A063007 -id:A063007 - cp's OEIS frontend

A262177 Decimal expansion of Q_5 = zeta(5) / (Sum_{k>=1} (-1)^(k+1) / (k^5 * binomial(2k, k))), a conjecturally irrational constant defined by an Apéry-like formula.

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

Offset: 1

Jean-François Alcover, Sep 14 2015

The similar constant Q_3 = zeta(3) / (Sum_{k>=1} (-1)^(k+1) / (k^3 * binomial(2k, k))) evaluates to 5/2.

			2.09486862201003699385024929373294163029675874856778182740127587837438...

G. C. Greubel, Table of n, a(n) for n = 1..5000
David Bailey, Jonathan Borwein, David Bradley, Experimental determination of Apéry-like identities for zeta(2n+2), arXiv:math/0505270 [math.NT], 2005.

Cf. A013663.

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.)

Q5 = Zeta[5]/Sum[(-1)^(k+1)/(k^5*Binomial[2k, k]), {k, 1, Infinity}]; RealDigits[Q5, 10, 103] // First

zeta(5)/suminf(k=1, (-1)^(k+1)/(k^5*binomial(2*k,k))) \\ Michel Marcus, Sep 14 2015

Equals 2*zeta(5)/6F5(1,1,1,1,1,1; 3/2,2,2,2,2; -1/4).

A089627 T(n,k) = binomial(n,2k)binomial(2*k,k) for 0 <= k <= n, triangle read by rows.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 0, 0, 1, 12, 6, 0, 0, 1, 20, 30, 0, 0, 0, 1, 30, 90, 20, 0, 0, 0, 1, 42, 210, 140, 0, 0, 0, 0, 1, 56, 420, 560, 70, 0, 0, 0, 0, 1, 72, 756, 1680, 630, 0, 0, 0, 0, 0, 1, 90, 1260, 4200, 3150, 252, 0, 0, 0, 0, 0, 1, 110, 1980, 9240, 11550, 2772, 0, 0, 0, 0, 0, 0, 1, 132, 2970, 18480, 34650, 16632, 924, 0, 0, 0, 0, 0, 0, 1, 156, 4290, 34320, 90090, 72072, 12012, 0, 0, 0, 0, 0, 0, 0, 1, 182, 6006, 60060, 210210, 252252, 84084, 3432, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 31 2003

The rows of this triangle are the gamma vectors of the n-dimensional type B associahedra (Postnikov et al., p.38 ). Cf. A055151 and A101280. - Peter Bala, Oct 28 2008
T(n,k) is the number of Grand Motzkin paths of length n having exactly k upsteps (1,1). Cf. A109189, A055151. - Geoffrey Critzer, Feb 05 2014
The result Sum_{k = 0..floor(n/2)} C(n,2*k)*C(2*k,k)*x^k = (sqrt(1 - 4*x))^n* P(n,1/sqrt(1 - 4*x)) expressing the row polynomials of this triangle in terms of the Legendre polynomials P(n,x) is due to Catalan. See Laden, equation 7.10, p. 56. - Peter Bala, Mar 18 2018

			Triangle begins:
  1
  1,   0
  1,   2,    0
  1,   6,    0,     0
  1,  12,    6,     0,     0
  1,  20,   30,     0,     0,     0
  1,  30,   90,    20,     0,     0,   0
  1,  42,  210,   140,     0,     0,   0, 0
  1,  56,  420,   560,    70,     0,   0, 0, 0
  1,  72,  756,  1680,   630,     0,   0, 0, 0, 0
  1,  90, 1260,  4200,  3150,   252,   0, 0, 0, 0, 0
  1, 110, 1980,  9240, 11550,  2772,   0, 0, 0, 0, 0, 0
  1, 132, 2970, 18480, 34650, 16632, 924, 0, 0, 0, 0, 0, 0
Relocating the zeros to be evenly distributed and interpreting the triangle as the coefficients of polynomials
                     1
                     1
                 1 + 2 q^2
                 1 + 6 q^2
            1 + 12 q^2 +  6 q^4
            1 + 20 q^2 + 30 q^4
       1 + 30 q^2 +  90 q^4 +  20 q^6
       1 + 42 q^2 + 210 q^4 + 140 q^6
  1 + 56 q^2 + 420 q^4 + 560 q^6 + 70 q^8
then the substitution q^k -> 1/(floor(k/2)+1) gives the Motzkin numbers A001006.
- _Peter Luschny_, Aug 29 2011

Alois P. Heinz, Rows n = 0..140, flattened
Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), #14.9.1.
Hyman N. Laden, An historical, and critical development of the theory of Legendre polynomials before 1900, Master of Arts Thesis, University of Maryland 1938.
Shi-Mei Ma, On gamma-vectors and the derivatives of the tangent and secant functions, arXiv:1304.6654 [math.CO], 2013.
A. Postnikov, V. Reiner, and L. Williams, Faces of generalized permutohedra, arXiv:math/0609184 [math.CO], 2006-2007.

