A002544 -id:A002544 - cp's OEIS frontend

A063007 T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.

1, 1, 2, 1, 6, 6, 1, 12, 30, 20, 1, 20, 90, 140, 70, 1, 30, 210, 560, 630, 252, 1, 42, 420, 1680, 3150, 2772, 924, 1, 56, 756, 4200, 11550, 16632, 12012, 3432, 1, 72, 1260, 9240, 34650, 72072, 84084, 51480, 12870, 1, 90, 1980, 18480, 90090, 252252, 420420, 411840, 218790, 48620

Offset: 0

Henry Bottomley, Jul 02 2001

T(n,k) is the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's Cluster algebra of finite type B_n. Take a row of this triangle regarded as a polynomial in x and rewrite as a polynomial in y := x+1. The coefficients of the polynomial in y give a row of triangle A008459 (squares of binomial coefficients). For example, x^2+6*x+6 = y^2+4*y+1. - Paul Boddington, Mar 07 2003
T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k N=(0,1) steps. E.g. T(2,0)=1 because we have DD; T(2,1) = 6 because we have NED, NDE, EDN, END, DEN and DNE; T(2,2)=6 because we have NNEE, NENE, NEEN, EENN, ENEN and ENNE. - Emeric Deutsch, Apr 20 2004
Another version of [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] = 1; 1, 0; 1, 2, 0; 1, 6, 6, 0; 1, 12, 30, 20, 0; ..., where DELTA is the operator defined in A084938. - Philippe Deléham Apr 15 2005
Terms in row n are the coefficients of the Legendre polynomial P(n,2x+1) with increasing powers of x.
From Peter Bala, Oct 28 2008: (Start)
Row n of this triangle is the f-vector of the simplicial complex dual to an associahedron of type B_n (a cyclohedron) [Fomin & Reading, p.60]. See A008459 for the corresponding h-vectors for associahedra of type B_n and A001263 and A033282 respectively for the h-vectors and f-vectors for associahedra of type A_n.
An alternative description of this triangle in terms of f-vectors is as follows. Let A_n be the root lattice generated as a monoid by {e_i - e_j: 0 <= i,j <= n+1}. Let P(A_n) be the polytope formed by the convex hull of this generating set. Then the rows of this array are the f-vectors of a unimodular triangulation of P(A_n) [Ardila et al.]. A008459 is the corresponding array of h-vectors for these type A_n polytopes. See A127674 (without the signs) for the array of f-vectors for type C_n polytopes and A108556 for the array of f-vectors associated with type D_n polytopes.
The S-transform on the ring of polynomials is the linear transformation of polynomials that is defined on the basis monomials x^k by S(x^k) = binomial(x,k) = x(x-1)...(x-k+1)/k!. Let P_n(x) denote the S-transform of the n-th row polynomial of this array. In the notation of [Hetyei] these are the Stirling polynomials of the type B associahedra. The first few values are P_1(x) = 2*x + 1, P_2(x) = 3*x^2 + 3*x + 1 and P_3(x) = (10*x^3 + 15*x^2 + 11*x + 3)/3. These polynomials have their zeros on the vertical line Re x = -1/2 in the complex plane, that is, the polynomials P_n(-x) satisfy a Riemann hypothesis. See A142995 for further details. The sequence of values P_n(k) for k = 0,1,2,3, ... produces the n-th row of A108625. (End)
This is the row reversed version of triangle A104684. - Wolfdieter Lang, Sep 12 2016
T(n, k) is also the number of (n-k)-dimensional faces of a convex n-dimensional Lipschitz polytope of real functions f defined on the set X = {1, 2, ..., n+1} which satisfy the condition f(n+1) = 0 (see Gordon and Petrov). - Stefano Spezia, Sep 25 2021
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial ((x+1)*(x+2)*(x+3)*...*(x+n) / n!)^2 in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 09 2022
Chapoton's observation above is correct: the precise expansion is ((x+1)*(x+2)*(x+3)*...*(x+n)/ n!)^2 = Sum_{k = 0..n} (-1)^k*T(n,n-k)*binomial(x+2*n-k, 2*n-k), as can be verified using the WZ algorithm. For example, n = 3 gives ((x+1)*(x+2)*(x+3)/3!)^2 = 20*binomial(x+6,6) - 30*binomial(x+5,5) + 12*binomial(x+4,4) - binomial(x+3,3). - Peter Bala, Jun 24 2023

			The triangle T(n, k) starts:
  n\k 0  1    2     3     4      5      6      7      8     9
  0:  1
  1:  1  2
  2:  1  6    6
  3:  1 12   30    20
  4:  1 20   90   140    70
  5:  1 30  210   560   630    252
  6:  1 42  420  1680  3150   2772    924
  7:  1 56  756  4200 11550  16632  12012   3432
  8:  1 72 1260  9240 34650  72072  84084  51480  12870
  9:  1 90 1980 18480 90090 252252 420420 411840 218790 48620
... reformatted by _Wolfdieter Lang_, Sep 12 2016
From _Petros Hadjicostas_, Jul 11 2020: (Start)
Its inverse (from Table II, p. 92, in Ser's book) is
   1;
  -1/2,  1/2;
   1/3, -1/2,    1/6;
  -1/4,  9/20,  -1/4,   1/20;
   1/5, -2/5,    2/7,  -1/10,  1/70;
  -1/6,  5/14, -25/84,  5/36, -1/28,  1/252;
   1/7, -9/28,  25/84, -1/6,   9/154, -1/84, 1/924;
   ... (End)

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 366.
J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, Table I, p. 92.
D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.

