A096789 -id:A096789 - cp's OEIS frontend

A010790 a(n) = n!*(n+1)!.

1, 2, 12, 144, 2880, 86400, 3628800, 203212800, 14631321600, 1316818944000, 144850083840000, 19120211066880000, 2982752926433280000, 542861032610856960000, 114000816848279961600000, 27360196043587190784000000, 7441973323855715893248000000

Offset: 0

N. J. A. Sloane

Let M_n be the symmetrical n X n matrix M_n(i,j)=1/min(i,j); then for n>=0 det(M_n)=(-1)^(n-1)/a(n-1). - Benoit Cloitre, Apr 27 2002
If n women and n men are to be seated around a circular table, with no two of the same sex seated next to each other, the number of possible arrangements is a(n-1). - Ross La Haye, Jan 06 2009
a(n-1) is also the number of (directed) Hamiltonian cycles in the complete bipartite graph K_{n,n}. - Eric W. Weisstein, Jul 15 2011
a(n) is also number of ways to place k nonattacking semi-bishops on an n X n board, sum over all k>=0 (for definition see A187235). - Vaclav Kotesovec, Dec 06 2011
a(n) is number of permutations of {1,2,3,...,2n} such that no odd numbers are adjacent. - Ran Pan, May 23 2015
a(n) is number of permutations of {1,2,3,...,2n+1} such that no odd numbers are adjacent. - Ran Pan, May 23 2015
a(n-1) is the number of elements of the wreath product of S_n and S_2 with cycle partition equal to (2n), where S_n is the symmetric group of order n. - Josaphat Baolahy, Mar 12 2024

			G.f. = 1 + 2*x + 12*x^2 + 144*x^3 + 2880*x^4 + 86400*x^5 + ...

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, pp. 63-65.
Kenneth H. Rosen, Editor-in-Chief, Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000, page 91. [Ross La Haye, Jan 06 2009]

T. D. Noe, Table of n, a(n) for n = 0..100
J. Agapito, On symmetric polynomials with only real zeros and nonnegative gamma-vectors, Linear Algebra and its Applications, Volume 451, 15 June 2014, Pages 260-289.
Steve Gadbois, 104.12 From calendar coincidence to factorials to Ramanujan, The Mathematical Gazette (2020) Vol. 104, Issue 560, 304-306.
Anatol N. Kirillov, On some quadratic algebras. I 1/2: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials, SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 002, 172 p. (2016).
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 268.
S. Tanimoto, Parity alternating permutations and signed Eulerian numbers, Ann. Comb. 14 (2010) 355 (total number of PAPs of [2n+1].)
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Shawn L. Witte, Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory, Ph. D. Dissertation, University of California-Davis (2020).
Index entries for sequences related to factorial numbers

Second column of triangle A129065.

Cf. A004737, A000290, A000108, A126120.

[Factorial(n)*Factorial(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 08 2014

Maple
```
f:= n-> n!*(n+1)!: seq(f(n), n=0..30);
```

s=1;lst={s};Do[s+=(s*=n)*n;AppendTo[lst, s], {n, 1, 4!, 1}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 15 2008 *)
Times@@@Partition[Range[0,25]!,2,1] (* Harvey P. Dale, Jun 17 2011 *)

a(n)= n!^2*(n+1) \\ Charles R Greathouse IV, Jul 31 2011

from math import factorial
def A010790(n): return factorial(n)**2*(n+1) # Chai Wah Wu, Apr 22 2024

[stirling_number1(n,1)*factorial (n-2) for n in range(2, 17)] # Zerinvary Lajos, Jul 07 2009

