A292219 -id:A292219 - cp's OEIS frontend

A000407 a(n) = (2*n+1)! / n!.

1, 6, 60, 840, 15120, 332640, 8648640, 259459200, 8821612800, 335221286400, 14079294028800, 647647525324800, 32382376266240000, 1748648318376960000, 101421602465863680000, 6288139352883548160000, 415017197290314178560000

Offset: 0

N. J. A. Sloane

The e.g.f. of 1/a(n) = n!/(2*n+1)! is (exp(sqrt(x)) - exp(-sqrt(x)))/(2*sqrt(x)). - Wolfdieter Lang, Jan 09 2012
Product of the larger parts of the partitions of 2n+2 into exactly two parts. - Wesley Ivan Hurt, Jun 15 2013
For n > 0, a(n-1) = (2n-1)!/(n-1)!, the number of ways n people can line up in n labeled queues. The derivation is straightforward. Person 1 has (2n-1) choices - be first in line in one of the queues or get behind one of the other people. Person 2 has (2n-2) choices - choose one of the n queues or get behind one of the remaining n-2 people. Continuing in this fashion, we finally find that person n has to choose one of the n queues. - Dennis P. Walsh, Mar 24 2016
For n > 0, a(n-1) is the number of functions f:[n]->[2n] that are acyclic and injective. Note that f is acyclic if, for all x in [n], x is not a member of the set {f(x),f(f(x)), f(f(f(x))), ...}. - Dennis P. Walsh, Mar 25 2016
a(n) is the number of labeled maximal outerplanar graphs with n-3 vertices. - Allan Bickle, Feb 19 2024

			G.f. = 1 + 6*x + 60*x^2 + 840*x^3 + 15120*x^4 + 332640*x^5 + 8648640*x^6 + ...
For n=1 the a(1)=6 ways for 2 people to line up in 2 queues are as follows: Q1<P1,P2> Q2<>, Q1<P2,P1> Q2<>, Q1<P1> Q2<P2>, Q1<P2> Q2<P1>, Q1<> Q2<P1,P2>, Q1<> Q2<P2,P1>. - _Dennis P. Walsh_, Mar 24 2016
For the unique maximal outerplanar graph with 4 vertices, there are C(4,2)=6 ways to label the two degree 3 vertices, and the other two labels are forced.  Thus a(1) = 6.

L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, pp. 12-26 of Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970. Reprinted with a slightly different title in Math. Annalen, 191 (1971), 87-98.
L. B. W. Jolley, Summation of Series, Dover, 1961.
Loren C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
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).

N. J. A. Sloane, Table of n, a(n) for n = 0..100
L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970, pp. 12-26. Reprinted with a slightly different title in Math. Annalen, Vol. 191 (1971), pp. 87-98.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Peter J. Cameron, Sequences Realized by Oligomorphic Permutation Groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
G. Hu, Catalan number and enumeration of maximal outerplanar graphs, Tsinghua Science and Technology 25 5 1 (2000), 109-114.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 139.
Loren C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Dan Levy and Lior Pachter, The Neighbor-Net Algorithm, arXiv:math/0702515 [math.CO], 2007-2008.
Lee A. Newberg, The Number of Clone Orderings, Discrete Applied Mathematics, Vol. 69, No. 3 (1996), pp. 233-245.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Robert W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
Herbert E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., Vol. 3, No. 23 (1948), pp. 167-169.
Herbert E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp. Vol. 9, No. 52 (1955), pp. 164-177.
Herbert E. Salzer, Orthogonal polynomials arising in the evaluation of inverse Laplace transforms, Math. Comp., Vol. 9, No. 52 (1955), 164-177. [Annotated scanned copy]
Maxie D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq., Vol. 13 (2010), Article 10.6.7, page 39.
Wikipedia, William Brouncker, 2nd Viscount Brouncker.
Index to divisibility sequences
Index entries for sequences related to factorial numbers

Cf. A001761-A001763, A007696, A024199, A073010.
A100622 is the "Number of topologically distinct solutions to the clone ordering problem for n clones" without the restriction that they be in a single contig (see [Newberg] for definition of contig).
Column m=0 of A292219.