T. D. Noe, Rows n = 0..100 of triangle, flattened
F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices, arXiv:0809.5123 [math.CO], 2008.
Peter Bala, Deformations of the Hadamard product of power series.
Cyril Banderier, Combinatoire analytique des chemins et des cartes, Thesis (2001), page 49.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009), #09.7.6.
H. J. Brothers, Pascal's Prism: Supplementary Material.
David Callan, A bijection for Delannoy paths, arXiv:2202.04649 [math.CO], 2022.
F. Chapoton, Enumerative properties of generalized associahedra, Séminaire Lotharingien de Combinatoire, B51b (2004), 16 pp.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Mark Dukes and Chris D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, Electronic Journal Of Combinatorics, 23(1) (2016), #P1.45.
S. Fomin and N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004; arXiv:math/0505518 [math.CO], 2005-2008.
S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15(2) (2002), 497-529.
S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
J. Gordon and F. Petrov, Combinatorics of the Lipschitz Polytope, Arnold Mathematical Journal (2016).
G. Hetyei, Face enumeration using generalized binomial coefficients. This is the draft version of Hetyei's paper referenced below. [Archived version]
Gabor Hetyei, The Stirling polynomial of a simplicial complex Discrete and Computational Geometry 35(3) (2006), 437-455.
Hsien-Kuei Hwang and Satoshi Kuriki, Integrated empirical measures and generalizations of classical goodness-of-fit statistics, arXiv:2404.06040 [math.ST], 2024. See p. 11.
C. Lanczos, Applied Analysis (Annotated scans of selected pages). See page 514.
T. Manneville and V. Pilaud, Compatibility fans for graphical nested complexes, arXiv preprint arXiv:1501.07152 [math.CO], 2015.
Thomas Selig, Combinatorial aspects of sandpile models on wheel and fan graphs, arXiv:2202.06487 [math.CO], 2022.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages) See Table I, page 92.
V. Strehl, Recurrences and Legendre transform, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.
R. A. Sulanke, Objects counted by the central Delannoy numbers., J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.
D. Zagier, Integral solutions of Apery-like recurrence equations.

See A331430 for an essentially identical triangle, except with signed entries.
Columns include A000012, A002378, A033487 on the left and A000984, A002457, A002544 on the right.
Main diagonal is A006480.
Row sums are A001850. Alternating row sums are A033999.
Cf. A007318, A008459, A009766, A046899, A104684, A109983, A127674.
Cf. A033282 (f-vectors type A associahedra), A108625, A080721 (f-vectors type D associahedra).
The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

a063007 n k = a063007_tabl !! n !! k
a063007_row n = a063007_tabl !! n
a063007_tabl = zipWith (zipWith (*)) a007318_tabl a046899_tabl
-- Reinhard Zumkeller, Nov 18 2014

/* As triangle: */ [[Binomial(n,k)*Binomial(n+k,k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 03 2015

p := (n,x) -> orthopoly[P](n,1+2*x): seq(seq(coeff(p(n,x),x,k), k=0..n), n=0..9);

Flatten[Table[Binomial[n, k]Binomial[n + k, k], {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Dec 24 2011 *)
Table[CoefficientList[Hypergeometric2F1[-n, n + 1, 1, -x], x], {n, 0, 9}] // Flatten
(* Peter Luschny, Mar 09 2018 *)

{T(n, k) = local(t); if( n<0, 0, t = (x + x^2)^n; for( k=1, n, t=t'); polcoeff(t, k) / n!)} /* Michael Somos, Dec 19 2002 */

{T(n, k) = binomial(n, k) * binomial(n+k, k)} /* Michael Somos, Sep 22 2013 */

{T(n, k) = if( k<0 || k>n, 0, (n+k)! / (k!^2 * (n-k)!))} /* Michael Somos, Sep 22 2013 */

T(n, k) = (n+k)!/(k!^2*(n-k)!) = T(n-1, k)*(n+k)/(n-k) = T(n, k-1)*(n+k)*(n-k+1)/k^2 = T(n-1, k-1)*(n+k)*(n+k-1)/k^2.
binomial(x, n)^2 = Sum_{k>=0} T(n,k) * binomial(x, n+k). - Michael Somos, May 11 2012
T(n, k) = A109983(n, k+n). - Michael Somos, Sep 22 2013
G.f.: G(t, z) = 1/sqrt(1-2*z-4*t*z+z^2). Row generating polynomials = P_n(1+2z), i.e., T(n, k) = [z^k] P_n(1+2*z), where P_n are the Legendre polynomials. - Emeric Deutsch, Apr 20 2004
Sum_{k>=0} T(n, k)*A000172(k) = Sum_{k>=0} T(n, k)^2 = A005259(n). - Philippe Deléham, Jun 08 2005
1 + z*d/dz(log(G(t,z))) = 1 + (1 + 2*t)*z + (1 + 8*t + 8*t^2)*z^2 + ... is the o.g.f. for a signed version of A127674. - Peter Bala, Sep 02 2015
If R(n,t) denotes the n-th row polynomial then x^3 * exp( Sum_{n >= 1} R(n,t)*x^n/n ) = x^3 + (1 + 2*t)*x^4 + (1 + 5*t + 5*t^2)*x^5 + (1 + 9*t + 21*t^2 + 14*t^3)*x^6 + ... is an o.g.f for A033282. - Peter Bala, Oct 19 2015
P(n,x) := 1/(1 + x)*Integral_{t = 0..x} R(n,t) dt are (modulo differences of offset) the row polynomials of A033282. - Peter Bala, Jun 23 2016
From Peter Bala, Mar 09 2018: (Start)
R(n,x) = Sum_{k = 0..n} binomial(2*k,k)*binomial(n+k,n-k)*x^k.
R(n,x) = Sum_{k = 0..n} binomial(n,k)^2*x^k*(1 + x)^(n-k).
n*R(n,x) = (1 + 2*x)*(2*n - 1)*R(n-1,x) - (n - 1)*R(n-2,x).
R(n,x) = (-1)^n*R(n,-1 - x).
R(n,x) = 1/n! * (d/dx)^n ((x^2 + x)^n). (End)
The row polynomials are R(n,x) = hypergeom([-n, n + 1], [1], -x). - Peter Luschny, Mar 09 2018
T(n,k) = C(n+1,k)*A009766(n,k). - Bob Selcoe, Jan 18 2020 (Connects this triangle with the Catalan triangle. - N. J. A. Sloane, Jan 18 2020)
If we let A(n,k) = (-1)^(n+k)*(2*k+1)*(n*(n-1)*...*(n-(k-1)))/((n+1)*...*(n+(k+1))) for n >= 0 and k = 0..n, and we consider both T(n,k) and A(n,k) as infinite lower triangular arrays, then they are inverses of one another. (Empty products are by definition 1.) See the example below. The rational numbers |A(n,k)| appear in Table II on p. 92 in Ser's (1933) book. - Petros Hadjicostas, Jul 11 2020
From Peter Bala, Nov 28 2021: (Start)
Row polynomial R(n,x) = Sum_{k >= n} binomial(k,n)^2 * x^(k-n)/(1+x)^(k+1) for x > -1/2.
R(n,x) = 1/(1 + x)^(n+1) * hypergeom([n+1, n+1], [1], x/(1 + x)).
R(n,x) = (1 + x)^n * hypergeom([-n, -n], [1], x/(1 + x)).
R(n,x) = hypergeom([(n+1)/2, -n/2], [1], -4*x*(1 + x)).
If we set R(-1,x) = 1, we can run the recurrence n*R(n,x) = (1 + 2*x)*(2*n - 1)*R(n-1,x) - (n - 1)*R(n-2,x) backwards to give R(-n,x) = R(n-1,x).
R(n,x) = [t^n] ( (1 + t)*(1 + x*(1 + t)) )^n. (End)
n*T(n,k) = (2*n-1)*T(n-1,k) + (4*n-2)*T(n-1,k-1) - (n-1)*T(n-2,k). - Fabián Pereyra, Jun 30 2022
From Peter Bala, Oct 07 2024: (Start)
n-th row polynomial R(n,x) = Sum_{k = 0..n} binomial(n, k) * x^k o (1 + x)^(n-k), where o denotes the black diamond product of power series as defined by Dukes and White (see Bala, Section 4.4, exercise 3).
Denote this triangle by T. Then T * transpose(T) = A143007, the square array of crystal ball sequences for the A_n X A_n lattices.
Let S denote the triangle ((-1)^(n+k)*T(n, k))n,k >= 0, a signed version of this triangle. Then S^(-1) * T = A007318, Pascal's triangle; it appears that T * S^(-1) = A110098.
T = A007318 * A115951. (End)