Row sums A002426. Antidiagonal sums A098479.

Cf. A055151, A063007, A109187, A101280.

for i from 0 to 12 do seq(binomial(i, j)*binomial(i-j, j), j=0..i) od; # Zerinvary Lajos, Jun 07 2006
# Alternatively:
R := (n, x) -> simplify(hypergeom([1/2 - n/2, -n/2], [1], 4*x)):
Trow := n -> seq(coeff(R(n,x), x, j), j=0..n):
seq(print(Trow(n)), n=0..9); # Peter Luschny, Mar 18 2018

nn=15;mxy=(1-x-(1-2x+x^2-4x^2y)^(1/2))/(2x^2 y);Map[Select[#,#>0&]&, CoefficientList[Series[1/(1-x-2y x^2mxy),{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Feb 05 2014 *)

T(n,k) = binomial(n,2*k)*binomial(2*k,k);
concat(vector(15, n, vector(n, k, T(n-1, k-1)))) \\ Gheorghe Coserea, Sep 01 2018

T(n,k) = n!/((n-2*k)!*k!*k!).
E.g.f.: exp(x)*BesselI(0, 2*x*sqrt(y)). - Vladeta Jovovic, Apr 07 2005
O.g.f.: ( 1 - x - sqrt(1 - 2*x + x^2 - 4*x^2*y))/(2*x^2*y). - Geoffrey Critzer, Feb 05 2014
R(n, x) = hypergeom([1/2 - n/2, -n/2], [1], 4*x) are the row polynomials. - Peter Luschny, Mar 18 2018
From Peter Bala, Jun 23 2023: (Start)
T(n,k) = Sum_{i = 0..k} (-1)^i*binomial(n, i)*binomial(n-i, k-i)^2. Cf. A063007(n,k) = Sum_{i = 0..k} binomial(n, i)^2*binomial(n-i, k-i).
T(n,k) = A063007(n-k,k); that is, the diagonals of this table are the rows of A063007. (End)

A104684 Triangle read by rows: 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 D=(1,1) steps.

Original entry on oeis.org

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

Offset: 0

Emeric Deutsch, Apr 24 2005

Row sums are the central Delannoy numbers (A001850). T(n,0)=A000984(n) (the central binomial numbers). Alternating row sums = 1 See the Bataille link.
  Row reversed version of A063007.
Another version of [0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] = 1; 0, 1; 0, 2, 1; 0, 6, 6, 1; 0, 20, 30, 12, 1; 0, 70, 140, 90, 20, 1; ..., where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 25 2005
Terms in row n are the coefficients of the Legendre polynomial P(n,2x+1) with decreasing powers of x.
Coefficient array of x^n*Legendre_P(n,2/x+1). - Paul Barry, Apr 19 2009

			T(2,1)=6 because we have NED, NDE, EDN, END, DEN and DNE.
The triangle T(n, k) begins:
n\k    0     1     2     3     4    5    6  7 8 ...
0:     1
1:     2     1
2:     6     6     1
3:    20    30    12     1
4:    70   140    90    20    1
5:   252   630   560   210   30     1
6:   924  2772  3150  1680  420    42    1
7:  3432 12012 16632 11550 4200   756   56  1
8: 12870 51480 84084 72072 34650 9240 1260 72 1
...
row n=9: 48620 218790 411840 420420 252252 90090 18480 1980 90 1,
row n=10: 184756 923780 1969110 2333760 1681680 756756 210210 34320 2970 110 1.
... reformatted by _Wolfdieter Lang_, Sep 11 2016

T. D. Noe, Rows n=0..100 of triangle, flattened
Michel Bataille, Quickie Q.944, Maths. Magazine, 77, No. 4, p. 321, Answer A.944, Maths. Magazine, 77, No. 4, p. 327.
H. J. Brothers, Pascal's Prism: Supplementary Material.
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri, Volume L, 2021.
Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Cf. A000984, A001850, A063007.

