A002457 -id:A002457 - cp's OEIS frontend

A000531 From area of cyclic polygon of 2n + 1 sides.

1, 7, 38, 187, 874, 3958, 17548, 76627, 330818, 1415650, 6015316, 25413342, 106853668, 447472972, 1867450648, 7770342787, 32248174258, 133530264682, 551793690628, 2276098026922, 9373521044908, 38546133661492

Offset: 1

Simon Plouffe

Expected number of matches remaining in Banach's original matchbox problem (counted when empty box is chosen), multiplied by 2^(2*n-1). - Michael Steyer, Apr 13 2001

A conjectured definition: Let 0 < a_1 < a_2 <...

a(n) = total weight of upsteps in all Dyck n-paths (A000108) when each upstep is weighted with its position in the path. For example, the Dyck path UDUUDUDD has upsteps in positions 1,3,4,6 and contributes 1+3+4+6=14 to the weight for Dyck 4-paths. The summand (n-k)*binomial(2*n+1, k) in the Maple formula below is the total weight of upsteps terminating at height n-k, 0<=k<=n-1. - David Callan, Dec 29 2006

Catalan transform of binomial transform of squares. - Philippe Deléham, Oct 31 2008

a(n) is also the number of walks of length 2n in the quarter plane starting and ending at the origin using steps {(1,1),(1,0),(-1,0), (-1,-1)} (which appear in Gessel's conjecture) in which the steps (1,0) and (-1,0) appear exactly once each. - Arvind Ayyer, Mar 02 2009

Equals the Catalan sequence, A000108, convolved with A002457: (1, 6, 30, 140, ...). - Gary W. Adamson, May 14 2009

Total number of occurrences of the pattern 213 (or 132) in all skew-indecomposable (n+2)-permutations avoiding the pattern 123. For example, a(1) = 1, since there is one occurrence of the pattern 213 in the set {213, 132}. - Cheyne Homberger, Mar 13 2013

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I.

T. D. Noe, Table of n, a(n) for n = 1..100
Arvind Ayyer, Towards a human proof of Gessel's conjecture, arXiv:0902.2329 [math.CO], 2009, JIS 12 (2009) 09.4.2
Hacène Belbachir, Toufik Djellal, Jean-Gabriel Luque, On the self-convolution of generalized Fibonacci numbers, arXiv:1703.00323 [math.CO], 2017.
F. Bowman, Cyclic pentagons, Math. Gaz. 36, (1952). 244-250. MR0051523.
A. Burstein and S. Elizalde, Total occurrence statistics on restricted permutations, arXiv preprint arXiv:1305.3177 [math.CO], 2013.
C. Homberger, Expected patterns in permutation classes, Electronic Journal of Combinatorics, 19(3) (2012), P43.
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
Yasuhiko Kamiyama, The Euler characteristic of the regular spherical polygon spaces, arXiv:1803.05559 [math.GT], 2018.
A. F. Moebius, Über die Gleichungen, mittelst welcher aus den Seiten eines in einen Kreis zu beschreibenden Vielecks der Halbmesser des Kreises und die Fläche des Vielecks gefunden werden, Gesammelte Werke, vol. 1., pp. 407-438.
D. P. Robbins, Areas of polygons inscribed in a circle, Amer. Math. Monthly, 102 (1995), 523-530.
Eric Weisstein's World of Mathematics, Cyclic Polygon
Y. Q. Zhao, Introduction to Probability with Applications

Cf. A002457 (Banach's modified matchbox problem), A135404, A002457, A258431.

f := proc(n) sum((n-k)*binomial(2*n+1,k),k=0..n-1); end;

a[n_] := ((2n+1)!/n!^2-4^n)/2; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Dec 07 2011, after Pari *)

PARI
```
a(n)=if(n<1,0,((2*n+1)!/n!^2-4^n)/2)
```

a(n) = ((2n+1)!/((n!)^2)-4^n)/2. - Simon Norton (simon(AT)dpmms.cam.ac.uk), May 14 2001
na(n) = (8n-2)a(n-1) - (16n-8)a(n-2), n>1. - Michael Somos, Apr 18 2003
E.g.f.: 1/2*((1+4*x)*exp(2*x)*BesselI(0, 2*x) + 4*x*exp(2*x)*BesselI(1, 2*x) - exp(4*x)). - Vladeta Jovovic, Sep 22 2003
a(n-1) = 4^n*sum_{k=0..n} binomial(2*k+1, k)*4^(-k) = (2*n+1)*(2*n+3)*C(n) - 2^(2*n+1) (C(n) = Catalan); g.f.: x*c(x)/(1-4*x)^(3/2), c(x): g.f. of Catalan numbers A000108. - Wolfdieter Lang
a(n) = Sum_{k=0..n} A039599(n,k)*k^2, for n>=1. - Philippe Deléham, Jun 10 2007
a(n) = Sum_{k=0..n} A106566(n,k)*A001788(k). - Philippe Deléham, Oct 31 2008
(Conjecture) a(n)=2^(2*n)*sum_{k=1..n} cos(k*Pi/(2*n+1))^2*n. - L. Edson Jeffery, Jan 21 2012

Moebius reference from Michael Somos

A002544 a(n) = binomial(2n+1,n)(n+1)^2.