From Karol A. Penson, Oct 23 2001: (Start)
Integral representation as n-th moment of a positive function f on the positive half axis, where f(x) = 2*sqrt(x)*BesselK(1, 2*sqrt(x)). Then:
a(n) = Integral_{x>=0} x^n * f(x) dx.
G.f.: a(0) = 1 and a(n) = subs(x=0, n!*diff(1/((x-1)^2), x$n)) for n >= 1. (End)
Sum_{i >=0} 1/a(i) = A096789. - Gerald McGarvey, Jun 10 2004
With b(n)=A002378(n) for n>0 and b(0)=1, a(n) = b(n)*b(n-1)...*b(0). - Tom Copeland, Sep 21 2011
a(n) = det(PS(i+1,j), 1 <= i,j <= n), where PS(n,k) are Legendre-Stirling numbers of the second kind. - Mircea Merca, Apr 04 2013
a(n) = (2*n)! / A000108(n) which implies that the e.g.f. of A126120 is Sum_{k>=0} x^(2*k) / a(k). - Michael Somos, Nov 15 2014
0 = a(n)*(+18*a(n+2) - 15*a(n+3) + a(n+4)) + a(n+1)*(-9*a(n+2) - 4*a(n+3)) + a(n+2)*(+3*a(n+2)) for all n>=0. - Michael Somos, Nov 15 2014
From Ilya Gutkovskiy, Jan 20 2017: (Start)
a(n) ~ 2*Pi*n^(2*n+2)/exp(2*n).
Sum_{n>=0} (-1)^n/a(n) = BesselJ(1,2) = 0.576724807756873387202448... = A348607 (End)
D-finite with recurrence: a(n) -n*(n+1)*a(n-1)=0. - R. J. Mathar, Jan 27 2020
a(n) = 1/([x^n] hypergeom([], [2], x)). - Peter Luschny, Sep 13 2024

A001053 a(n+1) = n*a(n) + a(n-1) with a(0)=1, a(1)=0.

Original entry on oeis.org

1, 0, 1, 2, 7, 30, 157, 972, 6961, 56660, 516901, 5225670, 57999271, 701216922, 9173819257, 129134686520, 1946194117057, 31268240559432, 533506283627401, 9634381345852650, 183586751854827751, 3681369418442407670, 77492344539145388821, 1708512949279640961732

Offset: 0

N. J. A. Sloane, R. K. Guy

Denominator of continued fraction given by C(n) = [ 1; 2,3,4,...n ]. Cf. A001040. - Amarnath Murthy, May 02 2001
If initial 1 is omitted, CONTINUANT transform of 0, 1, 2, 3, 4, 5, ...
Number of deco polyominoes of height n having no 1-cell columns. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=1 because the vertical and horizontal dominoes are the deco polyominoes of height 2, of which only the vertical domino does not have 1-cell columns. a(n)=A121554(n,0). - Emeric Deutsch, Aug 16 2006
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 1's along the superdiagonal and the subdiagonal, and consecutive integers from 0 to n-1 along the main diagonal (see Mathematica code below). - John M. Campbell, Jul 08 2011
Conjecture: 2*n!*a(n) is the number of open tours by a rook on an (n X 2) chessboard which starts and ends at the same line of length n. - Mikhail Kurkov, Nov 19 2019

			G.f. = 1 + x^2 + 2*x^3 + 7*x^4 + 30*x^5 + 157*x^6 + 972*x^7 + 6961*x^8 + ...
a(5) = 4*a(4) + a(3) = 4*7+2 = 30.
See A058279 and A058307 for similar recurrences and e.g.f.s. - _Wolfdieter Lang_, May 19 2010

Archimedeans Problems Drive, Eureka, 20 (1957), 15.
M. E. Larsen, Summa Summarum, A. K. Peters, Wellesley, MA, 2007; see p. 35. [From N. J. A. Sloane, Jan 29 2009]
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
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
S. B. Ekhad, Problem 10356, Amer. Math. Monthly, 101 (1994), 75. [From _N. J. A. Sloane_, Jan 29 2009]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Generalized Continued Fraction

A column of A058294.
The square roots of the terms of A144656.
See also the constant in A060997.
Cf. A121554, A001040. A058798

a:=[0,1];; for n in [3..25] do a[n]:=(n-1)*a[n-1]+a[n-2]; od; Concatenation([1], a); # G. C. Greubel, Sep 20 2019

a001053 n = a001053_list !! n
a001053_list = 1 : 0 :
   zipWith (+) a001053_list (zipWith (*) [1..] $ tail a001053_list)
-- Reinhard Zumkeller, Nov 02 2011