a104684 n k = a104684_tabl !! n !! k
a104684_row n = a104684_tabl !! n
a104684_tabl = map (map abs) $
               zipWith (zipWith (*)) a130595_tabl a092392_tabl
-- Reinhard Zumkeller, Dec 20 2013

T:=(n,k)->binomial(n,k)*binomial(2*n-k,n): for n from 0 to 9 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

T[n_, k_] := Binomial[n, k] Binomial[2n-k, n];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)

T(n, k) = binomial(n, k)*binomial(2n-k, n) (0 <= k <= n).
G.f.: G(t, z) = 1/sqrt((1-tz)^2 - 4z).
T(n,k) = binomial(2(n-k),n-k)*binomial(2n-k,k). - Paul Barry, Mar 14 2006
T(2n,n) = C(2n,n)*C(3n,n) = C(n,n)*C(2n,n)*C(3n,n) = A006480(n). - Paul Barry, Mar 14 2006
G.f.: 1/(1-xy-2x/(1-xy-x/(1-xy-x/(1-xy-x/(1-xy-x... (continued fraction). - Paul Barry, Jan 06 2009
T(n,k) = Sum_{j=0..n} C(n,j)^2*C(j,k). - Paul Barry, May 28 2009
T(n,k) = [x^k]F(-n,-n;1;1+x). - Paul Barry, Oct 05 2010
T(n,k) = (n-k+1)*A060693(n,k). - Peter Luschny, May 17 2011
T(n,k) = A054142(n,k)*A000984(n-k). - Philippe Deléham, Nov 19 2011.
T(n,k) = abs(A130595(n,k)*A092392(n,k)). - Reinhard Zumkeller, Dec 20 2013

A127674 Coefficient table for Chebyshev polynomials T(2*n,x) (increasing even powers x, without zeros).

Original entry on oeis.org

1, -1, 2, 1, -8, 8, -1, 18, -48, 32, 1, -32, 160, -256, 128, -1, 50, -400, 1120, -1280, 512, 1, -72, 840, -3584, 6912, -6144, 2048, -1, 98, -1568, 9408, -26880, 39424, -28672, 8192, 1, -128, 2688, -21504, 84480, -180224, 212992, -131072, 32768, -1, 162, -4320, 44352, -228096, 658944, -1118208

Offset: 0

Wolfdieter Lang Mar 07 2007

Let C_n be the root lattice generated as a monoid by {+-2*e_i: 1 <= i <= n; +-e_i +- e_j: 1 <= i not equal to j <= n}. Let P(C_n) be the polytope formed by the convex hull of this generating set. Then the rows of (the signless version of) this array are the f-vectors of a unimodular triangulation of P(C_n) [Ardila et al.]. See A086645 for the corresponding array of h-vectors for these type C_n polytopes. See A063007 for the array of f-vectors for type A_n polytopes and A108556 for the array of f-vectors associated with type D_n polytopes. - Peter Bala, Oct 23 2008

			[1];
[-1,2];
[1,-8,8];
[-1,18,-48,32];
[1,-32,160,-256,128];
...
See a link for the row polynomials.
The T-polynomial for row n=3, [-1,18,-48,32], is T(2*3,x) =  -1*x^0 + 18*x^2 - 48*x^4 + 32*x^6.

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990. p. 37, eq.(1.96) and p. 4. eq.(1.10).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 795.
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.
G. Boros, V. H. Moll, Landen transformations and the integration of rational functions, Math. Comp. 71 (238) (2001) 649-668, absolute values in Lemma A.3.
C. Lanczos, Applied Analysis (Annotated scans of selected pages) See p. 516.
Wolfdieter Lang, Row polynomials.
Richard J. Mathar, Chebyshev approximation of x^m (-log x)^l in the interval 0 <= x <= 1, arXiv:2408.15212 [math.CA], 2024. See p. 1.
Index entries for sequences related to Chebyshev polynomials.