Original entry on oeis.org

1, 12, 90, 560, 3150, 16632, 84084, 411840, 1969110, 9237800, 42678636, 194699232, 878850700, 3931426800, 17450721000, 76938289920, 337206098790, 1470171918600, 6379820115900, 27569305764000, 118685861314020, 509191949220240, 2177742427450200, 9287309860732800

Offset: 0

N. J. A. Sloane, Simon Plouffe

Coefficients for numerical differentiation.
Take the first n integers 1,2,3..n and find all combinations with repetitions allowed for the first n of them. Find the sum of each of these combinations to get this sequence. Example for 1 and 2: 1,2,1+1,1+2,2+2 gives sum of 12=a(2). - J. M. Bergot, Mar 08 2016
Let cos(x) = 1 -x^2/2 +x^4/4!-x^6/6!.. = Sum_i (-1)^i x^(2i)/(2i)! be the standard power series of the cosine, and y = 2*(1-cos(x)) = 4*sin^2(x/2) = x^2 -x^4/12 +x^6/360 ...= Sum_i 2*(-1)^(i+1) x^(2i)/(2i)! be a closely related series. Then this sequence represents the reversion x^2 = Sum_i 1/a(i) *y^(i+1). - R. J. Mathar, May 03 2022

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 514.
J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 92.
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
W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables) [Annotated scanned copy]
Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
A. Petojevic and N. Dapic, The vAm(a,b,c;z) function, Preprint 2013.
H. E. Salzer, Coefficients for numerical differentiation with central differences, J. Math. Phys., 22 (1943), 115-135.
H. E. Salzer, Coefficients for numerical differentiation with central differences, J. Math. Phys., 22 (1943), 115-135. [Annotated scanned copy]
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)
R. Shenton and A. W. Kemp, An S-fraction and ln^2(1+x), Journal of Computational and Applied Mathematics, 26 (1989) 367-370 North-Holland.
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 21.
Mats Vermeeren, A dynamical solution to the Basel problem, arXiv preprint arXiv:1506.05288 [math.CA], 2015.

Cf. A085373, A002457, A002674, A202543.
Equals A002736/2.
A diagonal of A331430.

seq((n+1)^2*(binomial(2*n+2, n+1))/2, n=0..29); # Zerinvary Lajos, May 31 2006

Table[Binomial[2n+1,n](n+1)^2,{n,0,20}] (* Harvey P. Dale, Mar 23 2011 *)

PARI
```
a(n)=binomial(2*n+1,n)*(n+1)^2
```

x='x+O('x^99); Vec((1+2*x)/(1-4*x)^(5/2)) \\ Altug Alkan, Jul 09 2016

from sympy import binomial
def a(n): return binomial(2*n + 1, n)*(n + 1)**2 # Indranil Ghosh, Apr 18 2017

G.f.: (1 + 2x)/(1 - 4x)^(5/2).
a(n-1) = sum(i_1 + i_2 + ... + i_n) where the sum is over 0 <= i_1 <= i_2 <= ... <= i_n <= n; a(n) = (n+1)^2 C(2n+1, n). - David Callan, Nov 20 2003
a(n) = (n+1)^2 * binomial(2*n+2,n+1)/2. - Zerinvary Lajos, May 31 2006
Asymptotics: a(n)-> (1/64) * (128*n^2+176*n+41) * 4^n * n^(-1/2)/(sqrt(Pi)), for n->infinity. - Karol A. Penson, Aug 05 2013
G.f.: 2F1(3/2,2;1;4x). - R. J. Mathar, Aug 09 2015
a(n) = A002457(n)*(n+1). - R. J. Mathar, Aug 09 2015
a(n) = A000217(n)*A000984(n). - J. M. Bergot, Mar 10 2016
a(n-1) = A001791(n)*n*(n+1)/2. - Anton Zakharov, Jul 04 2016
From Ilya Gutkovskiy, Jul 04 2016: (Start)
E.g.f.: ((1 + 2*x)*(1 + 8*x)*BesselI(0,2*x) + 2*x*(3 + 8*x)*BesselI(1,2*x))*exp(2*x).
Sum_{n>=0} 1/a(n) = Pi^2/9 = A100044. (End)
From Peter Bala, Apr 18 2017: (Start)
With x = y^2/(1 + y) we have log^2(1 + y) = Sum_{n >= 0} (-1)^n*x^(n+1)/a(n). See Shenton and Kemp.
Series reversion ( Sum_{n >= 0} (-1)^n*x^(n+1)/a(n) ) = Sum_{n >= 1} 2*x^n/(2*n)! = Sum_{n >= 1} x^n/A002674(n). (End)
D-finite with recurrence n^2*a(n) -2*(n+1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Feb 08 2021
Sum_{n>=0} (-1)^n/a(n) = 4*arcsinh(1/2)^2 = A202543^2. - Amiram Eldar, May 14 2022

A020918 Expansion of 1/(1-4*x)^(7/2).

