D. Hanna, Jul 09 2006
G.f.: A(x) = 1 + x/(1 - x*A'(x)/A(x)) = 1 + x/(1-x - x^2*d/dx((A(x) - 1)/x)).
G.f.: A(x) = 1 + x*F(x) where F(x) satisfies F(x) = 1 + x*F(x) + x^2*F(x)*F'(x) and F'(x) = d/dx F(x). - Paul D. Hanna, Sep 02 2008
a(n) ~ exp(-1) * n! * (1 - 1/n - 5/(2*n^2) - 32/(3*n^3) - 1643/(24*n^4) - 23017/(40*n^5) - 4215719/(720*n^6)). - Vaclav Kotesovec, Feb 22 2014
A003319(n+1) = coefficient of x^n in A(x)^n. - Michael Somos, Feb 23 2014

More terms from David Wasserman, Jan 26 2005

Minor edits by Vaclav Kotesovec, Aug 01 2015

A059438 Triangle T(n,k) (1 <= k <= n) read by rows: T(n,k) is the number of permutations of [1..n] with k components.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 13, 7, 3, 1, 71, 32, 12, 4, 1, 461, 177, 58, 18, 5, 1, 3447, 1142, 327, 92, 25, 6, 1, 29093, 8411, 2109, 531, 135, 33, 7, 1, 273343, 69692, 15366, 3440, 800, 188, 42, 8, 1, 2829325, 642581, 125316, 24892, 5226, 1146, 252, 52, 9, 1

Offset: 1

N. J. A. Sloane, Feb 01 2001

			Triangle begins:
[1] [     1]
[2] [     1,     1]
[3] [     3,     2,     1]
[4] [    13,     7,     3,    1]
[5] [    71,    32,    12,    4,   1]
[6] [   461,   177,    58,   18,   5,   1]
[7] [  3447,  1142,   327,   92,  25,   6,  1]
[8] [ 29093,  8411,  2109,  531, 135,  33,  7, 1]
[9] [273343, 69692, 15366, 3440, 800, 188, 42, 8, 1]

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 262 (#14).
Antonio Di Crescenzo, Barbara Martinucci, and Abdelaziz Rhandi, A linear birth-death process on a star graph and its diffusion approximation, arXiv:1405.4312 [math.PR], 2014.
FindStat - Combinatorial Statistic Finder, The decomposition number of a permutation.
Peter Hegarty and Anders Martinsson, On the existence of accessible paths in various models of fitness landscapes, arXiv:1210.4798 [math.PR], 2012-2014. - From _N. J. A. Sloane_, Jan 01 2013
Sergey Kitaev and Philip B. Zhang, Distributions of mesh patterns of short lengths, arXiv:1811.07679 [math.CO], 2018.

A version with reflected rows is A263484.
Diagonals give A003319, A059439, A059440, A055998.
Cf. A000007, A085771, A084938.
T(2n,n) gives A308650.

# Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
PMatrix(10, A003319); # Peter Luschny, Oct 09 2022

(* p = indecomposable permutations = A003319 *) p[n_] := p[n] = n! - Sum[ k!*p[n-k], {k, 1, n-1}]; t[n_, k_] /; n < k = 0; t[n_, 1] := p[n]; t[n_, k_] /; n >= k := t[n, k] = Sum[ t[n-j, k-1]*p[j], {j, 1, n}]; Flatten[ Table[ t[n, k], {n, 1, 10}, {k, 1, n}] ] (* Jean-François Alcover, Mar 06 2012, after Philippe Deléham *)

def A059438_triangle(dim) :
    R = PolynomialRing(ZZ, 'x')
    C = [R(0)] + [R(1) for i in range(dim+1)]
    A = [(i + 2) // 2 for i in range(dim+1)]
    A[0] = R.gen(); T = []
    for k in range(1, dim+1) :
        for n in range(k, 0, -1) :
            C[n] = C[n-1] + C[n+1] * A[n-1]
        T.append(list(C[1])[1::])
    return T
A059438_triangle(8) # Peter Luschny, Sep 10 2022

# Alternatively, using the function PartTrans from A357078:
# Adds a (0,0)-based column (1, 0, 0, ...) to the left of the triangle.
dim = 10
A = ZZ[['t']]; g = A([0]+[factorial(n) for n in range(1, 30)]).O(dim+2)
PartTrans(dim, lambda n: list(g / (1 +  g))[n]) # Peter Luschny, Sep 11 2022