Cf. A075733 (different signs and offset). A084930 (coefficients of odd-indexed T-polynomials).
Cf. A053120 (coefficients of T-polynomials, with interspersed zeros).
Cf. A086645, A063007, A108556.

a(n,m) = 0 if n < m, a(0,0) = 1; otherwise a(n,m) = ((-1)^(n-m))*(2^(2*m-1))*binomial(n+m,2*m)*(2*n)/(n+m).
O.g.f.: (1 + z*(1 - 2*x))/((1 + z)^2 - 4*x*z) = 1 + (-1 + 2*x)*z + (1 - 8*x + 8*x^2)*z^2 + ... . [Peter Bala, Oct 23 2008] For the t-polynomials actually with x -> x^2. - Wolfdieter Lang, Aug 02 2014
Denoting the row polynomials by R(n,x) we have exp( Sum_{n >= 1} R(n,x)*z^n/n ) = 1/sqrt( (1 + z)^2 - 4*x*z ) = 1 + (-1 + 2*x)*z + (1 - 6*x + 6*x^2)*z^2 + ..., the o.g.f. for a signed version of A063007. - Peter Bala, Sep 02 2015
The n-th row polynomial equals T(n, 2*x - 1). - Peter Bala, Jul 09 2023

A253283 Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.

Original entry on oeis.org

1, 0, 1, 0, 2, 3, 0, 3, 12, 10, 0, 4, 30, 60, 35, 0, 5, 60, 210, 280, 126, 0, 6, 105, 560, 1260, 1260, 462, 0, 7, 168, 1260, 4200, 6930, 5544, 1716, 0, 8, 252, 2520, 11550, 27720, 36036, 24024, 6435, 0, 9, 360, 4620, 27720, 90090, 168168, 180180, 102960, 24310

Offset: 0

Peter Luschny, Mar 20 2015

The rows give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)^2*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1) / (n!*(n+1)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 31 2022

This is related to the cluster fans of type B (see Fomin and Zelevinsky reference) - F. Chapoton, Nov 17 2022.

			[1]
[0, 1]
[0, 2,   3]
[0, 3,  12,   10]
[0, 4,  30,   60,   35]
[0, 5,  60,  210,  280,  126]
[0, 6, 105,  560, 1260, 1260,  462]
[0, 7, 168, 1260, 4200, 6930, 5544, 1716]
.
R_0(x) = 1/(x-1)^0.
R_1(x) = 0/(x-1)^1 + 1/(x-1)^2.
R_2(x) = 0/(x-1)^2 + 2/(x-1)^3 + 3/(x-1)^4.
R_3(x) = 0/(x-1)^3 + 3/(x-1)^4 + 12/(x-1)^5 + 10/(x-1)^6.
Then k!*[x^k] R_n(x) is A001286(k+2) and A001754(k+3) for n = 2, 3 respectively.
.
Seen as an array A(n, k) = binomial(n + k, k)*binomial(n + 2*k - 1, n + k):
[0] 1, 1,   3,   10,    35,    126,     462, ...
[1] 0, 2,  12,   60,   280,   1260,    5544, ...
[2] 0, 3,  30,  210,  1260,   6930,   36036, ...
[3] 0, 4,  60,  560,  4200,  27720,  168168, ...
[4] 0, 5, 105, 1260, 11550,  90090,  630630, ...
[5] 0, 6, 168, 2520, 27720, 252252, 2018016, ...
[6] 0, 7, 252, 4620, 60060, 630630, 5717712, ...

Seiichi Manyama, Rows n = 0..139, flattened
F. Chapoton Enumerative Properties of Generalized Associahedra, Sém. Loth. Comb. B51b (2004).
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 A. Zelevinsky Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3.
Yunlong Shen and Lixin Shen, Orthogonal Fourier-Mellin moments for invariant pattern recognition, J. Opt. Soc. Am. A 11 (6) (1994) p 1748-1757, eq. (6).

T(n, n) = C(2*n-1, n) = A001700(n-1).
T(n, n-1) = A005430(n-1) for n >= 1.
T(n, n-2) = A051133(n-2) for n >= 2.
T(n, 2) = A027480(n-1) for n >= 2.
T(2*n, n) = A208881(n) for n >= 0.
A002002 (row sums).
Cf. A000142, A001286, A001754, A001755, A001777, A182626, A266732 (row k=3), A008316, A033282, A063007, A345013, A269944.