Original entry on oeis.org

1, 14, 126, 924, 6006, 36036, 204204, 1108536, 5819814, 29745716, 148728580, 730122120, 3528923580, 16830250920, 79342611480, 370265520240, 1712478031110, 7857252142740, 35794148650260

Offset: 0

N. J. A. Sloane

Also convolution of A000984 with A038845, also convolution of A000302 with A002802, also convolution of A002457 with A002697. - Rui Duarte, Oct 08 2011

5*a(n) is the number of (n+3) X 2 Young tableaux with a four horizontal walls between the first and second column. If there is a wall between two cells, the entries may be decreasing; see [Banderier, Wallner 2021], A000984 for one horizontal wall, A002457 for two, and A002802 for three. - Michael Wallner, Mar 09 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Cyril Banderier and Michael Wallner, Young Tableaux with Periodic Walls: Counting with the Density Method, Séminaire Lotharingien de Combinatoire, 85B (2021), Art. 47, 12 pp.

Cf. A000302, A000984, A001622, A002457, A002697, A002802, A000292, A038845, A046521 (fourth column).

List([0..30], n-> Binomial(2*n+6, n+3)*Binomial(n+3, 3)/20); # G. C. Greubel, Jul 20 2019

[Binomial(2*n+6, n+3)*Binomial(n+3, 3)/20: n in [0..30]]; // G. C. Greubel, Jul 20 2019

seq(binomial(2*n,n)*binomial(n,(n-3))/20, n=2..21); # Zerinvary Lajos, May 05 2007
seq(simplify(4^n*hypergeom([-n,-5/2], [1], 1)),n=0..18); # Peter Luschny, Apr 26 2016

CoefficientList[Series[1/(1-4x)^(7/2), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 04 2013 *)

vector(30, n, n--; binomial(2*n+6, n+3)*binomial(n+3, 3)/20 ) \\ G. C. Greubel, Jul 20 2019

[binomial(2*n+6, n+3)*binomial(n+3, 3)/20 for n in (0..30)] # G. C. Greubel, Jul 20 2019

a(n) = binomial(n+3, 3)*A000984(n+3)/A000984(3), where A000984 are the central binomial coefficients. - Wolfdieter Lang
a(n) ~ 8/15*Pi^(-1/2)*n^(5/2)*2^(2*n)*{1 + 35/8*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
a(n) = Sum_{a+b+c+d+e+f+g=n} f(a)*f(b)*f(c)*f(d)*f(e)*f(f)*f(g) with f(n)=A000984(n). - Philippe Deléham, Jan 22 2004
a(n) = A000292(n)*A000984(n+2)/20. - Zerinvary Lajos, May 05 2007
From Rui Duarte, Oct 08 2011: (Start)
a(n) = ((2n+5)(2n+3)(2n+1)/(5*3*1)) * binomial(2n, n).
a(n) = binomial(2n+6, 6) * binomial(2n, n) / binomial(n+3, 3).
a(n) = binomial(n+3, 3) * binomial(2n+6, n+3) / binomial(6, 3). (End)
a(n) = 4^n*hypergeom([-n,-5/2], [1], 1). - Peter Luschny, Apr 26 2016
Boas-Buck recurrence: a(n) = (14/n)*Sum_{k=0..n-1} 4^(n-k-1)*a(k), n >= 1, a(0) = 1. Proof from a(n) = A046521(n+3, 3). See a comment there. - Wolfdieter Lang, Aug 10 2017
From Amiram Eldar, Apr 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 10*sqrt(3)*Pi - 160/3.
Sum_{n>=0} (-1)^n/a(n) = 10*sqrt(5)*log(phi) - 320/3, where phi is the golden ratio (A001622). (End)
D-finite with recurrence n*a(n) +2*(-2*n-5)*a(n-1)=0. - R. J. Mathar, Aug 01 2022

A040075 5-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^5.

Original entry on oeis.org

1, 20, 240, 2240, 17920, 129024, 860160, 5406720, 32440320, 187432960, 1049624576, 5725224960, 30534533120, 159719096320, 821412495360, 4161823309824, 20809116549120, 102821517066240, 502682972323840, 2434043865989120, 11683410556747776, 55635288365465600

Offset: 0

Wolfdieter Lang

Also convolution of A020920 with A000984 (central binomial coefficients).
With a different offset, number of n-permutations (n=5) of 5 objects u, v, w, z, x with repetition allowed, containing exactly four (4)u's. Example: a(1)=20 because we have uuuuv, uuuvu, uuvuu, uvuuu, vuuuu, uuuuw, uuuwu, uuwuu, uwuuu, wuuuu, uuuuz, uuuzu, uuzuu, uzuuu, zuuuu, uuuux, uuuxu, uuxuu, uxuuu and xuuuu. - Zerinvary Lajos, May 19 2008
Also convolution of A000302 with A038846, also convolution of A002457 with A020918, also convolution of A002697 with A038845, also convolution of A002802 with A002802. [Rui Duarte, Oct 08 2011]

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Idempotent Number.
Index entries for linear recurrences with constant coefficients, signature (20, -160, 640, -1280, 1024).