I:=[0,1]; [1] cat [n le 2 select I[n] else (n-1)*Self(n-1) + Self(n-2): n in [1..25]]; // G. C. Greubel, Sep 20 2019

a[0]:=1: a[1]:=0: for n from 2 to 23 do a[n]:=(n-1)*a[n-1]+a[n-2] od: seq(a[n],n=0..23); # Emeric Deutsch, Aug 16 2006

a[0]=1; a[1] =0; a[n_]:= (n-1)*a[n-1] + a[n-2]; Table[a[n], {n, 0, 21}] (* Robert G. Wilson v, Feb 24 2005 *)
a[0] = 1; a[1] = 0; a[n_] := Permanent[SparseArray[{{i_, i_} :> i-1, Band[{2, 1}] -> 1, Band[{1, 2}] -> 1}, {n, n}]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* John M. Campbell, Jul 08 2011, updated by Jean-François Alcover, Nov 14 2016 *)
RecurrenceTable[{a[0]==1,a[1]==0,a[n]==(n-1)a[n-1]+a[n-2]},a,{n,30}] (* Harvey P. Dale, Jan 31 2013 *)
a[ n_] := With[ {m = Abs@n}, If[ m < 2, Boole[m == 0],
Gamma[m] HypergeometricPFQ[{3/2 - m/2, 1 - m/2}, {2, 2 - m, 1 - m}, 4]]]; (* Michael Somos, Nov 30 2018 *)

{a(n) = contfracpnqn(vector(abs(n), i, i))[2, 2]}; /* Michael Somos, Sep 25 2005 */

def A001053(n):
    if n < 3: return 1 if n != 1 else 0
    return gamma(n)*hypergeometric([3/2-n/2,1-n/2], [2,2-n,1-n], 4)
[round(A001053(n).n(100)) for n in (0..23)] # Peter Luschny, Sep 11 2014

a(n) = a(-n). for all n in Z. - Michael Somos, Sep 25 2005
E.g.f.: -Pi*(BesselI(1,2)*BesselY(0, 2*I*sqrt(1-x)) + I*BesselY(1, 2*I)*BesselI(0, 2*sqrt(1-x))). Such e.g.f. computations were the result of an e-mail exchange with Gary Detlefs. After differentiation and putting x=0 one has to use simplifications. See the Abramowitz-Stegun handbook, p. 360, 9.1.16 and p. 375, 9.63. - Wolfdieter Lang, May 19 2010
a(n) = 2*K_1(2)*I_n(-2)+2*I_1(2)*K_n(2), where In(z) is the modified Bessel function of the first kind and Kn(x) is the modified Bessel function of the second kind. - Alexander R. Povolotsky, Jan 26 2011
Limit_{n->infinity} a(n)/(n-1)! = BesselI(1,2) = 1.590636854637329... (A096789). - Vaclav Kotesovec, Jan 05 2013, corrected Mar 02 2013
a(n+1) = Sum_{k = 0..floor((n-1)/2)} (n-2*k-1)!*binomial(n-k-1,k) * binomial(n-k,k+1). Cf. A058798. - Peter Bala, Aug 01 2013
a(n) = Gamma(n)*hypergeometric([3/2-n/2, 1-n/2], [2, 2-n, 1-n], 4) for n >= 3. - Peter Luschny, Sep 11 2014
0 = a(n)*(-a(n+2)) + a(n+1)*(a(n+1) + a(n+2) - a(n+3)) + a(n+2)*(a(n+2)) for all n in Z. - Michael Somos, Feb 09 2017
Observed: a(n) = A096789*(n-1)!*(1 + 1/(n-1) + 1/(2*(n-1)^2) + O((n-1)^-3)). - A.H.M. Smeets, Aug 19 2018

More terms from James Sellers, Sep 19 2000

A070910 Decimal expansion of BesselI(0,2).

Original entry on oeis.org

2, 2, 7, 9, 5, 8, 5, 3, 0, 2, 3, 3, 6, 0, 6, 7, 2, 6, 7, 4, 3, 7, 2, 0, 4, 4, 4, 0, 8, 1, 1, 5, 3, 3, 3, 5, 3, 2, 8, 5, 8, 4, 1, 1, 0, 2, 7, 8, 5, 4, 5, 9, 0, 5, 4, 0, 7, 0, 8, 3, 9, 7, 5, 1, 6, 6, 4, 3, 0, 5, 3, 4, 3, 2, 3, 2, 6, 7, 6, 3, 4, 2, 7, 2, 9, 5, 1, 7, 0, 8, 8, 5, 5, 6, 4, 8, 5, 8, 9, 8, 9, 8, 4, 5, 9

Offset: 1

Benoit Cloitre, May 20 2002

			2.2795853023360672674372044408115333532858411...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:5:5 at page 20.