Let f(x) = Sum_{n >= 0} n!*x^n, g(x) = 1 - 1/f(x). Then g(x) is g.f. for first diagonal A003319 and Sum_{n >= k} T(n, k)*x^n = g(x)^k.
Triangle T(n, k), n > 0 and k > 0, read by rows; given by [0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA A000007 where DELTA is Deléham's operator defined in A084938.
T(n+k, k) = Sum_{a_1 + a_2 + ... + a_k = n} A003319(a_1)*A003319(a_2)*...*A003319(a_k). T(n, k) = 0 if n < k, T(n, 1) = A003319(n) and for n >= k T(n, k)= Sum_{j>=1} T(n-j, k-1)* A003319(j). A059438 is formed from the self convolution of its first column (A003319). - Philippe Deléham, Feb 04 2004
Sum_{k>0} T(n, k) = n!; see A000142. - Philippe Deléham, Feb 05 2004
If g(x) = x + x^2 + 3*x^3 + 13*x^4 + ... is the generating function for the number of permutations with no global descents, then 1/(1-g(x)) is the generating function for n!. Setting t=1 in f(x, t) implies Sum_{k=1..n} T(n,k) = n!. Let g(x) be the o.g.f. for A003319. Then the o.g.f. for this table is given by f(x, t) = 1/(1 - t*g(x)) - 1 (i.e., the coefficient of x^n*t^k in f(x,t) is T(n,k)). - Mike Zabrocki, Jul 29 2004

More terms from Vladeta Jovovic, Mar 04 2001

A111528 Square table, read by antidiagonals, where the g.f. for row n+1 is generated by: xR_{n+1}(x) = (1+nx - 1/R_n(x))/(n+1) with R_0(x) = Sum_{n>=0} n!*x^n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 13, 24, 1, 1, 5, 22, 71, 120, 1, 1, 6, 33, 148, 461, 720, 1, 1, 7, 46, 261, 1156, 3447, 5040, 1, 1, 8, 61, 416, 2361, 10192, 29093, 40320, 1, 1, 9, 78, 619, 4256, 23805, 99688, 273343, 362880, 1, 1, 10, 97, 876, 7045, 48096, 263313

Offset: 0

Paul D. Hanna, Aug 06 2005

			Table begins:
  1, 1,  2,   6,   24,   120,    720,    5040,     40320, ...
  1, 1,  3,  13,   71,   461,   3447,   29093,    273343, ...
  1, 1,  4,  22,  148,  1156,  10192,   99688,   1069168, ...
  1, 1,  5,  33,  261,  2361,  23805,  263313,   3161781, ...
  1, 1,  6,  46,  416,  4256,  48096,  591536,   7840576, ...
  1, 1,  7,  61,  619,  7045,  87955, 1187845,  17192275, ...
  1, 1,  8,  78,  876, 10956, 149472, 2195208,  34398288, ...
  1, 1,  9,  97, 1193, 16241, 240057, 3804353,  64092553, ...
  1, 1, 10, 118, 1576, 23176, 368560, 6262768, 112784896, ...
Rows are generated by logarithms of factorial series:
log(1 + x + 2*x^2 + 6*x^3 + 24*x^4 + ... n!*x^n + ...) = x + (3/2)*x^2 + (13/3)*x^3 + (71/4)*x^4 + (461/5)*x^5 + ...
(1/2)*log(1 + 2*x + 6*x^2 + ... + ((n+1)!/1!)*x^n + ...) = x + (4/2)*x^2 + (22/3)*x^3 + (148/4)*x^4 + (1156/5)*x^5 + ...
(1/3)*log(1 + 3*x + 12*x^2 + 60*x^3 + ... + ((n+2)!/2!)*x^n + ...) = x + (5/2)*x^2 + (33/3)*x^3 + (261/4)*x^4 + (2361/5)*x^5 +...
G.f. of row n may be expressed by the continued fraction:
R_n(x) = 1/(1+n*x - (n+1)*x/(1+(n+1)*x - (n+2)*x/(1+(n+2)*x -...
or recursively by: R_n(x) = 1/(1+n*x - (n+1)*x*R_{n+1}(x)).

A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf: A003319 (row 1), A111529 (row 2), A111530 (row 3), A111531 (row 4), A111532 (row 5), A111533 (row 6), A111534 (diagonal).

Similar recurrences: A124758, A243499, A284005, A329369, A341392.