T_row := proc(n) local egf, k, F, t;
if n=0 then RETURN(1) fi;
egf := (x/(1-x))^n/n!; t := diff(egf,[x$n]);
F := convert(t,parfrac,x);
# print(seq(k!*coeff(series(F,x,20),x,k),k=0..7));
# gives A000142, A001286, A001754, A001755, A001777, ...
seq(coeff(F,(x-1)^(-k)),k=n..2*n) end:
seq(print(T_row(n)),n=0..7);
# 2nd version by R. J. Mathar, Dec 18 2016:
A253283 := proc(n,k)
    binomial(n,k)*binomial(n+k-1,k-1) ;
end proc:

Table[Binomial[n, k] Binomial[n + k - 1, k - 1], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 22 2017 *)

T(n,k) = binomial(n,k)*binomial(n+k-1,k-1);
tabl(nn) = for(n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Apr 29 2018

The exponential generating functions for the rows of the square array L(n,k) = ((n+k)!/n!)*C(n+k-1,n-1) (associated to the unsigned Lah numbers) are given by R_n(x) = Sum_{k=0..n} T(n,k)/(x-1)^(n+k).
T(n,k) = C(n,k)*C(n+k-1,k-1).
Sum_{k=0..n} T(n,k) = (-1)^n*hypergeom([-n,n],[1],2) = (-1)^n*A182626(n).
Row generating function: Sum_{k>=1} T(n,k)*z^k = z*n* 2F1(1-n,n+1 ; 2; -z). - R. J. Mathar, Dec 18 2016
From Peter Bala, Feb 22 2017: (Start)
G.f.: (1/2)*( 1 + (1 - t)/sqrt(1 - 2*(2*x + 1)*t + t^2) ) = 1 + x*t + (2*x + 3*x^2)*t^2 + (3*x + 12*x^2 + 10*x^3)*t^3 + ....
n-th row polynomial R(n,x) = (1/2)*(LegendreP(n, 2*x + 1) - LegendreP(n-1, 2*x + 1)) for n >= 1.
The row polynomials are the black diamond product of the polynomials x^n and x^(n+1) (see Dukes and White 2016 for the definition of this product).
exp(Sum_{n >= 1} R(n,x)*t^n/n) = 1 + x*t + x*(1 + 2*x)*t^2 + x*(1 + 5*x + 5*x^2)*t^3 + ... is a g.f. for A033282, but with a different offset.
The polynomials P(n,x) := (-1)^n/n!*x^(2*n)*(d/dx)^n(1 + 1/x)^n begin 1, 3 + 2*x , 10 + 12*x + 3*x^2, ... and are the row polynomials for the row reverse of this triangle. (End)
Let Q(n, x) = Sum_{j=0..n} (-1)^(n - j)*A269944(n, j)*x^(2*j - 1) and P(x, y) = (LegendreP(x, 2*y + 1) - LegendreP(x-1, 2*y + 1)) / 2 (see Peter Bala above). Then n!*(n - 1)!*[y^n] P(x, y) = Q(n, x) for n >= 1. - Peter Luschny, Oct 31 2022
From Peter Bala, Apr 18 2024: (Start)
G.f.: Sum_{n >= 0} binomial(2*n-1, n)*(x*t)^n/(1 - t)^(2*n) = 1 + x*t + (2*x + 3*x^2)*t^2 + (3*x + 12*x^2 + 10*x^3)*t^3 + ....
n-th row polynomial R(n, x) = [t^n] ( (1 - t)/(1 - (1 + x)*t) )^n.
It follows that for integer x, the sequence {R(n, x) : n >= 0} satisfies the Gauss congruences: R(n*p^r, x) == R(n*p^(r-1), x) (mod p^r) for all primes p and positive integers n and r.
R(n, -2) = (-1)^n * A002003(n) for n >= 1.
R(n, 3) = A299507(n). (End)

A243949 Squares of the central Delannoy numbers: a(n) = A001850(n)^2.

Original entry on oeis.org

1, 9, 169, 3969, 103041, 2832489, 80802121, 2365752321, 70611901441, 2139090528969, 65568745087209, 2029206892664961, 63300531617048961, 1987912809986437161, 62787371136571152009, 1992942254830520803329, 63531842302018973818881, 2033004661359005674887561