Michael Penn, An exponential trigonometric integral., YouTube video, 2020.
Eric Weisstein's World of Mathematics, Factorial Sums.
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind.

Cf. A096789, A070913 (continued fraction), A006040.

Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2))), this sequence (I(0,2)).

RealDigits[ BesselI[0, 2], 10, 110] [[1]] (* Robert G. Wilson v, Jul 09 2004 *)
(* Or *) RealDigits[ Sum[ 1/(n!n!), {n, 0, Infinity}], 10, 110][[1]]

besseli(0,2) \\ Charles R Greathouse IV, Feb 19 2014

Equals Sum_{k>=0} 1/k!^2.
From Peter Bala, Aug 19 2013: (Start)
Continued fraction expansion: 1/(1 - 1/(2 - 1/(5 - 4/(10 - 9/(17 - ... - (n-1)^2/(n^2+1 - ...)))))). See A006040. Cf. A096789.
This continued fraction is the particular case k = 0 of the result BesselI(k,2) = Sum_{n = 0..oo} 1/(n!*(n+k)!) = 1/(k! - k!/((k+2) - (k+1)/((2*k+5) - 2*(k+2)/((3*k+10) - ... - n*(n+k)/(((n+1)*(n+k+1)+1) - ...))))). See the remarks in A099597 for a sketch of the proof. (End)
From Amiram Eldar, May 29 2021: (Start)
Equals (1/e^2) * Sum_{k>=0} binomial(2*k,k)/k! = e^2 * Sum_{k>=0} (-1)^k*binomial(2*k,k)/k!.
Equal (1/(2*Pi)) * Integral_{x=0..2*Pi} exp(2*sin(x)) dx. (End)
Equals BesselJ(0,2*i). - Jianing Song, Sep 18 2021

A060997 Decimal representation of continued fraction 1, 2, 3, 4, 5, 6, 7, ...

Original entry on oeis.org

1, 4, 3, 3, 1, 2, 7, 4, 2, 6, 7, 2, 2, 3, 1, 1, 7, 5, 8, 3, 1, 7, 1, 8, 3, 4, 5, 5, 7, 7, 5, 9, 9, 1, 8, 2, 0, 4, 3, 1, 5, 1, 2, 7, 6, 7, 9, 0, 5, 9, 8, 0, 5, 2, 3, 4, 3, 4, 4, 2, 8, 6, 3, 6, 3, 9, 4, 3, 0, 9, 1, 8, 3, 2, 5, 4, 1, 7, 2, 9, 0, 0, 1, 3, 6, 5, 0, 3, 7, 2, 6, 4, 3, 5, 7, 8, 6, 1, 1, 4, 6, 5, 9, 5, 0

Offset: 1

Robert G. Wilson v, May 14 2001

The value of this continued fraction is the ratio of two Bessel functions: BesselI(0,2)/BesselI(1,2) = A070910/A096789. Or, equivalently, to the ratio of the sums: Sum_{n>=0} 1/(n!n!) and Sum_{n>=0} n/(n!n!). - Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 31 2003

1.43312...=[1,2,3,4,5,...] = shape of a rectangle which partitions into n squares at stage n; i.e., this is an example of the match between the continued fraction of a number r and a rectangle having shape r. See A188640. - Clark Kimberling, Apr 09 2011

			1.433127426722311758317183455775...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000 (corrected by _Sean A. Irvine_, Apr 29 2022)
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]

Cf. A052119, A001053.

With[{nn = 110}, RealDigits[FromContinuedFraction[Range[nn]], 10, nn][[1]]]
(* Or *) RealDigits[ BesselI[0, 2] / BesselI[1, 2], 10, 110] [[1]]
(* Or *) RealDigits[ Sum[1/(n!n!), {n, 0, Infinity}] / Sum[n/(n!n!), {n, 0, Infinity}], 10, 110] [[1]]

set_display('none)$fpprec:100$bfloat(cfdisrep(makelist(x,x,1,1000))); /* Dimitri Papadopoulos, Oct 25 2022 */

besseli(0,2)/besseli(1,2) \\ Charles R Greathouse IV, Feb 19 2014

1/A052119.

A052119 Decimal expansion of number with continued fraction expansion 0, 1, 2, 3, 4, 5, 6, ...