[Factorial(2*n+1) / Factorial(n): n in [0..20]]; // Vincenzo Librandi, Jun 16 2015

For Maple program see A000903.
a := n -> pochhammer(n+1,n+1); (for n>=0) # Peter Luschny, Feb 14 2009

Table[(2n + 1)!/n!, {n, 0, 30}] (* Stefan Steinerberger, Apr 08 2006 *)
a[ n_] := If[ n < 0, 1/2, 1] Pochhammer[ n + 1, n + 1]; (* Michael Somos, Jan 03 2015 *)
a[ n_] := Which[ n < -1, -(-1)^n / (4 a[-n - 2]), n == -1, 1/2, True, (2 n + 1)! / n!]; (* Michael Somos, Jan 03 2015 *)

A000407(n):=(2*n+1)!/n!$
makelist(A000407(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=(2*n+1)!/n! \\ Charles R Greathouse IV, Jan 12 2012

{a(n) = if( n<-1, -(-1)^n / (4 * a(-n-2)), n==-1, 1/2, (2*n + 1)! / n!)}; /* Michael Somos, Jan 03 2015 */

E.g.f.: (1 - 4*x)^(-3/2). - Michael Somos, Jan 03 2015
E.g.f.: Sum_{k>=0} a(k+2) * x^k / k! = (1 - 2*x - sqrt(1 - 4*x)) / 4.
E.g.f. for a(n-1), n >= 0, with a(-1) := 0 is (-1+1/(1-4*x)^(1/2))/2. 2*a(n) = (4*n+2)(!^4) := Product_{j=0..n} (4*j + 2), (one half of 4-factorial numbers). - Wolfdieter Lang
a(n) = C(n+1)*(n+2)!/2 for all n>=0. - Paul Barry, Feb 16 2005
For n>1, a(n) = (1/2)*A001813(n+1). - Zerinvary Lajos, Jun 06 2007
For asymptotics see the Robinson paper.
Sum_{n >=0} n!/a(n) = 2*Pi/3^(3/2) =  1.2091995761... = A248897 [Jolley eq 261]
G.f.: 1 / (1 - 6*x / (1 - 4*x / (1 - 10*x / (1 - 8*x / (1 - 14*x / ... ))))). - Michael Somos, May 12 2012
G.f.: 1/Q(0), where Q(k) = 1 + 2*(2*k-1)*x - 4*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 2*x/(2*x + 1/(2*k+3)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
a(n) = -(-1)^n / (4 * a(-2-n)) = a(n-1) * (4*n+2) for all n in Z. - Michael Somos, Jan 03 2015
a(n) = A087299(2*n + 1). - Michael Somos, Jan 03 2015
From Peter Bala, Feb 16 2015: (Start)
Recurrence equation: a(n) = 4*a(n-1) + 4*(2*n - 1)^2*a(n-2) with a(0) = 1 and a(1) = 6.
The integer sequence b(n) := a(n)*Sum_{k = 0..n} (-1)^k/(2*k + 1), beginning [1, 4, 52, 608, 12624, ...], satisfies the same second-order recurrence equation. This leads to Brouncker's generalized continued fraction expansion Sum_{k >= 0} (-1)^k/(2*k + 1) = Pi/4 = 1/(1 + 1^2/(2 + 3^2/(2 + 5^2/(2 + ... )))). Note b(n) = 2^n*A024199(n+1).
Recurrence equation: a(n) = (5*n + 2)*a(n-1) - 2*n*(2*n - 1)^2*a(n-2) with a(0) = 1 and a(1) = 6.
The integer sequence c(n) := a(n)*Sum_{k = 0..n} k!^2/(2*k + 1)!, beginning [1, 7, 72, 1014, 18276, ... ], satisfies the same second-order recurrence equation. This leads to the generalized continued fraction expansion Sum_{k >= 0} k!^2/(2*k + 1)! = 2*Pi/sqrt(27) = 2*A073010 = 1/(1 - 1/(7 - 12/(12 - 30/(17 - ... - 2*n*(2*n - 1)/((5*n + 2) - ... ))))). (End)
a(n) = Product_{k=n+1..(2*n+1)} k. - Carlos Eduardo Olivieri, Jun 03 2015
From Ilya Gutkovskiy, Jan 17 2017: (Start)
a(n) ~ 2^(2*n+3/2)*n^(n+1)/exp(n).
Sum_{n>=0} 1/a(n) = exp(1/4)*sqrt(Pi)*erf(1/2) = 1.184593072938653151..., where erf() is the error function. (End)
Sum_{n>=0} (-1)^n/a(n) = exp(-1/4)*sqrt(Pi)*erfi(1/2), where erfi() is the imaginary error function. - Amiram Eldar, Jan 18 2021
It follows from the comments above that we have a(n) = a(n-1)*(4*n+2), with a(1) = 6, a(0) = 1.
a(n) = A081125(2*n+1). - R. J. Mathar, Jun 07 2025