Cf. A000302, A020920, A000984.

Cf. A038231.

List([0..30], n-> 4^n*Binomial(n+4, 4)); # G. C. Greubel, Jul 20 2019

[4^n*Binomial(n+4, 4): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011

seq(seq(binomial(i, j)*4^(i-4), j =i-4), i=4..22); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+4,4)*4^n,n=0..30); # Zerinvary Lajos, May 19 2008
spec := [S, {B=Set(Z, 0 <= card), S=Prod(Z, Z, Z, Z, B, B, B, B)}, labeled]: seq(combstruct[count](spec, size=n)/24, n=4..34); # Zerinvary Lajos, Apr 05 2009

Table[Binomial[n+4,4]*4^n, {n,0,30}] (* Michael De Vlieger, Aug 21 2015 *)

vector(30, n, n--; 4^n*binomial(n+4, 4)) \\ G. C. Greubel, Jul 20 2019

[lucas_number2(n, 4, 0)*binomial(n,4)/2^8 for n in range(4, 34)] # Zerinvary Lajos, Mar 11 2009

a(n) = binomial(n+4, 4)*4^n.
G.f.: 1/(1-4*x)^5.
a(n) = Sum_{ i_1+i_2+i_3+i_4+i_5+i_6+i_7+i_8+i_9+i_10 = n } f(i_1)*f(i_2) *f(i_3)*f(i_4)*f(i_5)*f(i_6)*f(i_7)*f(i_8)*f(i_9)*f(i_10) with f(k)=A000984(k). - Rui Duarte, Oct 08 2011
E.g.f.: (3 + 48*x + 144*x^2 + 128*x^3 + 32*x^4)*exp(4*x)/3. - G. C. Greubel, Jul 20 2019
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 376/3 - 432*log(4/3).
Sum_{n>=0} (-1)^n/a(n) = 2000*log(5/4) - 1336/3. (End)

A002696 Binomial coefficients C(2n,n-3).

Original entry on oeis.org

1, 8, 45, 220, 1001, 4368, 18564, 77520, 319770, 1307504, 5311735, 21474180, 86493225, 347373600, 1391975640, 5567902560, 22239974430, 88732378800, 353697121050, 1408831480056, 5608233007146, 22314239266528, 88749815264600, 352870329957600, 1402659561581460

Offset: 3

N. J. A. Sloane

Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch or cross the line x-y=3. - Herbert Kociemba, May 23 2004

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 517.
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).

Robert Israel, Table of n, a(n) for n = 3..1497
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].
A. Claesson and T. Mansour, Counting patterns of type (1,2) or (2,1), arXiv:math/0110036 [math.CO], 2001.
Milan Janjic, Two Enumerative Functions
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
Toufik Mansour and Mark Shattuck, Counting occurrences of subword patterns in non-crossing partitions, Art Disc. Appl. Math. (2022).
R. Parviainen, Lattice Path Enumeration of Permutations with k Occurrences of the Pattern 2-13, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.2.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Hermann Stamm-Wilbrandt, Compute C(2n, n-k) based on C(n,...) animation
Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).

Diagonal 7 of triangle A100257.
Column k=1 of A263776.
Cf. A001622.
Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002054 (m = 3), A002694 (m = 4), A003516 (m = 5), A030053 - A030056, A004310 - A004318.

List([3..30], n-> Binomial(2*n, n-3)) # G. C. Greubel, Mar 21 2019

[ Binomial(2*n,n-3): n in [3..30] ]; // Vincenzo Librandi, Apr 13 2011

A002696:=n->binomial(2*n,n-3): seq(A002696(n), n=3..30); # Wesley Ivan Hurt, Aug 19 2015

CoefficientList[Series[64/(((Sqrt[1-4x] +1)^6)*Sqrt[1-4x]), {x,0,30}], x] (* Robert G. Wilson v, Aug 08 2011 *)

a(n)=binomial(n+n,n-3) \\ Charles R Greathouse IV, Aug 08 2011

[binomial(2*n, n-3) for n in (3..30)] # G. C. Greubel, Mar 21 2019

G.f.: (1-sqrt(1-4*z))^6/(64*z^3*sqrt(1-4*z)). - Emeric Deutsch, Jan 28 2004
a(n) = Sum_{k=0..n} C(n, k)*C(n, k+3). - Hermann Stamm-Wilbrandt, Aug 17 2015
From Robert Israel, Aug 19 2015: (Start)
(n-2)*(n+4)*a(n+1) = (2*n+2)*(2*n+1)*a(n).
E.g.f.: I_3(2*x) * exp(2*x) where I_3 is a modified Bessel function. (End)
From Amiram Eldar, Aug 27 2022: (Start)
Sum_{n>=3} 1/a(n) = 3/4 + 2*Pi/(9*sqrt(3)).
Sum_{n>=3} (-1)^(n+1)/a(n) = 444*log(phi)/(5*sqrt(5)) - 1093/60, where phi is the golden ratio (A001622). (End)
G.f.: 2F1([7/2,4],[7],4*x). - Karol A. Penson, Apr 24 2024
From Peter Bala, Oct 13 2024: (Start)
a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * (x^3 - 6*x^2 + 9*x - 2)/sqrt(x*(4 - x)).
G.f: x^3 * B(x) * C(x)^6, where B(x) = 1/sqrt(1 - 4*x) is the g.f. of the central binomial coefficients A000984 and C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

More terms from Emeric Deutsch, Feb 18 2004

A033876 Expansion of 1/(2x) (1/(1-4*x)^(3/2)-1).