Original entry on oeis.org

6, 9, 7, 7, 7, 4, 6, 5, 7, 9, 6, 4, 0, 0, 7, 9, 8, 2, 0, 0, 6, 7, 9, 0, 5, 9, 2, 5, 5, 1, 7, 5, 2, 5, 9, 9, 4, 8, 6, 6, 5, 8, 2, 6, 2, 9, 9, 8, 0, 2, 1, 2, 3, 2, 3, 6, 8, 6, 3, 0, 0, 8, 2, 8, 1, 6, 5, 3, 0, 8, 5, 2, 7, 6, 4, 6, 4, 1, 1, 1, 2, 9, 9, 6, 9, 6, 5, 6, 5, 4, 1, 8, 2, 6, 7, 6, 5, 6, 8, 7, 2, 3, 9, 8, 2

Offset: 0

Robert Lozyniak (11(AT)onna.com), Jan 21 2000

			0.697774657964007982006790592551752599486658...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.2, p. 423.

G. C. Greubel, Table of n, a(n) for n = 0..5000
F. Amoretti, Sur la fraction continue [0,1,2,3,4,...], Nouvelles annales de mathématiques, 1ère série, tome 14 (1855), pp. 40-44.
Simon Plouffe, 10000 digits.
Simon Plouffe, Bessell(1,2)/Bessell(0,2).
Eric Weisstein's World of Mathematics, Continued Fraction Constants.
Eric Weisstein's World of Mathematics, Generalized Continued Fraction.
Index entries for sequences related to Bessel functions or polynomials.

Equals 1/A060997.

evalf(BesselI(1, 2)/BesselI(0, 2), 120);  # Alois P. Heinz, Jan 25 2022

RealDigits[ FromContinuedFraction[ Range[0, 44]], 10, 110][[1]]
(* Or *) RealDigits[ BesselI[1, 2] / BesselI[0, 2], 10, 110] [[1]]
(* Or *) RealDigits[ Sum[n/(n!n!), {n, 0, Infinity}] / Sum[1/(n!n!), {n, 0, Infinity}], 10, 110] [[1]] (* Robert G. Wilson v, Jul 09 2004 *)

besseli(1,2)/besseli(0,2) \\ Charles R Greathouse IV, Feb 19 2014

BesselI(1, 2)/BesselI(0, 2) = A096789/A070910. - Henry Bottomley, Jul 13 2001

Equivalently, the value of this continued fraction is the ratio of the sums: sum_{n=0..inf} n/(n!n!) and sum_{n=0..inf} 1/(n!n!). - Robert G. Wilson v, Jul 09 2004

More terms from Vladeta Jovovic, Mar 30 2000

Entry revised by N. J. A. Sloane, Aug 13 2006

A065033 1 appears three times, other numbers twice.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35

Offset: 0

N. J. A. Sloane, Nov 04 2001

Gives the number of terms in n-th row of many common tables.
Number of partitions of the (n+1)-th Fibonacci number into distinct Fibonacci numbers: a(n) = A000119(A000045(n)), see also A098641. - Reinhard Zumkeller, Apr 24 2005
a(n) = length of run n+1 of consecutive 4s in A254338. - Reinhard Zumkeller, Feb 27 2015
This is the Engel expansion of A070910 + A096789. - Benedict W. J. Irwin, Dec 16 2016

Harry J. Smith, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004526, A008619.

Cf. A254338.

a065033 n = 0 ^ n + div (n + 1) 2  -- Reinhard Zumkeller, Feb 27 2015

Array[Floor[#/2] &, 61] /. 0 -> 1 (* Michael De Vlieger, Mar 10 2020 *)

a(n) = { if (n<1, n==0, (n+1)\2) } \\ Harry J. Smith, Oct 03 2009

From Philippe Deléham, Sep 28 2006: (Start)
a(n) = a(n-1)+a(n-2)-a(n-3) for n>3.
G.f.: (1-x^2+x^3)/(1-x-x^2+x^3). (End)
a(n) = floor((n+1)/2) + 0^n. - Reinhard Zumkeller, Feb 27 2015
E.g.f.: (2 + exp(x)*x + sinh(x))/2. - Stefano Spezia, Aug 05 2025

A000994 Shifts 2 places left under binomial transform.