A002674 a(n) = (2n)!/2.

Original entry on oeis.org

1, 12, 360, 20160, 1814400, 239500800, 43589145600, 10461394944000, 3201186852864000, 1216451004088320000, 562000363888803840000, 310224200866619719680000, 201645730563302817792000000, 152444172305856930250752000000, 132626429906095529318154240000000

Offset: 1

N. J. A. Sloane, Simon Plouffe

Right side of the binomial sum n-> sum( (-1)^i * (n-i)^(2*n) * binomial(2*n, i), i=0..n). - Yong Kong (ykong(AT)curagen.com), Dec 28 2000
a(n) is the number of ways to display n distinct flags on n distinct poles and then linearly order all (including any empty) poles. - Geoffrey Critzer, Dec 16 2009
Product of the partition parts of 2n into exactly two parts. - Wesley Ivan Hurt, Jun 03 2013
Let f(x) be a polynomial in x. The expansion (2*sinh(x/2))^2 = x^2 + (1/12)*x^4 + (1/360)*x^6 + ... leads to the second central difference formula f(x+1) - 2*f(x) + f(x-1) = (2*sinh(D/2))^2(f(x)) = D^2(f(x)) + (1/12)*D^4(f(x)) + (1/360)* D^6(f(x)) + ..., where D denotes the differential operator d/dx. - Peter Bala, Oct 03 2019

			a(3) = 360, since 2(3) = 6 has exactly 3 partitions into two parts: (5,1), (4,2), (3,3).  Multiplying all the parts in the partitions, we get 5! * 3 = 360. - _Wesley Ivan Hurt_, Jun 03 2013

A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.33)
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).

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Ronald P. Nordgren, Compound Lucas Magic Squares, arXiv:2103.04774 [math.GM], 2021. See Table 2 p. 12.
H. E. Salzer, Tables of coefficients for obtaining central differences from the derivatives, Journal of Mathematics and Physics (this journal is also called Studies in Applied Mathematics), 42 (1963), 162-165, plus several inserted tables.
H. E. Salzer, Annotated scanned copy of left side of Table II.
Eric Weisstein's World of Mathematics, Central Difference.

a(n) = A090438(n, 2), n >= 1 (first column of (4, 2)-Stirling2 array).

Cf. A000142, A010050, A090438, A002544. Cf. A002671, A002672, A002673, A002675, A002676, A002677.

[n*Factorial(2*n-1): n in [1..15]]; // Vincenzo Librandi, Aug 23 2013

seq((2*k)!/2, k=1..20); # Wesley Ivan Hurt, Aug 22 2013

Table[n! Pochhammer[n, n], {n, 0, 10}] (* Geoffrey Critzer, Dec 16 2009 *)
Table[(2 n)! / 2, {n, 1, 15}] (* Vincenzo Librandi, Aug 23 2013 *)

a(n) = (2*n)!/2; \\ Indranil Ghosh, Apr 18 2017

4*sinh(x/2)^2 = Sum_{k>=1} x^(2k)/a(k). - Benoit Cloitre, Dec 08 2002
E.g.f.: (hypergeom([1/2, 1], [], 4*x)-1)/2 (cf. A090438).
a(n) = n*(2n-1)!. - Geoffrey Critzer, Dec 16 2009
a(n) = A010050(n)/2. - Wesley Ivan Hurt, Aug 22 2013
a(n) = Product_{k=0..n-1} (n^2 - k^2). - Stanislav Sykora, Jul 14 2014
Series reversion ( Sum_{n >= 1} x^n/a(n) ) = Sum_{n >= 1} (-1)^n*x^n/b(n-1), where b(n) = A002544(n). - Peter Bala, Apr 18 2017
From Amiram Eldar, Jul 09 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*(cosh(1) - 1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(1 - cos(1)). (End)

A001879 a(n) = (2n+2)!/(n!*2^(n+1)).

Original entry on oeis.org

1, 6, 45, 420, 4725, 62370, 945945, 16216200, 310134825, 6547290750, 151242416325, 3794809718700, 102776096548125, 2988412653476250, 92854250304440625, 3070380543400170000, 107655217802968460625, 3989575718580595893750, 155815096120119939628125

Offset: 0

N. J. A. Sloane

From Wolfdieter Lang, Oct 06 2008: (Start)
a(n) is the denominator of the n-th approximant to the continued fraction 1^2/(6+3^2/(6+5^2/(6+... for Pi-3. W. Lang, Oct 06 2008, after an e-mail from R. Rosenthal. Cf. A142970 for the corresponding numerators.
The e.g.f. g(x)=(1+x)/(1-2*x)^(5/2) satisfies (1-4*x^2)*g''(x) - 2*(8*x+3)*g'(x) -9*g(x) = 0 (from the three term recurrence given below). Also g(x)=hypergeom([2,3/2],[1],2*x). (End)
Number of descents in all fixed-point-free involutions of {1,2,...,2(n+1)}. A descent of a permutation p is a position i such that p(i) > p(i+1). Example: a(1)=6 because the fixed-point-free involutions 2143, 3412, and 4321 have 2, 1, and 3 descents, respectively. - Emeric Deutsch, Jun 05 2009
First differences of A193651. - Vladimir Reshetnikov, Apr 25 2016
a(n-2) is the number of maximal elements in the absolute order of the Coxeter group of type D_n. - Jose Bastidas, Nov 01 2021

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77 (Problem 10, values of Bessel polynomials).
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See pp. 3, 33.
Selden Crary, Richard Diehl Martinez and Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 1.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Preprint 2016.
J. Riordan, Notes to N. J. A. Sloane, Jul. 1968

Cf. A002544, A001814, A001876, A001877, A001878.
Second column of triangle A001497. Equals (A001147(n+1)-A001147(n))/2.
Equals row sums of A163938.