A103327 Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).

Original entry on oeis.org

1, 3, 1, 5, 10, 1, 7, 35, 21, 1, 9, 84, 126, 36, 1, 11, 165, 462, 330, 55, 1, 13, 286, 1287, 1716, 715, 78, 1, 15, 455, 3003, 6435, 5005, 1365, 105, 1, 17, 680, 6188, 19448, 24310, 12376, 2380, 136, 1, 19, 969, 11628, 50388, 92378, 75582, 27132, 3876, 171, 1

Offset: 0

Ralf Stephan, Feb 06 2005

A subset of Pascal's triangle A007318.
Elements have the same parity as those of Pascal's triangle.
Matrix inverse is A104033. - Paul D. Hanna, Feb 28 2005
Row reverse of A091042. - Peter Bala, Jul 29 2013
Let E(y) = cosh(sqrt(y)) = 1 + 3*y/3! + 5*y^2/5! + 7*y^3/7! + .... Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence (2*n+1)! as defined in Wang and Wang. Cf. A086645. - Peter Bala, Aug 06 2013
The row polynomial P(d, x) = Sum_{k=0..d} T(d, k)*x^k, multiplied by (2*d)!/d! = A001813(d), gives the numerator polynomial of the o.g.f. of the sequence of the diagonal d, for d >= 0, of the Sheffer triangle Lah[4,3] given in A292219. - Wolfdieter Lang, Oct 12 2017

			The triangle T(n, k) begins:
n\k   0    1     2      3      4      5      6     7    8   9  10 ...
0:    1
1:    3    1
2:    5   10     1
3:    7   35    21      1
4:    9   84   126     36      1
5:   11  165   462    330     55      1
6:   13  286  1287   1716    715     78      1
7:   15  455  3003   6435   5005   1365    105     1
8:   17  680  6188  19448  24310  12376   2380   136    1
9:   19  969 11628  50388  92378  75582  27132  3876  171   1
10:  21 1330 20349 116280 293930 352716 203490 54264 5985 210   1
... reformatted and extended. - _Wolfdieter Lang_, Oct 12 2017
From _Peter Bala_, Aug 06 2013: (Start)
Viewed as the generalized Riordan array (cosh(sqrt(y)), y) with respect to the sequence (2*n+1)! the column generating functions begin
1st col: cosh(sqrt(y)) = 1 + 3*y/3! + 5*y^2/5! + 7*y^3/7! + 9*y^4/9! + ....
2nd col: 1/3!*y*cosh(sqrt(y)) = y/3! + 10*y^2/5! + 35*y^3/7! + 84*y^4/9! + ....
3rd col: 1/5!*y^2*cosh(sqrt(y)) = y^2/5! + 21*y^3/7!! + 126*y^4/9! + 462*y^5/11! + .... (End)

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224.

Indranil Ghosh, Rows 0..120 of triangle, flattened
W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Reflected version of A091042. Cf. A086645, A103328.

Cf. A104033, A086645, A292219.

Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n+1, 2*k+1) ))); # G. C. Greubel, Aug 01 2019

[Binomial(2*n+1, 2*k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019

Flatten[Table[Binomial[2n+1,2k+1],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jun 19 2014 *)

create_list(binomial(2*n+1,2*k+1),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

{T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1+X*(1-Y))/((1+X*(1-Y))^2-4*X),n,x),k,y)} \\ Paul D. Hanna, Feb 28 2005

T(n,k) = binomial(2*n+1, 2*k+1);
for(n=0, 12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Aug 01 2019

[[binomial(2*n+1, 2*k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

G.f. for column k: Sum_{j=0..k+1} C(2*(k+1), 2*j)*x^j/(1-x)^(2*(k+1)). - Paul Barry, Feb 24 2005
G.f.: A(x, y) = (1 + x*(1-y))/( (1 + x*(1-y))^2 - 4*x ). - Paul D. Hanna, Feb 28 2005
Sum_{k=0..n} T(n, k)*A000364(n-k) = A002084(n). - Philippe Deléham, Aug 27 2005
E.g.f.: 1/sqrt(x)*sinh(sqrt(x)*t)*cosh(t) = t + (3 + x)*t^3/3! + (5 + 10*x + x^2)*t^5/5! + .... - Peter Bala, Jul 29 2013
T(n+2,k+2) = 2*T(n+1,k+2) + 2*T(n+1,k+1) - T(n,k+2) + 2*T(n,k+1) - T(n,k). - Emanuele Munarini, Jul 05 2017

A292221 Expansion of the exponential generating function (1/2)(1 + 4x)(1 - (1 + 4x)^(-3/2))/x.

Original entry on oeis.org

3, -3, 20, -210, 3024, -55440, 1235520, -32432400, 980179200, -33522128640, 1279935820800, -53970627110400, 2490952020480000, -124903451312640000, 6761440164390912000, -393008709555221760000, 24412776311194951680000, -1613955767240110694400000, 113146793787569865523200000, -8384177419658927035269120000

Offset: 0

Wolfdieter Lang, Sep 23 2017

This gives one half of the z-sequence for the generalized unsigned Lah number Sheffer matrix Lah[4,3] = A292219.
For Sheffer a- and z-sequences see a W. Lang link under A006232 with the references for the Riordan case, and also the present link.
a(n) = (-1)^n * A006963(n+2) for all n >= 0. - Michael Somos, Jul 02 2018

			The sequence z(4,3;n) = 2*a(n) begins: {6, -6, 40, -420, 6048, -110880, 2471040, -64864800, 1960358400, -67044257280, 2559871641600, ...}.

G. C. Greubel, Table of n, a(n) for n = 0..365
Wolfdieter Lang, Note on a- and z-sequences of Sheffer number triangles for certain generalized Lah numbers.

Cf. A001147, A006232, A006963, A292219.

[3,-3] cat [(-1)^n*Factorial(2*n+1)/Factorial(n+1): n in [2..30]]; // G. C. Greubel, Jul 28 2018

seq(coeff(series(factorial(n)*(1/2)*(1+4*x)*(1-(1+4*x)^(-3/2))/x, x,n+1),x,n),n=0..20); # Muniru A Asiru, Jul 29 2018

a[ n_] := If[ n < 1, 3 Boole[n == 0], (-2)^n (2 n + 1)!! / (n + 1)]; (* Michael Somos, Jul 02 2018 *)
With[{nn=20},CoefficientList[Series[1/2 (1+4x) (1-(1+4x)^(-3/2))/x,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 24 2018 *)

{a(n) = if( n<1, 3*(n==0), (-1)^n * (2*n + 1)! / (n + 1)!)}; /* Michael Somos, Jul 02 2018 */

a(n) = [x^n/n!] (1/2)*(1 + 4*x)*(1 - (1 + 4*x)^(-3/2))/x.
a(0) = 3, a(n) = (-2)^n*Product_{j=1..n} (1 + 2*j)/(n+1) = ((-2)^n/(n+1))*A001147(n+1), n >= 1.
0 = a(n)*(-2880*a(n+2) +2760*a(n+3) +952*a(n+4) +45*a(n+5)) +a(n+1)*(+216*a(n+2) -328*a(n+3) -81*a(n+4) -2*a(n+5)) +a(n+2)*(+12*a(n+3) +2*a(n+4)) for all n>0. - Michael Somos, Jul 02 2018

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A000407 a(n) = (2*n+1)! / n!.

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A103327 Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).

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A292221 Expansion of the exponential generating function (1/2)*(1 + 4*x)*(1 - (1 + 4*x)^(-3/2))/x.

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A292221 Expansion of the exponential generating function (1/2)(1 + 4x)(1 - (1 + 4x)^(-3/2))/x.