Original entry on oeis.org

3, 15, 70, 315, 1386, 6006, 25740, 109395, 461890, 1939938, 8112468, 33801950, 140408100, 581690700, 2404321560, 9917826435, 40838108850, 167890003050, 689232644100, 2825853840810, 11572544300460, 47342226683700, 193485622098600, 790066290235950, 3223470464162676

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

a(n) is the trace of the zigzag matrix Z(n+1) (see A088961). - Paul Boddington, Nov 03 2003
The number of edges in the odd graph O_k (for k >= 2) can be computed as 0.5*(2k-1)*C(2k-2,k-1). This sequence gives the number of edges in O_k for integer values of k from k=2. - K.V.Iyer, Mar 04 2009
Apparently the number of peaks in all symmetric Dyck paths with semilength 2n+2. - David Scambler, Apr 29 2013
For n > 0, also the number of maximal and maximum cliques in the (n+2)-odd graph. - Eric W. Weisstein, Nov 30 2017

			G.f. = 3 + 15*x + 70*x^2 + 315*x^3 + 1386*x^4 + 6006*x^5 + 25740*x^6 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Maximal Clique.
Eric Weisstein's World of Mathematics, Maximum Clique.
Eric Weisstein's World of Mathematics, Odd Graph.

Cf. A000984, A001622, A001700, A001803, A002457, A037965, A088961.

a033876 n = sum $ zipWith (!!) zss [0..n] where
   zss = take (n+1) $ g (take (n+1) (1 : [0,0..])) where
       g us = (take (n+1) $ g' us) : g (0 : init us)
       g' vs = last $ take (2 * n + 3) $
                      map snd $ iterate h (0, vs ++ reverse vs)
   h (p,ws) = (1 - p, drop p $ zipWith (+) ([0] ++ ws) (ws ++ [0]))
-- Reinhard Zumkeller, Oct 25 2013

[(2*n+3)*Binomial(2*n+1, n) : n in [0..40]]; // Wesley Ivan Hurt, Nov 30 2017

[seq((n+2)*binomial(2*(n+2),n+2)/4, n=0..22)]; # Zerinvary Lajos, Jan 04 2007

Table[nn = 2 n + 1; (2 n + 1)! Coefficient[Series[Exp[x] (x^n/n!)^2/2, {x, 0, nn}], x^(2 n + 1)], {n, 30}] (* Geoffrey Critzer, Apr 19 2017 *)
Table[n Binomial[2 n, n]/4, {n, 2, 20}] (* Eric W. Weisstein, Nov 30 2017 *)
Table[(4^n Gamma[n + 3/2])/(Sqrt[Pi] Gamma[n + 1]), {n, 20}] (* Eric W. Weisstein, Nov 30 2017 *)
CoefficientList[Series[((1 - 4 x)^(-3/2) - 1)/(2 x), {x, 0, 20}], x] (* Eric W. Weisstein, Nov 30 2017 *)

x='x+O('x^66); Vec( 1/(2*x) * (1/(1-4*x)^(3/2)-1) ) \\ Joerg Arndt, May 01 2013

a(n) = (2*n+3)*binomial(2*n+1, n). - Paul Boddington, Nov 03 2003
Equals n*A000984/4, n >= 2. - Zerinvary Lajos, Jan 04 2007
For n >= 1, 1/a(n-1) = Sum_{k>=0} binomial(2*k,k)/(4^(n+k)*(n+k+1)) = int(4*t^n/sqrt(1-4*t), t=0..1/4). - Groux Roland, Jan 17 2011
G.f.: - 1/(2*x) + G(0)/(4*x), where G(k)= 1 + 1/(1 - 2*x*(2*k+3)/(2*x*(2*k+3) + (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 18 2013
a(n) = 2^(2*n+1)*binomial(n+3/2, 1/2). - Peter Luschny, May 06 2014
0 = a(n)*(16*a(n+1) - 2*a(n+2)) + a(n+1)*(-6*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Sep 17 2014
a(n-2) = n*binomial(2*n, n)/4 for n > 1. - Eric W. Weisstein, Nov 30 2017
G.f.: ((1 - 4*x)^(-3/2) - 1)/2 (by definition). - Eric W. Weisstein, Nov 30 2017
D-finite with recurrence: (n+1)*a(n) +2*(-2*n-3)*a(n-1)=0. - R. J. Mathar, Jan 28 2020
G.f.: (1F0(3/2;;4*x)-1)/(2*x). - R. J. Mathar, Jan 28 2020
From Amiram Eldar, Mar 04 2023: (Start)
Sum_{n>=0} 1/a(n) = 4*Pi/(3*sqrt(3)) - 2.
Sum_{n>=0} (-1)^n/a(n) = 2 - 8*log(phi)/sqrt(5), where phi is the golden ratio (A001622). (End)
From Mélika Tebni, Sep 04 2024: (Start)
a(n) = A037965(n+2) - A001700(n).
E.g.f.: exp(2*x)*((3+8*x)*BesselI(0, 2*x) + (1+8*x)*BesselI(1, 2*x)). (End)
a(n) = 2^n*JacobiP(n+1, 1/2, -n-1, 3). - Peter Luschny, Jan 22 2025