T := (n, k) -> coeff(series(hypergeom([n+1, 1], [], x)/hypergeom([n, 1], [], x), x, 21), x, k):
#display as a sequence
seq(seq(T(n-k, k), k = 0..n), n = 0..10);
# display as a square array
seq(print(seq(T(n, k), k = 0..10)), n = 0..10); # Peter Bala, Jul 16 2022

T[n_, k_] := T[n, k] = Which[n < 0 || k < 0, 0, k == 0 || k == 1, 1, n == 0, k!, True, (T[n - 1, k + 1] - T[n - 1, k])/n - Sum[T[n, j]*T[n - 1, k - j], {j, 1, k - 1}]]; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2018 *)

{T(n,k)=if(n<0||k<0,0,if(k==0||k==1,1,if(n==0,k!, (T(n-1,k+1)-T(n-1,k))/n-sum(j=1,k-1,T(n,j)*T(n-1,k-j)))))}
for(n=0,10,for(k=0,10,print1(T(n,k),", ")); print(""))

{T(n,k)=if(n<0||k<0,0,if(k==0,1,if(n==0,k!, k/n*polcoeff(log(sum(m=0,k,(n-1+m)!/(n-1)!*x^m)),k))))}
for(n=0,10,for(k=0,10,print1(T(n,k),", ")); print(""))

T(n, 0) = 1, T(0, k) = k!, otherwise for n>=1 and k>=1:
T(n, k) = (T(n-1, k+1) - T(n-1, k))/n - Sum_{j=1..k-1} T(n, j)*T(n-1, k-j).
T(n, k) = (k/n)*[x^k] log(Sum_{m=0..k} (n-1+m)!/(n-1)!*x^m).
T(n, k) = Sum_{j = 0..k} A089949(k, j)*n^(k-j). - Philippe Deléham, Aug 08 2005
R_n(x) = -((n-1)!/n)/Sum_{i>=1} (i+n-2)!*x^i, n > 0. - Vladeta Jovovic, May 06 2006
G.f. of row R may be expressed by the continued fraction: W(0), where W(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+1+R)/( x*(k+1+R) - 1/W(k+1) ))). - Sergei N. Gladkovskii, Aug 26 2013
Conjecture: T(n, k) = b(2^(k-1) - 1, n) for k > 0 with T(n, 0) = 1 where b(n, m) = b(floor(n/2), m) + b(floor((2n - 2^A007814(n))/2), m) + m*b(A025480(n-1), m) for n > 0 with b(0, m) = 1. - Mikhail Kurkov, Dec 16 2021
From Peter Bala, Jul 11 2022: (Start)
O.g.f. for row n, n >= 1: R(n,x) = ( Sum_{k >= 0} (n+k)!/n!*x^k )/( Sum_{k >= 0} (n-1+k)!/(n-1)!*x^k ).
R(n,x)/(1 - n*x*R(n,x)) = Sum_{k >= 0} (n+k)!/n!*x^k.
For n >= 0, R(n,x) satisfies the Riccati equation x^2*d/dx(R(n,x)) + n*x*R(n,x)^2 - (1 + (n-1)*x)*R(n,x) + 1 = 0 with R(n,0) = 1.
Apply Stokes 1982 to find that for n >= 0, R(n,x) = 1/(1 - x/(1 - (n+1)*x/(1 - 2*x/(1 - (n+2)*x/(1 - 3*x/(1 - (n+3)*x/(1 - 4*x/(1 - (n+4)*x/(1 - ...))))))))), a continued fraction of Stieltjes type. (End)

A104980 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p(column p+1 of T), or [T^p](m,0) = pT(p+m,p+1) for all m>=1 and p>=-1.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 13, 7, 3, 1, 71, 33, 13, 4, 1, 461, 191, 71, 21, 5, 1, 3447, 1297, 461, 133, 31, 6, 1, 29093, 10063, 3447, 977, 225, 43, 7, 1, 273343, 87669, 29093, 8135, 1859, 353, 57, 8, 1, 2829325, 847015, 273343, 75609, 17185, 3251, 523, 73, 9, 1

Offset: 0

Paul D. Hanna, Apr 10 2005

Column 0 equals A003319 (indecomposable permutations). Amazingly, column 1 (A104981) equals SHIFT_LEFT(column 0 of log(T)), where the matrix logarithm, log(T), equals the integer matrix A104986.
From Paul D. Hanna, Feb 17 2009: (Start)
Square array A156628 has columns found in this triangle T:
Column 0 of A156628 = column 0 of T = A003319;
Column 1 of A156628 = column 1 of T = A104981;
Column 2 of A156628 = column 2 of T = A003319 shifted;
Column 3 of A156628 = column 1 of T^2 (A104988);
Column 5 of A156628 = column 2 of T^2 (A104988). (End)