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 13, 36, 109, 359, 1266, 4731, 18657, 77464, 337681, 1540381, 7330418, 36301105, 186688845, 995293580, 5491595645, 31310124067, 184199228226, 1116717966103, 6968515690273, 44710457783760, 294655920067105, 1992750830574681, 13817968813639426

Offset: 0

N. J. A. Sloane

a(n) is the number of permutations of [n-1] that avoid both of the dashed patterns 1-23 and 3-12 and start with a descent (or are a singleton). For example, a(5)=5 counts 2143, 3142, 3214, 3241, 4321. - David Callan, Nov 21 2011

			A(x) = 1 + x^2/(1-x) + x^4/((1-x)^2*(1-2x)) + x^6/((1-x)^2*(1-2x)^2*(1-3x)) +...

Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
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).

Alois P. Heinz, Table of n, a(n) for n = 0..650 (first 101 terms from T. D. Noe)
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
N. J. A. Sloane, Transforms
S. Tauber, On generalizations of the exponential function, Amer. Math. Monthly, 67 (1960), 763-767.

Cf. A000995, A051139, A051140. Cf. A007318.

Column k=2 of A143983. Cf. A007476, A088022, A086880.

a000994 n = a000994_list !! n
a000994_list = 1 : 0 : us where
  us = 1 : 1 : f 2 where
    f x = (1 + sum (zipWith (*) (map (a007318' x) [2..x]) us)) : f (x + 1)
-- Reinhard Zumkeller, Jun 02 2013

A000994 := proc(n) local k; option remember; if n <= 1 then 1 else 1 + add(binomial(n, k)*A000994(k - 2), k = 2 .. n); fi; end;

a[n_] := a[n] = 1 + Sum[Binomial[n, k]*a[k-2], {k, 2, n}]; Join[{1, 0}, Table[a[n], {n, 0, 24}]] (* Jean-François Alcover, Oct 11 2011, after Maple *)

a(n)=polcoeff(sum(k=0, n, x^(2*k)*(1-k*x)/prod(j=0, k, 1-j*x+x*O(x^n))^2), n) \\ Paul D. Hanna, Nov 02 2006

Since this satisfies a recurrence similar to that of the Bell numbers (A000110), the asymptotic behavior is presumably just as complicated - see A000110 for details.
However, a(n)/A000995(n) (e.g., 77464/63117) -> 1.228..., the constant in A051148 and A051149.
O.g.f.: A(x) = Sum_{n>=0} x^(2*n)*(1-n*x)/Product_{k=0..n} (1-k*x)^2. - Paul D. Hanna, Nov 02 2006
Let S(n) = Sum_{k >= 1} k^n/k!^2. Then S(n) = a(n)*S(0) + A000995(n)*S(1) is stated in A086880, where S(0) = 2.279585302... (see A070910) and S(1) = 1.590636854... (see A096789). Cf. A088022. - Peter Bala, Jan 27 2015
G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - x)) / (1 - x). - Ilya Gutkovskiy, Aug 09 2020

A228229 Recurrence a(n) = n(n + 1)a(n-1) + 1 with a(0) = 1.

Original entry on oeis.org

1, 3, 19, 229, 4581, 137431, 5772103, 323237769, 23273119369, 2094580743211, 230403881753211, 30413312391423853, 4744476733062121069, 863494765417306034559, 181333900737634267257391, 43520136177032224141773841, 11837477040152764966562484753

Offset: 0

Peter Bala, Aug 19 2013

Main subdiagonal (and main superdiagonal) of A099597. Cf. A006040 and A228230.

Seiichi Manyama, Table of n, a(n) for n = 0..252
E. W. Weisstein, Modified Bessel Function of the First Kind

Cf. A006040, A096789, A099597, A228230, A368837, A368838.