A000515 a(n) = (2n)!(2n+1)!/n!^4, or equally (2n+1)*binomial(2n,n)^2.

Original entry on oeis.org

1, 12, 180, 2800, 44100, 698544, 11099088, 176679360, 2815827300, 44914183600, 716830370256, 11445589052352, 182811491808400, 2920656969720000, 46670906271240000, 745904795339462400, 11922821963004219300, 190600129650794094000, 3047248986392325330000

Offset: 0

N. J. A. Sloane

a(n) is also the (n,n)-th entry in the inverse of the n-th Hilbert matrix. - Asher Auel, May 20 2001
a(n) is also the ratio of the determinants of the n-th Hilbert matrix to the (n+1)-th Hilbert matrix (see A005249), for n>0. Thus the determinant of the inverse of the n-th Hilbert matrix is the product of a(i) for i from 1 to n. (Claimed by Jud McCranie without proof, Jul 17 2000)
a(n) is the right side of the binomial sum: 2^(4*n) * Sum_{i=0..n} binomial(-1/2, i)*binomial(1/2, i). - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
Right-hand side of Sum_{i=0..n} Sum_{j=0..n} binomial(i+j,j)^2 * binomial(4n-2i-2j,2n-2j).

E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 96.
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.
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
G. E. Andrews and P. Paule, Some questions concerning computer-generated proofs of a binomial double-sum identity, J. Symbolic Computation 11(1994), 1-7.
D. Galakhov, A. Mironov and A. Morozov, Wall Crossing Invariants: from quantum mechanics to knots, arXiv preprint arXiv:1410.8482 [hep-th], 2014. See Eq. (A.15).
R. K. Guy, Letter to N. J. A. Sloane, Sep 1986
J. E. Lauer, Letter to N. J. A. Sloane, Dec 1980
D. H. Lehmer, Review of A. N. Lowan et al., "Table of the zeros of the Legendre polynomials of order 1-16...", in Math. Tables Aids Computation (MTAC), 1 (1943-1945), 52-53.
Pedro J. Miana and Natalia Romero, Moments of combinatorial and Catalan numbers, Journal of Number Theory, Volume 130, Issue 8, August 2010, Pages 1876-1887. See Omega3. Remark 3 p. 1882.
I. Nemes et al., How to do Monthly problems with your computer, Amer. Math. Monthly, 104 (1997), 505-519.
Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013 (see Omega_3).
Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33

Cf. A002894, A005249, A002457, A000108, A039598, A024492, A000894, A228329, A000515, A228330, A228331, A228332, A228333.

[(2*n+1)*Binomial(2*n,n)^2: n in [0..25]]; // Vincenzo Librandi, Oct 08 2015

with(linalg): for n from 1 to 24 do print(det(hilbert(n))/det(hilbert(n+1))): od;

A000515[n_] := (2*n + 1)*Binomial[2 n, n]^2 (* Enrique Pérez Herrero, Mar 31 2010 *)
Table[(2 n + 1) (n + 1)^2 CatalanNumber[n]^2, {n, 0, 18}] (* Jan Mangaldan, Sep 23 2021 *)

vector(100, n, n--; (2*n+1)*binomial(2*n,n)^2) \\ Altug Alkan, Oct 08 2015

a(n) ~ 2*Pi^-1*2^(4*n). - Joe Keane (jgk(AT)jgk.org), Jun 07 2002
O.g.f.: (2/Pi)*EllipticE(4*sqrt(x))/(1-16*x). - Vladeta Jovovic, Jun 15 2005
E.g.f.: Sum_{n>=0} a(n)*x^(2n)/(2n)! = BesselI(0, 2*x)*(BesselI(0, 2*x) + 4*x*BesselI(1, 2*x)). - Vladeta Jovovic, Jun 15 2005
E.g.f.: Sum_{n>=0} a(n)*x^(2n+1)/(2n+1)! = BesselI(0, 2x)^2*x. - Michael Somos, Jun 22 2005
E.g.f.: x*(BesselI(0, 2*x))^2 = x+(2*x^3)/(U(0)-2*x^2); U(k) = (2*x^2)*(2*k+1) + (k+1)^3 - (2*x^2)*(2*k+3)*((k+1)^3)/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 23 2011
n^2*a(n) - 4*(2*n-1)*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Sep 08 2013
O.g.f.: hypergeom([1/2, 3/2], [1], 16*x). - Peter Luschny, Oct 08 2015

A054335 A convolution triangle of numbers based on A000984 (central binomial coefficients of even order).