			SHIFT_LEFT(column 0 of T^-1) = -1*(column 0 of T);
SHIFT_LEFT(column 0 of T^1) = 1*(column 2 of T);
SHIFT_LEFT(column 0 of T^2) = 2*(column 3 of T);
where SHIFT_LEFT of column sequence shifts 1 place left.
Triangle T begins:
        1;
        1,      1;
        3,      2,      1;
       13,      7,      3,     1;
       71,     33,     13,     4,     1;
      461,    191,     71,    21,     5,    1;
     3447,   1297,    461,   133,    31,    6,   1;
    29093,  10063,   3447,   977,   225,   43,   7,  1;
   273343,  87669,  29093,  8135,  1859,  353,  57,  8, 1;
  2829325, 847015, 273343, 75609, 17185, 3251, 523, 73, 9, 1; ...
Matrix inverse T^-1 is A104984 which begins:
     1;
    -1,   1;
    -1,  -2,   1;
    -3,  -1,  -3,  1;
   -13,  -3,  -1, -4,  1;
   -71, -13,  -3, -1, -5,  1;
  -461, -71, -13, -3, -1, -6, 1; ...
Matrix T also satisfies:
[I + SHIFT_LEFT(T)] = [I - SHIFT_DOWN(T)]^-1, which starts:
    1;
    1,  1;
    2,  1,  1;
    7,  3,  1, 1;
   33, 13,  4, 1, 1;
  191, 71, 21, 5, 1, 1; ...
where SHIFT_DOWN(T) shifts columns of T down 1 row,
and SHIFT_LEFT(T) shifts rows of T left 1 column,
with both operations leaving zeros in the diagonal.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.

Cf. A003319 (column 0), A104981 (column 1), A104983 (row sums), A104984 (matrix inverse), A104988 (matrix square), A104990 (matrix cube), A104986 (matrix log), A156628.

T[n_, k_]:= T[n, k]= If[nJean-François Alcover, Aug 09 2018, from PARI *)

PARI
```
{T(n,k) = if(n
				
```
PARI
```
{T(n,k) = if(n
				
```

@CachedFunction
def T(n,k):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (k==n-1): return n
    else: return k*T(n, k+1) + sum( T(j, 0)*T(n, j+k+1) for j in (0..n-k-1) )
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 07 2021

T(n, k) = k*T(n, k+1) + Sum_{j=0..n-k-1} T(j, 0)*T(n, j+k+1) for n>k>0, with T(n, n) = 1, T(n+1, n) = n+1, T(n+1, 2) = T(n, 0) for n>=0.

A111529 Row 2 of table A111528.

Original entry on oeis.org

1, 1, 4, 22, 148, 1156, 10192, 99688, 1069168, 12468208, 157071424, 2126386912, 30797423680, 475378906432, 7793485765888, 135284756985472, 2479535560687360, 47860569736036096, 970606394944476160, 20635652201785613824, 459015456156148876288, 10662527360021306782720

Offset: 0

Paul D. Hanna, Aug 06 2005

			(1/2)*log(1 + 2*x + 6*x^2 + ... + ((n+1)!/1!)*x^n + ...)
= x + (4/2)*x^2 + (22/3)*x^3 + (148/4)*x^4 + (1156/5)*x^5 + ...

Robert Israel, Table of n, a(n) for n = 0..410
Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.
Richard J. Martin and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf. A111528 (table), A003319 (row 1), A111530 (row 3), A111531 (row 4), A111532 (row 5), A111533 (row 6), A111534 (diagonal).

Cf. A077607, A089949, A260491.

N:= 30: # to get a(0) to a(N)
g:= 1/2*log(add((n+1)!*x^n,n=0..N+1)):
S:= series(g,x,N+1);
1, seq(j*coeff(S,x,j),j=0..N); # Robert Israel, Jul 10 2015

T[n_, k_] := T[n, k] = Which[n<0 || k<0, 0, k==0 || k==1, 1, n==0, k!, True, (T[n-1, k+1]-T[n-1, k])/n - Sum[T[n, j] T[n-1, k-j], {j, 1, k-1}]];
a[n_] := T[2, n];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 09 2018 *)