[Factorial(2*n+2)/(Factorial(n)*2^(n+1)): n in [0..20]]; // Vincenzo Librandi, Nov 22 2011

restart: G(x):=(1-x)/(1-2*x)^(1/2): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od:x:=0:seq(f[n],n=2..20); # Zerinvary Lajos, Apr 04 2009

Table[(2n+2)!/(n!2^(n+1)),{n,0,20}] (* Vincenzo Librandi, Nov 22 2011 *)

PARI
```
a(n)=if(n<0,0,(2*n+2)!/n!/2^(n+1))
```

E.g.f.: (1+x)/(1-2*x)^(5/2).
a(n)*n = a(n-1)*(2n+1)*(n+1); a(n) = a(n-1)*(2n+4)-a(n-2)*(2n-1), if n>0. - Michael Somos, Feb 25 2004
From Wolfdieter Lang, Oct 06 2008: (Start)
a(n) = (n+1)*(2*n+1)!! with the double factorials (2*n+1)!!=A001147(n+1).
D-finite with recurrence a(n) = 6*a(n-1) + ((2*n-1)^2)*a(n-2), a(-1)=0, a(0)=1. (End)
With interpolated 0's, e.g.f.: B(A(x)) where B(x)= x exp(x) and A(x)=x^2/2.
E.g.f.: -G(0)/2 where G(k) = 1 - (2*k+3)/(1 - x/(x - (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2012
G.f.: (1-x)/(2*x^2*Q(0)) - 1/(2*x^2), where Q(k) = 1 - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 20 2013
From Karol A. Penson, Jul 12 2013: (Start)
Integral representation as n-th moment of a signed function w(x) of bounded variation on (0,infinity),
  w(x) = -(1/4)*sqrt(2)*sqrt(x)*(1-x)*exp(-x/2)/sqrt(Pi):
  a(n) = Integral_{x>=0} x^n*w(x), n>=0.
  For x>1, w(x)>0. w(0)=w(1)=limit(w(x),x=infinity)=0. For x<1, w(x)<0.
Asymptotics: a(n)->(1/576)*2^(1/2+n)*(1152*n^2+1680*n+505)*exp(-n)*(n)^(n), for n->infinity. (End)
G.f.: 2F0(3/2,2;;2x). - R. J. Mathar, Aug 08 2015

Entry revised Aug 31 2004 (thanks to Ralf Stephan and Michael Somos)

E.g.f. in comment line corrected by Wolfdieter Lang, Nov 21 2011

A091476 Decimal expansion of Pi^2/4.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jan 13 2004

			2.46740110027233965470862274996903778...

Muniru A Asiru, Table of n, a(n) for n = 1..2000
Ben Hambrecht and Grant Sanderson, The stunning geometry behind this surprising equation, 3Blue1Brown video (2018).
Josef Hofbauer, A simple proof of 1 + 1/2^2 + 1/3^2 + ... = Pi^2/6 and related identities, The American Mathematical Monthly 109:2 (2002), pp. 196-200.
Michael Penn, Neat ways to solve complicated limits., YouTube video, 2023.
T. Piezas III, Golden ratio and nested radicals
Johan Wästlund, Summing inverse squares by euclidean geometry
Eric Weisstein's World of Mathematics, Definite Integral
H. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics and Theoretical Computer Science 3(4) (1999), 189-192.
Index entries for transcendental numbers

Cf. A002388, A013661, A019669.

evalf(Pi^2/4, 120); # Muniru A Asiru, Sep 18 2018

First[RealDigits[Pi^2/4,10,100]] (* Paolo Xausa, Oct 31 2023 *)

Pi^2/4 \\ Charles R Greathouse IV, Mar 02 2018

2*sumpos(n=1,(2*n-1)^-2) \\ Charles R Greathouse IV, Mar 02 2018

Equals Integral_{x=0..Pi} x*sin(x)/(1+cos(x)^2) dx.
Equals Integral_{x=0..1} log((1+x)/(1-x))/x dx. - Jean-François Alcover, May 13 2013
Equals Integral_{x=0..oo} K_0(x)^2 dx, where K_0 is a modified Bessel function (see Gradstein-Ryshik 6.576.4). - R. J. Mathar, Oct 09 2015
Equals A003881 * A000796. - R. J. Mathar, Oct 09 2015
Equals ... + (-5)^-2 + (-3)^-2 + (-1)^-2 + 1^-2 + 3^-2 + 5^-2 + .... - Charles R Greathouse IV, Mar 02 2018
From A.H.M. Smeets, Sep 18 2018: (Start)
Equals A102753/2.
Equals 2*Sum_{k > 0} 1/(2*k - 1)^2. (End)
Pi^2/4 = Integral_{x = 0..oo} x/sinh(x) dx. More generally, Pi^2/4 = 2*(1 + 1/3^2 + ... + 1/(2*n-1)^2) + Integral_{x = 0..oo} exp(-2*n*x)*x/sinh(x). - Peter Bala, Nov 05 2019
Equals Integral_{x=0..oo} log(x)/(x^2 - 1) dx. - Amiram Eldar, Aug 12 2020
Equals Sum_{n >= 0} 2^(n+1)/((n+1)^2*binomial(2*n+1,n)). See my entry in A002544 dated Apr 18 2017. Cf. A253191. - Peter Bala, Jan 30 2023
From Peter Bala, Nov 16 2023: (Start)
Pi^2/4 = 16*Sum_{k >= 1} k^2/(4*k^2 - 1)^2 = (2*16^2)*Sum_{k >= 1} k^2/((4*k^2 - 1)*(4*k^2 - 9))^2.
The general result, which can be proved using the WZ method (see Wilf for examples of this method), is that for n >= 0 there holds
Pi^2/4 = 16^(n+1)*(2*n + 1)*(2*n)!^4/(4*n)! * Sum_{k >= 1} k^2/( (4*k^2 - 1)*(4*k^2 - 9)*...*(4*k^2 - (2*n+1)^2) )^2. (End)
Equals Re(Polylog(2, 2)). - Mohammed Yaseen, Jul 03 2024
From A.H.M. Smeets, Apr 10 2025: (Start)
Let X(p,q) be the p-th smallest zero of the Laguerre polynomial of order q.
Equals lim_{k -> oo} X(k,k^2).
Equals lim_{q -> oo} X(1,q)*q.
Equals lim_{k -> oo} X(k,k^4)*sqrt(k).
Equals lim_{k -> oo} X(k^3,k^4)/sqrt(k).
More general, let P = log_q(p^2/q), then, for any p, 0 < p <= q, equals lim_{q -> oo} X(p,q)/q^P. (End)
Equals Integral_{x=-1..1} -log(abs(x))/(1 - x^2) dx. - Kritsada Moomuang, May 28 2025