Original entry on oeis.org

1, 2, 1, 6, 4, 1, 20, 16, 6, 1, 70, 64, 30, 8, 1, 252, 256, 140, 48, 10, 1, 924, 1024, 630, 256, 70, 12, 1, 3432, 4096, 2772, 1280, 420, 96, 14, 1, 12870, 16384, 12012, 6144, 2310, 640, 126, 16, 1, 48620, 65536, 51480, 28672, 12012, 3840, 924, 160, 18, 1

Offset: 0

Wolfdieter Lang, Mar 13 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. The g.f. for the row polynomials p(n,x) (increasing powers of x) is 1/(sqrt(1-4*z)-x*z).
The column sequences are for m=0..20: A000984, A000302 (powers of 4), A002457, A002697, A002802, A038845, A020918, A038846, A020920, A040075, A020922, A045543, A020924, A054337, A020926, A054338, A020928, A054339, A020930, A054340, A020932.
Riordan array (1/sqrt(1-4*x),x/sqrt(1-4*x)). - Paul Barry, May 06 2009
The matrix inverse is apparently given by deleting the leftmost column from A206022. - R. J. Mathar, Mar 12 2013

			Triangle begins:
    1;
    2,    1;
    6,    4,   1;
   20,   16,   6,   1;
   70,   64,  30,   8,  1;
  252,  256, 140,  48, 10,  1;
  924, 1024, 630, 256, 70, 12, 1; ...
Fourth row polynomial (n=3): p(3,x) = 20 + 16*x + 6*x^2 + x^3.
From _Paul Barry_, May 06 2009: (Start)
Production matrix begins
    2,   1;
    2,   2,  1;
    0,   2,  2,  1;
   -2,   0,  2,  2,  1;
    0,  -2,  0,  2,  2,  1;
    4,   0, -2,  0,  2,  2, 1;
    0,   4,  0, -2,  0,  2, 2, 1;
  -10,   0,  4,  0, -2,  0, 2, 2, 1;
    0, -10,  0,  4,  0, -2, 0, 2, 2, 1; (End)

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Paul Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv preprint arXiv:1312.0583 [math.CO], 2013.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

Cf. A000984, A035324, A054336.

Row sums: A026671.

T:= function(n, k)
    if k mod 2=0 then return Binomial(2*n-k, n-Int(k/2))*Binomial(n-Int(k/2),Int(k/2))/Binomial(k,Int(k/2));
    else return 4^(n-k)*Binomial(n-Int((k-1)/2)-1, Int((k-1)/2));
    fi;
  end;
Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Jul 20 2019

T:= func< n, k | (k mod 2) eq 0 select Binomial(2*n-k, n-Floor(k/2))* Binomial(n-Floor(k/2),Floor(k/2))/Binomial(k,Floor(k/2)) else 4^(n-k)*Binomial(n-Floor((k-1)/2)-1, Floor((k-1)/2)) >;
[[T(n,k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Jul 20 2019

A054335 := proc(n,k)
    if k <0 or k > n then
        0 ;
    elif type(k,odd) then
        kprime := floor(k/2) ;
        binomial(n-kprime-1,kprime)*4^(n-k) ;
    else
        kprime := k/2 ;
        binomial(2*n-k,n-kprime)*binomial(n-kprime,kprime)/binomial(k,kprime) ;
    end if;
end proc: # R. J. Mathar, Mar 12 2013
# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, n -> binomial(2*(n-1), n-1)); # Peter Luschny, Oct 19 2022

Flatten[ CoefficientList[#1, x] & /@ CoefficientList[ Series[1/(Sqrt[1 - 4*z] - x*z), {z, 0, 9}], z]] (* or *)
a[n_, k_?OddQ] := 4^(n-k)*Binomial[(2*n-k-1)/2, (k-1)/2]; a[n_, k_?EvenQ] := (Binomial[n-k/2, k/2]*Binomial[2*n-k, n-k/2])/Binomial[k, k/2]; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 08 2011, updated Jan 16 2014 *)

T(n, k) = if(k%2==0, binomial(2*n-k, n-k/2)*binomial(n-k/2,k/2)/binomial(k,k/2), 4^(n-k)*binomial(n-(k-1)/2-1, (k-1)/2));
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 20 2019

def T(n, k):
    if (mod(k,2)==0): return binomial(2*n-k, n-k/2)*binomial(n-k/2,k/2)/binomial(k,k/2)
    else: return 4^(n-k)*binomial(n-(k-1)/2-1, (k-1)/2)
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jul 20 2019

a(n, 2*k+1) = binomial(n-k-1, k)*4^(n-2*k-1), a(n, 2*k) = binomial(2*(n-k), n-k)*binomial(n-k, k)/binomial(2*k, k), k >= 0, n >= m >= 0; a(n, m) := 0 if n

Column recursion: a(n, m)=2*(2*n-m-1)*a(n-1, m)/(n-m), n>m >= 0, a(m, m) := 1.