{a(n)=if(n<0,0,if(n==0,1, (n/2)*polcoeff(log(sum(m=0,n,(m+1)!/1!*x^m)),n)))}

G.f.: (1/2)*log(Sum_{n >= 0} (n+1)!*x^n) = Sum_{n >= 1} a(n)*x^n/n.
G.f.: 1/(1+2*x - 3*x/(1+3*x - 4*x/(1+4*x - ... (continued fraction).
a(n) = Sum_{k = 0..n} 2^(n-k)*A089949(n,k). - Philippe Deléham, Oct 16 2006
G.f. 1/(2*x-G(0)) where G(k) = 2*x - 1 - k*x - x*(k+1)/G(k+1); G(0)=x (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 14 2012
G.f.: 1/(2*x) - 1/(G(0) - 1) where G(k) = 1 + x*(k+1)/(1 - 1/(1 + 1/G(k+1)));(continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 20 2012
G.f.: 1 + x/(G(0)-2*x) where G(k) = 1 + (k+1)*x - x*(k+3)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 26 2012
G.f.: (1 + 1/Q(0))/2, where Q(k) = 1 + k*x - x*(k+2)/Q(k+1); (continued fraction). In general, the g.f. for row (r+2) is (r + 1 + 1/Q(0))/(r + 2). - Sergei N. Gladkovskii, May 04 2013
G.f.: W(0), where W(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+3)/( x*(k+3) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
a(n) ~ n! * n^2/2 * (1 - 1/n - 2/n^2 - 8/n^3 - 52/n^4 - 436/n^5 - 4404/n^6 - 51572/n^7 - 683428/n^8 - 10080068/n^9 - 163471284/n^10), where the coefficients are given by (n+2)*(n+1)/n^2 * Sum_{k>=0} A260491(k)/(n+2)^k. - Vaclav Kotesovec, Jul 27 2015
a(n) = -A077607(n+2)/2. - Vaclav Kotesovec, Jul 29 2015
From Peter Bala, Jul 12 2022: (Start)
O.g.f: A(x) = ( Sum_{k >= 0} ((k+2)!/2!)*x^k )/( Sum_{k >= 0} (k+1)!*x^k ).
A(x)/(1 - 2*x*A(x)) = Sum_{k >= 0} ((k+2)!/2!)*x^k.
Riccati differential equation: x^2*A'(x) + 2*x*A^2(x) - (1 + x)*A(x) + 1 = 0.
Apply Stokes 1982 to find that A(x) = 1/(1 - x/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 3*x/(1 - 5*x/(1 - ... - n*x/(1 - (n+2)*x/(1 - ...))))))))), a continued fraction of Stieltjes type. (End)

A111530 Row 3 of table A111528.

Original entry on oeis.org

1, 1, 5, 33, 261, 2361, 23805, 263313, 3161781, 40907241, 567074925, 8385483393, 131787520101, 2194406578521, 38605941817245, 715814473193073, 13956039627763221, 285509132504621001, 6116719419966460365

Offset: 0

Paul D. Hanna, Aug 06 2005

			(1/3)*(log(1 + 3*x + 12*x^2 + 60*x^3 + ... + (n+2)!/2!)*x^n + ...)
= x + 5/2*x^2 + 33/3*x^3 + 261/4*x^4 + 2361/5*x^5 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf: A111528 (table), A003319 (row 1), A111529 (row 2), A111531 (row 4), A111532 (row 5), A111533 (row 6), A111534 (diagonal).

Cf. A001710, A001715.

T[n_, k_] := T[n, k] = Which[n<0 || k<0, 0, k==0 || k==1, 1, n==0, k!, True, (T[n-1, k+1]-T[n-1, k])/n - Sum[T[n, j]*T[n-1, k-j], {j, 1, k-1}]];
a[n_] := T[3, n];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Aug 09 2018 *)

{a(n)=if(n<0,0,if(n==0,1, (n/3)*polcoeff(log(sum(m=0,n,(m+2)!/2!*x^m) + x*O(x^n)),n)))} \\ fixed by Vaclav Kotesovec, Jul 27 2015