Offset: 0

Paul D. Hanna, Aug 17 2014

In general, we have the binomial identity:
if b(n) = Sum_{k=0..n} t^k * C(2*k, k) * C(n+k, n-k), then b(n)^2 = Sum_{k=0..n} (t^2+t)^k * C(2*k, k)^2 * C(n+k, n-k), where the g.f. of b(n) is 1/sqrt(1 - (4*t+2)*x + x^2), and the g.f. of b(n)^2 is 1 / AGM(1-x, sqrt((1+x)^2 - (4*t+2)^2*x)), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.
Note that the g.f. of A001850 is 1/sqrt(1 - 6*x + x^2).
Limit_{n -> oo} a(n+1)/a(n) = (3 + 2*sqrt(2))^2 = 17 + 12*sqrt(2).
From Gheorghe Coserea, Jul 05 2016: (Start)
Diagonal of the rational function 1/(1 - x - y - z - x*y + x*z - y*z - x*y*z).
Annihilating differential operator: x*(x-1)*(x+1)*(x^2-34*x+1)*Dx^2 + (3*x^4-66*x^3-70*x^2+70*x-1)*Dx + x^3-7*x^2-35*x+9.
(End).
The sequence b(n) mentioned above is the sequence of shifted Legendre polynomials P(n,2*t + 1) (see A063007). See Zudilin for a g.f. for the sequence b(n)^2. - Peter Bala, Mar 02 2017

			G.f.: A(x) = 1 + 9*x + 169*x^2 + 3969*x^3 + 103041*x^4 + 2832489*x^5 +...

Seiichi Manyama, Table of n, a(n) for n = 0..500
A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"
W. Zudilin, A generating function of the squares of Legendre polynomials, arXiv:1210.2493v2 [math.CA], 2012.

Sequences of the form LegendreP(n, 2*m+1)^2: A000012 (m=0), this sequence (m=1), A243943 (m=2), A243944 (m=3), A243007 (m=4).
Related to diagonal of rational functions: A268545 - A268555.
Cf. A001850, A243945, A245925.

[Evaluate(LegendrePolynomial(n), 3)^2 : n in [0..40]]; // G. C. Greubel, May 17 2023

Table[Sum[2^k *Binomial[2*k, k]^2 *Binomial[n+k, n-k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 18 2014 *)
a[n_]:= HypergeometricPFQ[{1/2, -n, n+1}, {1, 1}, -8];
Table[a[n], {n, 0, 17}] (* Peter Luschny, Mar 14 2018 *)
LegendreP[Range[0, 30], 3]^2 (* G. C. Greubel, May 17 2023 *)

{a(n) = sum(k=0, n, 2^k * binomial(2*k, k)^2 * binomial(n+k, n-k) )}
for(n=0, 20, print1(a(n), ", "))

{a(n)=polcoeff( 1 / agm(1-x, sqrt((1+x)^2 - 36*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

from math import comb
def A243949(n): return sum(comb(n,k)*comb(n+k,k) for k in range(n+1))**2 # Chai Wah Wu, Mar 23 2023

[gen_legendre_P(n,0,3)^2 for n in range(41)] # G. C. Greubel, May 17 2023

G.f.: 1 / AGM(1-x, sqrt(1-34*x+x^2)). - Paul D. Hanna, Aug 30 2014
a(n) = Sum_{k=0..n} 2^k * C(2*k, k)^2 * C(n+k, n-k).
a(n)^(1/2) = Sum_{k=0..n} C(2*k, k) * C(n+k, n-k).
Recurrence: n^2*(2*n-3)*a(n) = (2*n-1)*(35*n^2 - 70*n + 26)*a(n-1) - (2*n-3)*(35*n^2 - 70*n + 26)*a(n-2) + (n-2)^2*(2*n-1)*a(n-3). - Vaclav Kotesovec, Aug 18 2014
a(n) ~ (4 + 3*sqrt(2)) * (3 + 2*sqrt(2))^(2*n) / (8*Pi*n). - Vaclav Kotesovec, Aug 18 2014
From Gheorghe Coserea, Jul 05 2016: (Start)
G.f.: hypergeom([1/12, 5/12],[1],27648*x^4*(x^2-34*x+1)*(x-1)^2/(1-36*x+134*x^2-36*x^3+x^4)^3)/(1-36*x+134*x^2-36*x^3+x^4)^(1/4).
0 = x*(x-1)*(x+1)*(x^2-34*x+1)*y'' + (3*x^4-66*x^3-70*x^2+70*x-1)*y' + (x^3-7*x^2-35*x+9)*y, where y is g.f.
(End)
a(n) = Sum_{k = 0..n} 4^k*binomial(n+k,2*k)^2*binomial(2*k,k). - Peter Bala, Mar 02 2017
a(n) = hypergeom([1/2, -n, n + 1], [1, 1], -8). - Peter Luschny, Mar 14 2018
G.f.: Sum_{n >= 0} (2^n)*binomial(2*n,n)^2 *x^n/(1-x)^(2*n+1). - Peter Bala, Feb 07 2022

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.