A228229 :=proc(n) option remember
    if n = 0 then 1
    else n*(n+1)*procname(n-1) + 1
    end if:
end proc:
seq(A228229(n), n = 0..20);

RecurrenceTable[{a[n] == n*(n + 1)*a[n-1] + 1, a[0] == 1},a,{n,0,20}] (* Vaclav Kotesovec, May 06 2015 *)

a(n) = n!*(n + 1)!*sum {k = 0..n} 1/(k!*(k + 1)!).
Generating function: 1/(1 - x)*1/sqrt(x)*BesselI(1, 2*sqrt(x)) = sum {n >= 0} a(n)*x^n/(n!*(n + 1)!).
Defining recurrence equation: a(n) = n*(n + 1)*a(n-1) + 1 with a(0) = 1.
Alternative recurrence equation: a(0) = 1, a(1) = 3, and for n >= 2, a(n) = (n*(n + 1) + 1)*a(n-1) - n*(n - 1)*a(n-2).
The sequence b(n) := n!*(n + 1)! satisfies the same recurrence with the initial conditions b(0) = 1, b(1) = 2. It follows that we have the finite continued fraction expansion a(n) = n!*(n + 1)!*(1/(1 - 1/(3 - 2/(7 - 6/(13 - … - n*(n - 1)/(n^2 + n + 1)))))). Taking the limit yields the continued fraction expansion for the modified Bessel function value BesselI(1,2) = sum {k = 0..inf} 1/(k!*(k + 1)!) = 1/(1 - 1/(3 - 2/(7 - 6/(13 - ...- n*(n - 1)/(n^2 + n + 1 - ...))))) = 1.590636... (see A096789).
a(n) ~ BesselI(1,2) * n!*(n+1)!. - Vaclav Kotesovec, May 06 2015

A229020 Decimal expansion of 1 - 1/(12) + 1/(122) - 1/(1223) + ...

Original entry on oeis.org

6, 8, 8, 9, 4, 8, 4, 4, 7, 6, 9, 8, 7, 3, 8, 2, 0, 4, 0, 5, 4, 9, 5, 0, 0, 1, 5, 8, 1, 1, 8, 6, 7, 1, 0, 5, 3, 3, 1, 3, 6, 2, 9, 4, 3, 2, 8, 9, 9, 2, 2, 4, 0, 6, 9, 3, 8, 5, 5, 1, 7, 6, 7, 0, 5, 5, 7, 6, 0, 3, 0, 5, 6, 9, 7, 3, 1, 5, 1, 5, 7, 6, 1, 3, 3, 9, 4, 9, 4, 0, 9, 6, 2, 2, 5, 6, 9, 7, 3, 7, 4, 6, 8, 3, 9, 1, 0, 7, 1, 3, 2, 5, 5

Offset: 0

Ralf Stephan, Sep 11 2013

From Peter Bala, Jan 28 2015: (Start)
As a sum of positive terms, the constant equals Sum_{k >= 1} k/(k!*(k+1)!). If we set S(n) = Sum_{k >= 0} k^n/(k!*(k+1)!) for n >= 0, so this constant is S(1), then S(n) is an integral linear combination of S(0) and S(1). For example S(7) = 16*S(0) + 11*S(1). Cf. A086880. S(0) is A096789.
The Pierce expansion of this constant begins [1, 3, 14, 15, 26, 40, 43, 71, 83, 8120, ...] giving the alternating series representation for this constant 1 - 1/3 + 1/(3*14) - 1/(3*14*15) + 1/(3*14*15*26) - .... (End)

			0.68894844769873820405495001581186710536...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Eric Weisstein's World of Mathematics, Pierce Expansion.

Cf. A130820.

digits = 113; NSum[(-1)^(n+1)*1/Product[1+Floor[k/2], {k, 1, n}], {n, 1, Infinity}, NSumTerms -> digits, Method -> "AlternatingSigns", WorkingPrecision -> digits+5] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 21 2014 *)
RealDigits[BesselI[2, 2], 10, 113][[1]] (* Jean-François Alcover, Nov 19 2015, after Peter Bala *)

suminf(n=1,(-1)^(n+1)*1./prod(i=1,n,1+floor(i/2)))

suminf(k=1, k/(k!*(k+1)!)) \\ Michel Marcus, Feb 03 2015

besseli(2, 2) \\ Altug Alkan, Nov 19 2015

Equals exp(-2) * Sum_{k>=0} binomial(2*k,k)/(k+1)!. - Amiram Eldar, Jun 12 2021

A099597 Array T(n,k) read by antidiagonals: expansion of exp(x+y)/(1-xy).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 9, 4, 1, 1, 5, 19, 19, 5, 1, 1, 6, 33, 82, 33, 6, 1, 1, 7, 51, 229, 229, 51, 7, 1, 1, 8, 73, 496, 1313, 496, 73, 8, 1, 1, 9, 99, 919, 4581, 4581, 919, 99, 9, 1, 1, 10, 129, 1534, 11905, 32826, 11905, 1534, 129, 10, 1, 1, 11, 163, 2377, 25733, 137431, 137431, 25733, 2377, 163, 11, 1