G.f.: (1/3)*log(Sum_{n>=0} (n+2)!/2!*x^n) = Sum_{n>=1} a(n)*x^n/n.
G.f.: A(x) = 1/(1 + 3*x - 4*x/(1 + 4*x - 5*x/(1 + 5*x - ... (continued fraction).
a(n) = Sum_{k=0..n} 3^(n-k)*A089949(n,k). - Philippe Deléham, Oct 16 2006
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k-1/2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013
G.f.: W(0), where W(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+1+R)/( x*(k+1+R) - 1/W(k+1) ))); R=3 is Row R of table A111528 (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
a(n) ~ n! * n^3/6 * (1 - 4/n^2 - 15/n^3 - 99/n^4 - 882/n^5 - 9531/n^6 - 119493/n^7 - 1693008/n^8 - 26638245/n^9 - 459682047/n^10). - Vaclav Kotesovec, Jul 27 2015
From Peter Bala, May 24 2017: (Start)
O.g.f. A(x) = ( Sum_{n >= 0} (n+3)!/3!*x^n ) / ( Sum_{n >= 0} (n+2)!/2!*x^n ).
1/(1 - 3*x*A(x)) = Sum_{n >= 0} (n+2)!/2!*x^n. Cf. A001710.
A(x)/(1 - 3*x*A(x)) = Sum_{n >= 0} (n+3)!/3!*x^n. Cf. A001715.
A(x) satisfies the Riccati equation x^2*A'(x) + 3*x*A^2(x) - (1 + 2*x)*A(x) + 1 = 0.
G.f. as an S-fraction: A(x) = 1/(1 - x/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 3*x/(1 - 6*x/(1 - ... - n*x/(1 - (n+3)*x/(1 - ... ))))))))), by Stokes 1982.
A(x) = 1/(1 + 3*x - 4*x/(1 - x/(1 - 5*x/(1 - 2*x/(1 - 6*x/(1 - 3*x/(1 - ... - (n + 3)*x/(1 - n*x/(1 - ... ))))))))). (End)

A111531 Row 4 of table A111528.

Original entry on oeis.org

1, 1, 6, 46, 416, 4256, 48096, 591536, 7840576, 111226816, 1680157056, 26918720896, 455971214336, 8143926373376, 153013563734016, 3017996904928256, 62369444355076096, 1348096649995841536, 30426167700424728576, 715935203128235401216

Offset: 0

Paul D. Hanna, Aug 06 2005

			(1/4)*(log(1 + 4*x + 20*x^2 + 120*x^3 + ... + (n+3)!/3!)*x^n + ...)
= x + 6/2*x^2 + 46/3*x^3 + 416/4*x^4 + 4256/5*x^5 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf: A111528 (table), A003319 (row 1), A111529 (row 2), A111530 (row 3), A111532 (row 5), A111533 (row 6), A111534 (diagonal).

Cf. A001715, A001720.

T[n_, k_] := T[n, k] = Which[n<0 || k<0, 0, k==0 || k==1, 1, n==0, k!, True, (T[n-1, k+1]-T[n-1, k])/n-Sum[T[n, j]*T[n-1, k-j], {j, 1, k-1}]];
a[n_] := T[4, n];
a /@ Range[0, 19] (* Jean-François Alcover, Oct 01 2019 *)

{a(n)=if(n<0,0,if(n==0,1, (n/4)*polcoeff(log(sum(m=0,n,(m+3)!/3!*x^m) +x*O(x^n)),n)))}
for(n=0,20,print1(a(n),", "))

G.f.: (1/4)*log(Sum_{n>=0} (n+3)!/3!*x^n) = Sum_{n>=1} a(n)*x^n/n.
G.f.: A(x) = 1/(1 + 4*x - 5*x/(1 + 5*x - 6*x/(1 + 6*x - ... (continued fraction).
a(n) = Sum_{k=0..n} 4^(n-k)*A089949(n,k). - Philippe Deléham, Oct 16 2006
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k-1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 05 2013
G.f.: W(0)/4 + 3/4, where W(k) = 1 - x*(k+4)/( x*(k+4) - 1/(1 - x*(k+2)/( x*(k+2) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
a(n) ~ n! * n^4/24 * (1 + 2/n - 5/n^2 - 30/n^3 - 184/n^4 - 1664/n^5 - 18688/n^6 - 245120/n^7 - 3641280/n^8 - 60090368/n^9 - 1086985152/n^10). - Vaclav Kotesovec, Jul 27 2015
From Peter Bala, May 25 2017: (Start)
O.g.f. A(x) = ( Sum_{n >= 0} (n+4)!/4!*x^n ) / ( Sum_{n >= 0} (n+3)!/3!*x^n ).
1/(1 - 4*x*A(x)) = Sum_{n >= 0} (n+3)!/3!*x^n. Cf. A001715.
A(x)/(1 - 4*x*A(x)) = Sum_{n >= 0} (n+4)!/4!*x^n. Cf. A001720.
A(x) satisfies the Riccati equation x^2*A'(x) + 4*x*A^2(x) - (1 + 3*x)*A(x) + 1 = 0.
G.f. as an S-fraction: A(x) = 1/(1 - x/(1 - 5*x/(1 - 2*x/(1 - 6*x/(1 - 3*x/(1 - 7*x/(1 - ... - n*x/(1 - (n+4)*x/(1 - ... ))))))))), by Stokes 1982.
A(x) = 1/(1 + 4*x - 5*x/(1 - x/(1 - 6*x/(1 - 2*x/(1 - 7*x/(1 - 3*x/(1 - ... - (n + 4)*x/(1 - n*x/(1 - ... ))))))))). (End)