A109983 Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.

Original entry on oeis.org

1, 0, 1, 2, 0, 0, 1, 6, 6, 0, 0, 0, 1, 12, 30, 20, 0, 0, 0, 0, 1, 20, 90, 140, 70, 0, 0, 0, 0, 0, 1, 30, 210, 560, 630, 252, 0, 0, 0, 0, 0, 0, 1, 42, 420, 1680, 3150, 2772, 924, 0, 0, 0, 0, 0, 0, 0, 1, 56, 756, 4200, 11550, 16632, 12012, 3432

Offset: 0

Emeric Deutsch, Jul 07 2005

A Delannoy path of length n is a path from (0, 0) to (n, n), consisting of steps E = (1,0), N = (0,1) and D = (1,1).
Row n has 2*n+1 terms, the first n of which are 0.
Row sums are the central Delannoy numbers (A001850).
Column sums are the central trinomial coefficients (A002426).

			T(2, 3) = 6 because we have DNE, DEN, NED, END, NDE and EDN.
Triangle begins
  1;
  0,1,2;
  0,0,1,6,6;
  0,0,0,1,12,30,20;
  ...

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
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.
Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Cf. A001850, A002426, A000984, A063007, A104684, A109984.

a109983 n k = a109983_tabf !! n !! k
a109983_row n = a109983_tabf !! n
a109983_tabf = zipWith (++) (map (flip take (repeat 0)) [0..]) a063007_tabl
-- Reinhard Zumkeller, Nov 18 2014

T := (n,k)->binomial(n,2*n-k)*binomial(k,n):
for n from 0 to 8 do seq(T(n,k),k=0..2*n) od; # yields sequence in triangular form
# Alternative:
gf := ((1 - x*y)^2 - 4*x^2*y)^(-1/2):
yser := series(gf, y, 12): ycoeff := n -> coeff(yser, y, n):
row := n -> seq(coeff(expand(ycoeff(n)), x, k), k=0..2*n):
seq(row(n), n=0..7); # Peter Luschny, Oct 28 2020

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

T(n, k) = binomial(n, 2*n-k) binomial(k, n).
T(n, k) = A104684(n, 2*n-k).
G.f.: 1/sqrt((1 - t*z)^2 - 4*z*t^2).
T(n, 2*n) = binomial(2*n, n) (A000984).
Sum_{k=0..n} k*T(n, k) = A109984(n).
T(n, k) = A063007(n, k-n). - Michael Somos, Sep 22 2013

A105868 Triangle read by rows, T(n,k) = C(n,k)*C(k,n-k).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 6, 1, 0, 0, 6, 12, 1, 0, 0, 0, 30, 20, 1, 0, 0, 0, 20, 90, 30, 1, 0, 0, 0, 0, 140, 210, 42, 1, 0, 0, 0, 0, 70, 560, 420, 56, 1, 0, 0, 0, 0, 0, 630, 1680, 756, 72, 1, 0, 0, 0, 0, 0, 252, 3150, 4200, 1260, 90, 1, 0, 0, 0, 0, 0, 0, 2772, 11550, 9240, 1980, 110, 1, 0, 0

Offset: 0

Paul Barry, Apr 23 2005

Row sums are the central trinomial coefficients A002426.
Product of A007318 and this sequence is A008459.
Coefficient array for polynomials P(n,x) = x^n*F(1/2-n/2,-n/2;1;4/x). - Paul Barry, Oct 04 2008
Column sums give A001850. It appears that the sums along the antidiagonals of the triangle produce A182883. - Peter Bala, Mar 06 2013

			Triangle begins
  1;
  0,  1;
  0,  2,  1;
  0,  0,  6,  1;
  0,  0,  6, 12,  1;
  0,  0,  0, 30, 20, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows

Cf. A063007. A001850 (column sums), A182883.