G.f. for column m: cbie(x)*(x*cbie(x))^m, with cbie(x) := 1/sqrt(1-4*x).

G.f.: 1/(1-x*y-2*x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction). - Paul Barry, May 06 2009

Sum_{k>=0} T(n,2*k)*(-1)^k*A000108(k) = A000108(n+1). - Philippe Deléham, Jan 30 2012

Sum_{k=0..floor(n/2)} T(n-k,n-2*k) = A098615(n). - Philippe Deléham, Feb 01 2012

T(n,k) = 4*T(n-1,k) + T(n-2,k-2) for k>=1. - Philippe Deléham, Feb 02 2012

Vertical recurrence: T(n,k) = 1*T(n-1,k-1) + 2*T(n-2,k-1) + 6*T(n-3,k-1) + 20*T(n-4,k-1) + ... for k >= 1 (the coefficients 1, 2, 6, 20, ... are the central binomial coefficients A000984). - Peter Bala, Oct 17 2015

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

A020920 Expansion of 1/(1-4*x)^(9/2).

Original entry on oeis.org

1, 18, 198, 1716, 12870, 87516, 554268, 3325608, 19122246, 106234700, 573667380, 3024791640, 15628090140, 79342611480, 396713057400, 1957117749840, 9540949030470, 46021048264620, 219878341708740, 1041528987041400, 4895186239094580, 22844202449108040

Offset: 0

N. J. A. Sloane

Also convolution of A000984 with A038846, also convolution of A000302 with A020918, also convolution of A002457 with A038845, also convolution of A002697 with A002802. - Rui Duarte, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000302, A000332, A000984, A001622, A002457, A002697, A002802, A020918, A038845, A038846, A046521 (fifth column).

List([0..30], n-> Binomial(n+4, 4)*Binomial(2*(n+4), n+4)/70) # G. C. Greubel, Jul 20 2019

[(2*n+7)*(2*n+5)*(2*n+3)*(2*n+1)*Binomial(2*n, n)/105: n in [0..30]]; // Vincenzo Librandi, Jul 05 2013

seq(binomial(2*n+8, n+4)*binomial(n+4, n)/70, n=0..30); # Zerinvary Lajos, May 05 2007

CoefficientList[Series[1/(1-4x)^(9/2), {x,0,30}], x] (* Vincenzo Librandi, Jul 05 2013 *)

vector(30, n, n--; m=n+4; binomial(m, 4)*binomial(2*m, m)/70) \\ G. C. Greubel, Jul 20 2019

[binomial(n+4, 4)*binomial(2*(n+4), n+4)/70 for n in (0..30)] # G. C. Greubel, Jul 20 2019

a(n) = binomial(n+4, 4)*A000984(n+4)/A000984(4), where A000984 are the central binomial coefficients. - Wolfdieter Lang
a(n) = Sum_{ a+b+c+d+e+f+g+h+i=n} f(a)*f(b)*f(c)*f(d)*f(e)*f(f)*f(g) *f(h)*f(i) with f(n)=A000984(n). - Philippe Deléham, Jan 22 2004
a(n) = A000332(n+4)*A000984(n+4)/70. - Zerinvary Lajos, May 05 2007
From Rui Duarte, Oct 08 2011: (Start)
a(n) = ((2n+7)(2n+5)(2n+3)(2n+1)/(7*5*3*1)) * binomial(2n, n).
a(n) = binomial(2n+8, 8) * binomial(2n, n) / binomial(n+4, 4).
a(n) = binomial(n+4, 4) * binomial(2n+8, n+4) / binomial(8, 4). (End)
Boas-Buck recurrence: a(n) = (18/n)*Sum_{k=0..n-1} 4^(n-k-1)*a(k), n >= 1, a(0) = 1. Proof from a(n) = A046521(n+4, 4). See a comment there.
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 1148/5 - 42*sqrt(3)*Pi.
Sum_{n>=0} (-1)^n/a(n) = 700*sqrt(5)*log(phi) - 11284/15, where phi is the golden ratio (A001622). (End)

Previous Showing 31-40 of 146 results. Next

cp's OEIS Frontend

A000531 From area of cyclic polygon of 2n + 1 sides.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A002544 a(n) = binomial(2*n+1,n)*(n+1)^2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

A020918 Expansion of 1/(1-4*x)^(7/2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A040075 5-fold convolution of A000302 (powers of 4); expansion of 1/(1-4*x)^5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A002696 Binomial coefficients C(2n,n-3).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A033876 Expansion of 1/(2*x) * (1/(1-4*x)^(3/2)-1).

Views

Author

A002544 a(n) = binomial(2n+1,n)(n+1)^2.

A033876 Expansion of 1/(2x) (1/(1-4*x)^(3/2)-1).