A090238 Triangle T(n, k) read by rows. T(n, k) is the number of lists of k unlabeled permutations whose total length is n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 24, 16, 6, 1, 0, 120, 72, 30, 8, 1, 0, 720, 372, 152, 48, 10, 1, 0, 5040, 2208, 828, 272, 70, 12, 1, 0, 40320, 14976, 4968, 1576, 440, 96, 14, 1, 0, 362880, 115200, 33192, 9696, 2720, 664, 126, 16, 1, 0, 3628800, 996480, 247968, 64704, 17312, 4380, 952, 160, 18, 1

Offset: 0

Philippe Deléham, Jan 23 2004, Jun 14 2007

T(n,k) is the number of lists of k unlabeled permutations whose total length is n. Unlabeled means each permutation is on an initial segment of the positive integers. Example: with dashes separating permutations, T(3,2) = 4 counts 1-12, 1-21, 12-1, 21-1. - David Callan, Nov 29 2007
For n > 0, -Sum_{i=0..n} (-1)^i*T(n,i) is the number of indecomposable permutations A003319. - Peter Luschny, Mar 13 2009
Also the convolution triangle of the factorial numbers for n >= 1. - Peter Luschny, Oct 09 2022

			Triangle begins:
  1;
  0,       1;
  0,       2,      1;
  0,       6,      4,      1;
  0,      24,     16,      6,     1;
  0,     120,     72,     30,     8,     1;
  0,     720,    372,    152,    48,    10,     1;
  0,    5040,   2208,    828,   272,    70,    12,    1;
  0,   40320,  14976,   4968,  1576,   440,    96,   14,   1;
  0,  366880, 115200,  33192,  9696,  2720,   664,  126,  16,   1;
  0, 3628800, 996480, 247968, 64704, 17312,  4380,  952, 160,  18,  1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 171, #34.

Another version: A059369.
Row sums: A051296, A003319 (n>0).
Diagonals: A000007, A000142, A059371, A000012, A005843, A054000.
Cf. A084938.

T := proc(n,k) option remember; if n=0 and k=0 then return 1 fi;
if n>0 and k=0 or k>0 and n=0 then return 0 fi;
T(n-1,k-1)+(n+k-1)*T(n-1,k)/k end:
for n from 0 to 10 do seq(T(n,k),k=0..n) od; # Peter Luschny, Mar 03 2016
# Uses function PMatrix from A357368.
PMatrix(10, factorial); # Peter Luschny, Oct 09 2022

T[n_, k_] := T[n, k] = T[n-1, k-1] + ((n+k-1)/k)*T[n-1, k]; T[0, 0] = 1; T[, 0] = T[0, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2018 *)

T(n, k) is given by [0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
T(n, k) = T(n-1, k-1) + ((n+k-1)/k)*T(n-1, k); T(0, 0)=1, T(n, 0)=0 if n > 0, T(0, k)=0 if k > 0.
G.f. for the k-th column: (Sum_{i>=1} i!*t^i)^k = Sum_{n>=k} T(n, k)*t^n.
Sum_{k=0..n} T(n, k)*binomial(m, k) = A084938(m+n, m). - Philippe Deléham, Jan 31 2004
T(n, k) = Sum_{j>=0} A090753(j)*T(n-1, k+j-1). - Philippe Deléham, Feb 18 2004
From Peter Bala, May 27 2017: (Start)
Conjectural o.g.f.: 1/(1 + t - t*F(x)) = 1 + t*x + (2*t + t^2)*x^2 + (6*t + 4*t^2 + t^3)*x^3 + ..., where F(x) = Sum_{n >= 0} n!*x^n.
If true then a continued fraction representation of the o.g.f. is 1 - t + t/(1 - x/(1 - t*x - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - 4*x/(1 - 4*x/(1 - ... ))))))))). (End)

New name using a comment from David Callan by Peter Luschny, Sep 01 2022

A111532 Row 5 of table A111528.