A002736 Apéry numbers: a(n) = n^2*C(2n,n).

Original entry on oeis.org

0, 2, 24, 180, 1120, 6300, 33264, 168168, 823680, 3938220, 18475600, 85357272, 389398464, 1757701400, 7862853600, 34901442000, 153876579840, 674412197580, 2940343837200, 12759640231800, 55138611528000, 237371722628040, 1018383898440480

Offset: 0

N. J. A. Sloane

Let H be the n X n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row n-1 of B equals -a(n-1). - T. D. Noe, May 01 2011

J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933, p. 93.
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..200
Kunle Adegoke, Robert Frontczak, and Taras Goy, Fibonacci-Catalan Series, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 22 (2022), #A110.
Paul S. Bruckman, Problem B-871, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 37, No. 1 (1999), p. 85.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 (Annotated scans of some selected pages).
Indulis Strazdins, Solution to problem B-871, Fibonacci Quartely, Vol. 38, No. 1 (2000), pp. 86-87.
Hans J. H. Tuenter, Walking into an absolute sum, arXiv:math/0606080 [math.NT], 2006. Published version on Walking into an absolute sum, The Fibonacci Quarterly, Vol. 40, No. 2 (May 2002), pp. 175-180.
Alfred J. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3), Math. Intelligencer, Vol. 1 (1978/1979), pp. 195-203.

Cf. A000108, A002544, A005258, A005259, A005429, A005430, A086467.

A diagonal of A331431.

[n^2*Binomial(2*n, n): n in [0..30]]; // Vincenzo Librandi, Aug 08 2014

seq(n^2*binomial(2*n,n), n=0..50); # Robert Israel, Aug 07 2014

CoefficientList[ Series[x (4 x + 2)/(1 - 4 x)^(5/2), {x, 0, 20}], x] (* Robert G. Wilson v, Aug 08 2011 *)
Table[n^2 Binomial[2n,n],{n,0,30}] (* Harvey P. Dale, Jun 21 2017 *)

combinat::catalan(n)*(n+1)*n^2 $ n = 0..36 // Zerinvary Lajos, Apr 17 2007

my(x='x+O('x^100)); concat(0, Vec(x*(4*x+2)/((1-4*x)^(5/2)))) \\ Altug Alkan, Mar 21 2016

a(n) = n^2*binomial(2*n, n); \\ Michel Marcus, Mar 21 2016

[n^2*(n+1)*catalan_number(n) for n in (0..30)] # G. C. Greubel, Mar 23 2022

G.f.: x*(4*x+2)/((1-4*x)^(5/2)). - Marco A. Cisneros Guevara, Jul 25 2011
Sum_{n>=1} 1/a(n) = Pi^2/18 (Euler). - Benoit Cloitre, Apr 07 2002
From Ilya Gutkovskiy, Jan 17 2017: (Start)
a(n) ~ 4^n*n^(3/2)/sqrt(Pi).
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(phi)^2 = A086467, where phi is the golden ratio. (End)
D-finite with recurrence: (-n+1)*a(n) +2*(n+4)*a(n-1) +4*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 21 2020
a(n) = (2n)!/(Gamma(n))^2. - Diego Rattaggi, Mar 30 2020
a(n) = Sum_{k=0..2*n} binomial(2*n,k)*abs(n-k)^3 (Bruckman, 1999; Strazdins, 2000). - Amiram Eldar, Jan 12 2022
Sum_{n>=1} x^n/a(n) = 2*arcsin(sqrt(x)/2)^2, for abs(x) < 4 (Adegoke et al., 2022, section 5, p. 10). - Amiram Eldar, Dec 07 2024
From Peter Bala, Aug 02 2025: (Start)
For n >= 1,
a(n) = 2*n*(2*n-1)/(n-1)^2 * a(n-1) with a(1) = 2 and
1/a(n) = Sum_{k = 0..n} (-1)^(n+k+1) * binomial(n, k)*binomial(n+k, k)/(n+k)^2. (End)
a(n) = 2 * A002544(n-1) for n>=1. - Alois P. Heinz, Aug 03 2025

A253191 Decimal expansion of log(2)^2.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Mar 24 2015

			0.480453013918201424667102526326664971730552951594545586866864...

David H. Bailey, Jonathan M. Borwein and Richard E. Crandall, On the Khintchine Constant, Mathematics of Computation, Vol. 66, No. 217 (1997), pp. 417-431, see p. 419; alternative link, p. 4.
Index entries for transcendental numbers

Cf. A001008, A002162, A002805, A175478.