[[Binomial(n,k)*Binomial(k,n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jun 14 2015

gf := 1/((1 - x*y)^2 - 4*y^2*x)^(1/2):
yser := series(gf, y, 12): ycoeff := n -> coeff(yser, y, n):
row := n -> seq(coeff(expand(ycoeff(n)), x, k), k=0..n):
seq(row(n), n=0..7); # Peter Luschny, Oct 28 2020

Flatten[Table[Binomial[n,k]Binomial[k,n-k],{n,0,20},{k,0,n}]] (* Harvey P. Dale, Nov 12 2014 *)

G.f.: 1/(sqrt((1-x*y)^2-4*x^2*y)). - Vladimir Kruchinin, Oct 28 2020

A108556 Triangle, read by rows, where row n equals the inverse binomial transform of the crystal ball sequence for D_n lattice.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 12, 30, 20, 1, 24, 120, 192, 96, 1, 40, 330, 940, 1080, 432, 1, 60, 732, 3200, 6240, 5568, 1856, 1, 84, 1414, 8708, 25200, 37184, 27104, 7744, 1, 112, 2480, 20352, 80960, 173824, 206080, 126976, 31744, 1, 144, 4050, 42588, 221544, 643824, 1096032, 1085760, 579456, 128768

Offset: 0

Paul D. Hanna, Jun 10 2005

Row n equals the inverse binomial transform of row n of the square array A108553.

Array of f-vectors for type D root polytopes [Ardila et al.]. See A063007 and A127674 for the arrays of f-vectors for type A and type C root polytopes respectively. - Peter Bala, Oct 23 2008

			Triangle begins:
1;
1,2;
1,4,4;
1,12,30,20;
1,24,120,192,96;
1,40,330,940,1080,432;
1,60,732,3200,6240,5568,1856;
1,84,1414,8708,25200,37184,27104,7744;
1,112,2480,20352,80960,173824,206080,126976,31744; ...

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.

Cf. A108553, A108557 (row sums), A108558, Rows are inverse binomial transforms of: A001844 (row 2), A005902 (row 3), A007204 (row 4), A008356 (row 5), A008358 (row 6), A008360 (row 7), A008362 (row 8), A008377 (row 9), A008379 (row 10).

Cf. A063007, A127674.

T[n_, k_] := Module[{A}, A = Table[Table[If[r - 1 == 0 || c - 1 == 0, 1, If[r - 1 == 1, 2c - 1, Sum[Binomial[r + c - j - 2, c - j - 1] (Binomial[2r - 2, 2j] - 2(r - 1) Binomial[r - 3, j - 1]), {j, 0, c - 1}]]], {c, 1, n + 1}], {r, 1, n + 1}]; SeriesCoefficient[((A[[n + 1]]. x^Range[0, n]) /. x -> x/(1 + x))/(1 + x), {x, 0, k}]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 26 2018, from PARI *)

T(n,k)=local(A=vector(n+1,r,vector(n+1,c,if(r-1==0 || c-1==0,1,if(r-1==1,2*c-1, sum(j=0,c-1,binomial(r+c-j-2,c-j-1)*(binomial(2*r-2,2*j)-2*(r-1)*binomial(r-3,j-1)))))))); polcoeff(subst(Ser(A[n+1]),x,x/(1+x))/(1+x),k)

Main diagonal equals A008353: 2^(n-1)*(2^n-n) for n>1.

O.g.f. : rational function N(x,z)/D(x,z), where N(x,z) = 1 - 3*(1 + 2*x)*z + (3 + 8*x + 8*x^2)*z^2 - (1 + 2*x)*(1 - 6*x - 6*x^2)z^3 - 8*x*(1 + x)(1 + 2*x + 2*x^2)*z^4 + 2*x*(1 + x)*(1 + 2*x)*z^5 and D(x,z) = ((1-z)^2 - 4*x*z)*(1 - z*(1 + 2*x))^2. - Peter Bala, Oct 23 2008

cp's OEIS Frontend

A262177 Decimal expansion of Q_5 = zeta(5) / (Sum_{k>=1} (-1)^(k+1) / (k^5 * binomial(2k, k))), a conjecturally irrational constant defined by an Apéry-like formula.

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A089627 T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.

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A104684 Triangle read by rows: 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 D=(1,1) steps.

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A127674 Coefficient table for Chebyshev polynomials T(2*n,x) (increasing even powers x, without zeros).

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A253283 Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.

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A243949 Squares of the central Delannoy numbers: a(n) = A001850(n)^2.

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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).

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A089627 T(n,k) = binomial(n,2k)binomial(2*k,k) for 0 <= k <= n, triangle read by rows.

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