Original entry on oeis.org

1, 1, 7, 61, 619, 7045, 87955, 1187845, 17192275, 264940405, 4326439075, 74593075525, 1353928981075, 25809901069525, 515683999204675, 10779677853137125, 235366439343773875, 5359766538695291125

Offset: 0

Paul D. Hanna, Aug 06 2005

			(1/5)*(log(1 + 5*x + 30*x^2 + 210*x^3 + ... + (n+4)!/4!)*x^n + ...)
= x + 7/2*x^2 + 61/3*x^3 + 619/4*x^4 + 7045/5*x^5 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..440
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Cf: A111528 (table), A003319 (row 1), A111529 (row 2), A111530 (row 3), A111531 (row 4), A111533 (row 6), A111534 (diagonal).

Cf. A001720, A001725.

m = 18; (-1/(5x)) ContinuedFractionK[-i x, 1 + i x, {i, 5, m+4}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 02 2019 *)

{a(n)=if(n<0,0,if(n==0,1, (n/5)*polcoeff(log(sum(m=0,n,(m+4)!/4!*x^m) + x*O(x^n)),n)))} \\ fixed by Vaclav Kotesovec, Jul 27 2015

G.f.: (1/5)*log(Sum_{n>=0} (n+4)!/4!*x^n) = Sum_{n>=1} a(n)*x^n/n.
G.f.: 1/(1 + 5*x - 6*x/(1 + 6*x - 7*x/(1 + 7*x - ... (continued fraction).
a(n) = Sum_{k=0..n} 5^(n-k)*A089949(n,k). - Philippe Deléham, Oct 16 2006
G.f.: (4 + 1/Q(0))/5, where Q(k) = 1 - 3*x + k*x - x*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 04 2013
a(n) ~ n! * n^5/5! * (1 + 5/n - 55/n^3 - 356/n^4 - 3095/n^5 - 35225/n^6 - 475000/n^7 - 7293775/n^8 - 124710375/n^9 - 2339428250/n^10). - Vaclav Kotesovec, Jul 27 2015
From Peter Bala, May 25 2017: (Start)
O.g.f.: A(x) = ( Sum_{n >= 0} (n+5)!/5!*x^n ) / ( Sum_{n >= 0} (n+4)!/4!*x^n ).
1/(1 - 5*x*A(x)) = Sum_{n >= 0} (n+4)!/4!*x^n. Cf. A001720.
A(x)/(1 - 5*x*A(x)) = Sum_{n >= 0} (n+5)!/5!*x^n. Cf. A001725.
A(x) satisfies the Riccati equation x^2*A'(x) + 5*x*A^2(x) - (1 + 4*x)*A(x) + 1 = 0.
G.f. as an S-fraction: A(x) = 1/(1 - x/(1 - 6*x/(1 - 2*x/(1 - 7*x/(1 - 3*x/(1 - 8*x/(1 - ... - n*x/(1 - (n+5)*x/(1 - ... ))))))))), by Stokes 1982.
A(x) = 1/(1 + 5*x - 6*x/(1 - x/(1 - 7*x/(1 - 2*x/(1 - 8*x/(1 - 3*x/(1 - ... - (n + 5)*x/(1 - n*x/(1 - ... ))))))))). (End)

cp's OEIS Frontend

A003149 a(n) = Sum_{k=0..n} k!*(n - k)!.

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A075834 Coefficients of power series A(x) such that n-th term of A(x)^n = n! x^(n-1) for n > 0.

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A059438 Triangle T(n,k) (1 <= k <= n) read by rows: T(n,k) is the number of permutations of [1..n] with k components.

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A111528 Square table, read by antidiagonals, where the g.f. for row n+1 is generated by: x*R_{n+1}(x) = (1+n*x - 1/R_n(x))/(n+1) with R_0(x) = Sum_{n>=0} n!*x^n.

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A104980 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p*(column p+1 of T), or [T^p](m,0) = p*T(p+m,p+1) for all m>=1 and p>=-1.

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A111529 Row 2 of table A111528.

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A111528 Square table, read by antidiagonals, where the g.f. for row n+1 is generated by: xR_{n+1}(x) = (1+nx - 1/R_n(x))/(n+1) with R_0(x) = Sum_{n>=0} n!*x^n.

A104980 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^p) = p(column p+1 of T), or [T^p](m,0) = pT(p+m,p+1) for all m>=1 and p>=-1.