Mathematica
```
RealDigits[Log[2]^2, 10, 103] // First
```

log(2)^2 \\ Charles R Greathouse IV, Apr 20 2016

Integral_{0..1} log(1-x^2)/(x*(1+x)) dx = -log(2)^2.
Integral_{0..1} log(log(1/x))/(x+sqrt(x)) dx = log(2)^2.
Equals Sum_{k>=1} H(k)/(2^k * (k+1)) = 2 * Sum_{k>=1} (-1)^(k+1) * H(k)/(k+1), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Aug 05 2020
Equals Sum_{n >= 0} (-1)^n/(2^(n+1)*(n+1)^2*binomial(2*n+1,n)). See my entry in A002544 dated Apr 18 2017. Cf. A091476. - Peter Bala, Jan 30 2023
Equals 2*Integral_{x=-1..1} (abs(x)*log(x^2 + 1))/(x^2 + 1) dx. - Kritsada Moomuang, May 27 2025

A063746 Triangle read by rows giving number of partitions of k (k=0 .. n^2) with Ferrers plot fitting in an n X n box.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 7, 9, 11, 14, 16, 18, 19, 20, 20, 19, 18, 16, 14, 11, 9, 7, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 7, 11, 13, 18, 22, 28, 32, 39, 42, 48, 51, 55, 55, 58, 55, 55, 51, 48, 42, 39, 32, 28

Offset: 0

Wouter Meeussen, Aug 14 2001

Seems to approximate a Gaussian distribution, the sum of all 1+n^2 terms in a row equals the central binomial coefficients.
a(n,k) is the number of sequences of n 0's and n 1's having major index equal to k (the major index is the sum of the positions of the 1's that are immediately followed by 0's). Equivalently, a(n,k) is the number of Grand Dyck paths of length 2n for which the sum of the positions of the valleys is k. Example: a(3,7)=2 because the only sequences of three 0's and three 1's with major index 7 are 010110 and 110010. The corresponding Grand Dyck paths are obtained by replacing a 0 by a U=(1,1) step and a 1 by a D=(1,-1) step. - Emeric Deutsch, Oct 02 2007
Also, number of n-multisets in [0..n] whose elements sum up to n. - M. F. Hasler, Apr 12 2012
Let P be the poset [n] X [n] ordered by the product order.  Let J(P) be the set of all order ideals of P, ordered by inclusion.  Then J(P) is a finite sublattice of Young's lattice and T(n,k) is the number of elements in J(P) that have rank k. - Geoffrey Critzer, Mar 26 2020

			From _M. F. Hasler_, Apr 12 2012: (Start)
The table reads:
n=0: 1  _  (k=0)
n=1: 1 1  _  (k=0..1)
n=2: 1 1 2 1 1  _  (k=0..4)
n=3: 1 1 2 3 3 3 3  2  1  1  _  (k=0..9)
n=4: 1 1 2 3 5 5 7  7  8  7  7  5  5  3  2  1  1  _  (k=0..16)
n=5: 1 1 2 3 5 7 9 11 14 16 18 19 20 20 19 18 16 ...  _  (k=0..25)
etc. (End)
Cycle index of S(3) is (1/6)*(x(1)^3+3*x(1)*x(2)+2*x(3)), so g.f. for 3rd row is (1/6)*((1+x+x^2+x^3)^3+3*(1+x+x^2+x^3)*(1+x^2+x^4+x^6)+2*(1+x^3+x^6+x^9)) = x^9+x^8+2*x^7+3*x^6+3*x^5+3*x^4+3*x^3+2*x^2+x+1.
a(3,7)=2 because the only partitions of 7 with Ferrers plot fitting into a 3 X 3 box are [3,3,1] and [3,2,2].

G. E. Andrews and K. Eriksson, Integer partitions, Cambridge Univ. Press, 2004, pp. 67-69.
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; exercise 3.2.3.
A. V. Yurkin, New binomial and new view on light theory, (book), 2013, 78 pages, no publisher listed.

Alois P. Heinz, Rows k = 0..31, flattened
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, page 45.
A. V. Yurkin, On similarity of systems of geometrical and arithmetic triangles, in Mathematics, Computing, Education Conference XIX, 2012.
A. V. Yurkin, New view on the diffraction discovered by Grimaldi and Gaussian beams, arXiv preprint arXiv:1302.6287 [physics.optics], 2013.

Cf. A000041, A000070, A002544, A008968, A047971, A047993, A126869, A182738.
Row lengths are given by A002522. - M. F. Hasler, Apr 14 2012
Antidiagonal sums are given by A260894.
Row sums give A000984.

for n from 0 to 15 do QBR[n]:=sum(q^i,i=0..n-1) od: for n from 0 to 15 do QFAC[n]:=product(QBR[j],j=1..n) od: qbin:=(n,k)->QFAC[n]/QFAC[k]/QFAC[n-k]: for n from 0 to 7 do P[n]:=sort(expand(simplify(qbin(2*n,n)))) od: for n from 0 to 7 do seq(coeff(P[n],q,j),j=0..n^2) od; # yields sequence in triangular form - Emeric Deutsch, Apr 23 2007
# second Maple program:
b:= proc(n, i, k) option remember;
      `if`(n=0, 1, `if`(i<1 or k<1, 0, b(n, i-1, k)+
      `if`(i>n, 0, b(n-i, i, k-1))))
    end:
T:= n-> seq(b(k, min(n, k), n), k=0..n^2):
seq(T(n), n=0..8); # Alois P. Heinz, Apr 05 2012

Table[nn=n^2;CoefficientList[Series[Product[(1-x^(n+i))/(1-x^i),{i,1,n}],{x,0,nn}],x],{n,0,6}]//Grid (* Geoffrey Critzer, Sep 27 2013 *)
Table[CoefficientList[QBinomial[2n,n,q] // FunctionExpand, q], {n,0,6}] // Flatten (* Peter Luschny, Jul 22 2016 *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1 || k < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k - 1]]]];
T[n_] := Table[b[k, Min[n, k], n], {k, 0, n^2}];
Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

T(n,k)=polcoeff(prod(i=0,n,sum(j=0,n,x^(j*i*(n^2+n+1)+j),O(x^(k*(n^2+n+1)+n+1)))),k*(n^2+n+1)+n)  /* Based on a more general formula due to R. Gerbicz. M. F. Hasler, Apr 12 2012 */