Offset: 0

Ralf Stephan, Oct 28 2004

Rows are polynomials in n whose coefficients are in A099599.
From Peter Bala, Aug 19 2013: (Start)
The k-th superdiagonal sequence of this square array occurs as the sequence of numerators in the convergents to a certain continued fraction representation of the constant BesselI(k,2), where BesselI(k,x) is a modified Bessel function of the first kind:
Let d_k(n) = T(n,n+k) = n! * (n+k)! * Sum_{i=0..n} 1/(i!*(i+k)!) denote the sequence of entries on the k-th superdiagonal. It satisfies the first-order recurrence equation d_k(n) = n*(n+k)*d_k(n-1) + 1 with d_k(0) = 1 and also the second-order recurrence d_k(n) = (n*(n+k)+1)*d_k(n-1) - (n-1)*(n-1+k)*d_k(n-2) with initial conditions d_k(0) = 1 and d_k(1) = k+2. This latter recurrence is also satisfied by the sequence n!*(n+k)!. From this observation we obtain the finite continued fraction expansion d_k(n) = n!*(n+k)!*(1/(k! - k!/((k+2) - (k+1)/((2*k+5) - 2*(k+2)/((3*k+10) - ... - n*(n+k)/(((n+1)*(n+k+1)+1) ))))).
Taking the limit as n -> infinity produces a continued fraction representation for the modified Bessel function value BesselI(k,2) = Sum_{i=0..inf} 1/(i!*(i+k)!) = 1/(k! - k!/((k+2) -(k+1)/((2*k+5) - 2*(k+2)/((3*k+10) - ... - n*(n+k)/(((n+1)*(n+k+1)+1) - ...))))). See A070910 for the case k = 0 and A096789 for the case k = 1. (End)

			1, 1,  1,   1,    1,     1,
1, 2,  3,   4,    5,     6,
1, 3,  9,  19,   33,    51,
1, 4, 19,  82,  229,   496,
1, 5, 33, 229, 1313,  4581,
1, 6, 51, 496, 4581, 32826,

E. W. Weisstein, Modified Bessel Function of the First Kind

Rows include A000012, A000027, A058331. Main diagonal is A006040. Antidiagonal sums are in A099598. Cf. A099599.

Cf. A088699. A228229 (main super and subdiagonal).

#A099597
T := proc(n,k) option remember;
if n = 0 then 1 elif k = 0 then 1
else n*k*thisproc(n-1,k-1) + 1
fi
end:
# Diplay entries by antidiagonals
seq(seq(T(n-k,k), k = 0..n), n = 0..10);
# Peter Bala, Aug 19 2013

T[, 0] = T[0, ] = 1;
T[n_, k_] := T[n, k] = n k T[n - 1, k - 1] + 1;
Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 02 2019 *)

T(n,k) = Sum_{i=0..min(n,k)} C(n,i)*C(k,i)*i!^2. The LDU factorization of this square array is P * D * transpose(P), where P is Pascal's triangle A007318 and D = diag(0!^2, 1!^2, 2!^2, ... ). Compare with A088699. - Peter Bala, Nov 06 2007
Recurrence equation: T(n,k) = n*k*T(n-1,k-1) + 1 with boundary conditions T(n,0) = T(0,n ) = 1.
Main subdiagonal and main superdiagonal [1, 3, 19, 229, ...] is A228229. - Peter Bala, Aug 19 2013
nth row/column o.g.f.: HypergeometricPFQ[{1,1,-n},{},x/(x-1)]/(1-x) (see comment in A099599). - Natalia L. Skirrow, Jul 18 2025

cp's OEIS Frontend

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A060997 Decimal representation of continued fraction 1, 2, 3, 4, 5, 6, 7, ...

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A052119 Decimal expansion of number with continued fraction expansion 0, 1, 2, 3, 4, 5, 6, ...

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A065033 1 appears three times, other numbers twice.

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A228229 Recurrence a(n) = n(n + 1)a(n-1) + 1 with a(0) = 1.

A229020 Decimal expansion of 1 - 1/(12) + 1/(122) - 1/(1223) + ...