Table[T[k, n, n], {n, 0, 9}, {k, 0, n^2}] with T[ ] defined as in A047993.
G.f.: Consider a function; f(n) = 1 + sum(i_1=1, n, sum(i_2=0, i_1, ..., sum(i_n=0, i_(n-1), x^(sum(j=1, n, i_j))*(1+...+x^i_n))...)) Then the GF is f(1)+x^3.f(2)+x^8.f(3)+..., where after x^3 the increase is n^2+1 from f(n). - Jon Perry, Jul 13 2004
G.f. for n-th row is obtained if we set x(i) = 1+x^i+x^(2*i)+...+x^(n*i), i=1, 2, ..., n, in the cycle index Z(S(n);x(1), x(2), ..., x(n)) of the symmetric group S(n) of degree n. - Vladeta Jovovic, Dec 17 2004
G.f. of row n: the q-binomial coefficient [2n,n]. - Emeric Deutsch, Apr 23 2007
T(n,k)=1 for k=0,1,n^2-1,n^2. For all m>n, T(m,n)=T(n,n)=A000041(n), i.e., below the diagonal the columns remain constant, because there cannot be more than n nonzero elements with sum <= n. - M. F. Hasler, Apr 12 2012
T(n,2n) = A128552(n-2). - Geoffrey Critzer, Sep 27 2013
From Alois P. Heinz, Jan 09 2025: (Start)
Sum_{k=0..n} T(n,k) = A000070(n).
Sum_{k=0..n} k * T(n,k) = A182738(n).
Sum_{k=0..n^2} k * T(n,k) = A002544(n-1) for n>=1.
Sum_{k=0..n^2} (-1)^k * T(n,k) = A126869(n). (End)

A110505 Numerators of unsigned columns of triangle A110504: a(n) = n!A110504(n,0) = (-1)^kn!*A110504(n+k,k) for all k >= 0.

Original entry on oeis.org

0, 1, 3, 7, 30, 144, 876, 6084, 48816, 438624, 4389120, 48263040, 579242880, 7529552640, 105417365760, 1581231456000, 25299906508800, 430096581734400, 7741753102540800, 147093162635059200, 2941864569520128000

Offset: 0

Paul D. Hanna, Jul 23 2005

Triangle A110504 equals the matrix logarithm of triangle A110503.
Triangle A110503 shifts one column left under matrix inverse.
Lim_{n->infinity} a(n)/n! = Pi*2*sqrt(3)/9 = 1.209199576...

			E.g.f.: A(x) = x + 3*x^2/2! + 7*x^3/3! + 30*x^4/4! + 144*x^5/5! + 876*x^6/6! + ...
where A(x) satisfies: A(x)*A(-x) = -arccos(1-1/2*x^2)^2, and
arccos(1-1/2*x^2)^2 = Sum_{n>=0} x^(2*n+2)/( C(2*n+1, n)*(n+1)^2 ) = x^2 + 1/12*x^4 + 1/90*x^6 + 1/560*x^8 + 1/3150*x^10 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A110503 (triangular matrix), A110504 (matrix logarithm), A002544.

/* From relation to unsigned columns of triangle A110504: */
{a(n)=local(M=matrix(n+1,n+1,r,c,if(r>=c, if(r==c || c%2==1,1,if(r%2==0 && r==c+2,-2,-1))))); n!*sum(i=1,#M,-(M^0-M)^i/i)[n+1,1]}
for(n=0,30,print1(a(n),", "))

/* As Partial Sums of Series: */
a(n)=if(n<1,0,n!*(1+sum(n=2,n,(-1)^n/(binomial(n-2,n\2-1)*n*(n-1)/((n+1)\2)))))
for(n=0,30,print1(a(n),", "))

E.g.f.: (2+x-x^2)/(2*(1-x)) * arccos(1-x^2/2) / sqrt(1-x^2/4).
E.g.f. A(x) satisfies:
(1) A(x)*A(-x) = -arccos(1-1/2*x^2)^2 = Sum_{n>=0} -x^(2*n+2)/( C(2*n+1, n)*(n+1)^2 ).
(2) 1/(1-x) = Sum_{n>=1} A(x)^floor((n+1)/2) * A(-x)^floor(n/2)/n!.
a(2*n+1) = (2*n+1)!*(1 + Sum_{k=1..n} (1/binomial(2*k+1, k))/(k+1)).
a(2*n+2) = (2*n+2)!*(1 + 1/2 - Sum_{k=1..n} 1/binomial(2*k+2, k)/k) = n!*(1 + 1/2 - 1/3 + 1/12 - 1/20 + 1/60 - 1/105 + 1/280 -+ ...).
Recurrence: 4*a(n) = 2*(2*n-1)*a(n-1) + (n-2)*(n+1)*a(n-2) - (n-3)*(n-2)*n*a(n-3). - Vaclav Kotesovec, May 09 2014

A085373 a(n) = binomial(2n+1, n+1)*binomial(n+2, 2).

Original entry on oeis.org

1, 9, 60, 350, 1890, 9702, 48048, 231660, 1093950, 5080790, 23279256, 105462084, 473227300, 2106121500, 9307051200, 40873466520, 178520875830, 775924068150, 3357800061000, 14473885526100, 62168784497820, 266168518910580

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jun 26 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000984, A002544, A119578.

[Binomial(2*n+1, n+1)*Binomial(n+2, 2): n in [0..30]]; // G. C. Greubel, Feb 12 2019

Table[Binomial[2n+1, n+1]Binomial[n+2, 2], {n, 0, 30}]

a(n)=binomial(2*n+1,n+1)*binomial(n+2,2)

[binomial(2*n+1, n+1)*binomial(n+2, 2) for n in (0..30)] # G. C. Greubel, Feb 12 2019

From David Callan, Nov 20 2003: (Start)
a(n-1) = Sum_{1<=i1<=i2<=...<=in<=n} (i1 + i2 + ... + in).
G.f.: (1 - x)/(1 - 4*x)^(5/2). (End)
a(n) = A119578(n+1)/2. - Zerinvary Lajos, Jun 19 2008
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=0} 1/a(n) = 4*Pi^2/9 - 4*Pi/sqrt(3) + 4.
Sum_{n>=0} (-1)^n/a(n) = 8*sqrt(5)*log(phi) - 16*log(phi)^2 - 4, where phi = A001622 is the golden ratio. (End)

A331430 Triangle read by rows: T(n, k) = (-1)^(k+1)binomial(n,k)binomial(n+k,k) (n >= k >= 0).

Original entry on oeis.org

-1, -1, 2, -1, 6, -6, -1, 12, -30, 20, -1, 20, -90, 140, -70, -1, 30, -210, 560, -630, 252, -1, 42, -420, 1680, -3150, 2772, -924, -1, 56, -756, 4200, -11550, 16632, -12012, 3432, -1, 72, -1260, 9240, -34650, 72072, -84084, 51480, -12870, -1, 90, -1980, 18480, -90090, 252252, -420420, 411840, -218790, 48620, -1, 110, -2970, 34320, -210210, 756756, -1681680, 2333760, -1969110, 923780, -184756

Offset: 0

N. J. A. Sloane, Jan 17 2020

This is Table I of Ser (1933), page 92.
From Petros Hadjicostas, Jul 09 2020: (Start)
Essentially Ser (1933) in his book (and in particular for Tables I-IV) finds triangular arrays that allow him to express the coefficients of various kinds of series in terms of the coefficients of other series.
He uses Newton's series (or some variation of it), factorial series, and inverse factorial series. Unfortunately, he uses unusual notation, and as a result it is difficult to understand what he is actually doing.
Rivoal (2008, 2009) essentially uses factorial series and transformations to other kinds of series to provide new proofs of the irrationality of log(2), zeta(2), and zeta(3). As a result, the triangular array T(n,k) appears in various parts of his papers.
We believe Table I (p. 92) in Ser (1933), regarding the numbers T(n,k), corresponds to four different formulas. We have deciphered the first two of them. (End)

			Triangle T(n,k) (with rows n >= 0 and columns k=0..n) begins:
  -1;
  -1,  2;
  -1,  6,   -6;
  -1, 12,  -30,   20;
  -1, 20,  -90,  140,    -70;
  -1, 30, -210,  560,   -630,   252;
  -1, 42, -420, 1680,  -3150,  2772,   -924;
  -1, 56, -756, 4200, -11550, 16632, -12012, 3432;
  ...
From _Petros Hadjicostas_, Jul 11 2020: (Start)
Its inverse (from Table II, p. 92) is
  -1;
  -1/2, 1/2;
  -1/3, 1/2,   -1/6;
  -1/4, 9/20,  -1/4,  1/20;
  -1/5, 2/5,   -2/7,  1/10, -1/70;
  -1/6, 5/14, -25/84, 5/36, -1/28,  1/252;
  -1/7, 9/28, -25/84, 1/6,  -9/154,  1/84, -1/924;
   ... (End)

J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, pp. 92-93.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Buhl, Book review: J. Ser - Les calculs formels des séries de factorielles, L'Enseignement Mathématique, 32 (1933), p. 275.
L. A. MacColl, Review: J. Ser, Les calculs formels des séries de factorielles, Bull. Amer. Math. Soc., 41(3) (1935), p. 174.
L. M. Milne-Thomson, Review of Les calculs formels des séries de factorielles. By J. Ser. Pp. vii, 98. 20 fr. 1933. (Gauthier-Villars), The Mathematical Gazette, Vol. 18, No. 228 (May, 1934), pp. 136-137.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)
Tanguy Rivoal, Applications arithmétiques de l'interpolation lagrangienne, preprint (2008); see pp. 1 and 15.
Tanguy Rivoal, Applications arithmétiques de l'interpolation lagrangienne, Int. J. Number Theory 5.2 (2009), 185-208; see pp. 185 and 199.

A063007 is the same triangle without the minus signs, and has much more information.
Columns 1 and 2 are A002378 and A033487; the last three diagonals are A002544, A002457, A000984.
Cf. A331431, A331432.

/* As triangle: */ [[(-1)^(k+1) * Factorial(n+k) / (Factorial(k) * Factorial(k) * Factorial(n-k)): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jan 19 2020

Table[CoefficientList[-Hypergeometric2F1[-n, n + 1, 1, x], x], {n, 0, 9}] // Flatten (* Georg Fischer, Jan 18 2020 after Peter Luschny in A063007 *)

def T(n,k): return (-1)^(k+1)*falling_factorial(n+k,2*k)/factorial(k)^2
flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # Peter Luschny, Jul 09 2020

T(n,k) can also be written as (-1)^(k+1)*(n+k)!/(k!*k!*(n-k)!).
From Petros Hadjicostas, Jul 09 2020: (Start)
Ser's first formula from his Table I (p. 92) is the following:
Sum_{k=0..n} T(n,k)*k!/(x*(x+1)*...*(x+k)) = -(x-1)*(x-2)*...*(x-n)/(x*(x+1)*...*(x+n)).
As a result, Sum_{k=0..n} T(n,k)/binomial(m+k, k) = 0 for m = 1..n.
Ser's second formula from his Table I appears also in Rivoal (2008, 2009) in a slightly different form:
Sum_{k=0..n} T(n,k)/(x + k) = (-1)^(n+1)*(x-1)*(x-2)*...*(x-n)/(x*(x+1)*...*(x+n)).
As a result, for m = 1..n, Sum_{k=0..n} T(n,k)/(m + k) = 0. (End)
T(n,k) = (-1)^(k+1)*FallingFactorial(n+k,2*k)/(k!)^2. - Peter Luschny, Jul 09 2020
From Petros Hadjicostas, Jul 10 2020: (Start)
Peter Luschny's formula above is essentially the way the numbers T(n,k) appear in Eq. (7) on p. 86 of Ser's (1933) book. Eq. (7) is essentially equivalent to the first formula above (related to Table I on p. 92).
By inverting that formula (in some way), he gets
n!/(x*(x+1)*...*(x+n)) = Sum_{p=0..n} (-1)^p*(2*p+1)*f_p(n+1)*f_p(x), where f_p(x) = (x-1)*...*(x-p)/(x*(x+1)*...*(x+p)). This is equivalent to Eq. (8) on p. 86 of Ser's book.
The rational coefficients A(n,p) = (2*p+1)*f_p(n+1) = (2*p+1)*(n*(n-1)*...*(n+1-p))/((n+1)*...*(n+1+p)) appear in Table II on p. 92 of Ser's book.
If we consider the coefficients T(n,k) and (-1)^(p+1)*A(n,p) as infinite lower triangular matrices, then they are inverses of one another (see the example below). This means that, for m >= s,
Sum_{k=s..m} T(m,k)*(-1)^(s+1)*A(k,s) = I(s=m) = Sum_{k=s..m} (-1)^(k+1)*A(m,k)*T(k,s), where I(s=m) = 1, if s = m, and = 0, otherwise.
Without the (-1)^p, we get the formula
1/(x+n) = Sum_{p=0..n} (2*p+1)*f_p(n+1)*f_p(x),
which apparently is the inversion of the second of Ser's formulas (related to Table I on p. 92).
In all of the above formulas, an empty product is by definition 1, so f_0(x) = 1/x. (End)

Thanks to Bob Selcoe, who noticed a typo in one of the entries, which, when corrected, led to an explicit formula for the whole of Ser's Table I.

cp's OEIS Frontend

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A331430 Triangle read by rows: T(n, k) = (-1)^(k+1)binomial(n,k)binomial(n+k,k) (n >= k >= 0).