A002894 -id:A002894 - cp's OEIS frontend

A364510 a(n) = binomial(4*n, n)^2.

1, 16, 784, 48400, 3312400, 240374016, 18116083216, 1401950721600, 110634634890000, 8862957169158400, 718528370729238784, 58818762721626513424, 4853704694918904043024, 403242220875862752160000, 33694913171561404510440000, 2829611125043050701300998400

Offset: 0

Peter Bala, Jul 28 2023

Paolo Xausa, Table of n, a(n) for n = 0..500
Wikipedia, Dixon's identity

Cf. A002894, A005810, A364506, A364511, A364512.

Row 2 of A364509.

Maple
```
seq( binomial(4*n,n)^2, n = 0..15);
```

A364510[n_]:=Binomial[4n,n]^2;Array[A364510,15,0] (* Paolo Xausa, Oct 05 2023 *)

a(n) = Sum_{i = -n..n} (-1)^i * binomial(2*n, n+i)^2 * binomial(4*n, 2*n+i).
Compare with Dixon's identity: Sum_{i = -n..n} (-1)^i * binomial(2*n, n+i)^3 = (3*n)!/n!^3.
a(n) = A005810(n)^2.
P-recursive: a(n) = 16 * ( (4*n - 1)*(4*n - 2)*(4*n - 3)/(3*n*(3*n - 1)*(3*n - 2)) )^2 * a(n-1) with a(0) = 1.
a(n) ~ c^n * 2/(3*Pi*n), where c = (2^16)/(3^6).
a(n) = [x^n] G(x)^(16*n), where the power series G(x) = 1 + x + 9*x^2 + 225*x^3 + 7525*x^4 + 295228*x^5 + 12787152*x^6 + 592477457*x^7 + 28827755219*x^8 + ... appears to have integer coefficients.
exp( Sum_{n > = 1} a(n)*x^n/n ) = F(x)^16, where the power series F(x) =  1 + x + 25*x^2 + 1033*x^3 + 53077*x^4 + 3081944*x^5 + 193543624*x^6 + 12835533333*x^7 + 886092805699*x^8 + ... appears to have integer coefficients.
The supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3^r)) hold for all primes p >= 5 and all positive integers n and r.
a(n) = [x^(3*n)] ( (1 - x)^(6*n)*Legendre_P(2*n, (1 + x)/(1 - x)) ). - Peter Bala, Aug 14 2023

A071801 a(n) = binomial(2n, n) - binomial(n, floor(n/2))^2.

Original entry on oeis.org

0, 1, 2, 11, 34, 152, 524, 2207, 7970, 32744, 121252, 491988, 1850380, 7455944, 28337976, 113708295, 435443490, 1742630120, 6711230900, 26811568916, 103711749284, 413849297784, 1606464657096, 6405315809516, 24935144010764, 99367486347752

Offset: 0

T. D. Noe, Jun 06 2002

Number of lattice paths in the lattice [0..n] X [0..n] which do not pass through the point (floor(n/2),floor(n/2)). In this case, the "hole" in the lattice is at the point closest to the lattice center.

T. D. Noe, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Lattice path.

Cf. A000984, A001405, A002894, A071800, A071803.

[Binomial(2*n, n) - Binomial(n, Floor(n/2))^2 : n in [0..40]]; // Wesley Ivan Hurt, Jan 03 2017

A071801:=n->binomial(2*n, n) - binomial(n, floor(n/2))^2: seq(A071801(n), n=0..30); # Wesley Ivan Hurt, Jan 03 2017

Table[Binomial[2n, n] - Binomial[n, Floor[n/2]]^2, {n, 0, 20}]

a(n) = A000984(n) - A001405(n)^2.
Also, a(n) = Sum_{m=0..n} binomial(n, m)^2 - binomial(n, floor(n/2))^2.
G.f.: 1/sqrt(1-4*x) + 1/(4*x) - (4*x+1)*EllipticK(4*x)/(2*x*Pi). - Mark van Hoeij, May 01 2013

More terms from Roger L. Bagula, Aug 28 2006

Edited by N. J. A. Sloane, Oct 08 2006

A232607 G.f. A(x) satisfies: (A(x) + xA'(x)) / (A(x) - xA(x)^2) = Sum_{n>=0} binomial(2n,n)^2x^n.

Original entry on oeis.org

1, 3, 19, 159, 1546, 16517, 188246, 2248863, 27844369, 354576634, 4618570090, 61289049293, 826064774033, 11281763625102, 155834042142463, 2173801434825011, 30585769379262567, 433633765794690539, 6189637467948022825, 88886796123324352030, 1283443017706197910489, 18623352714450226405962

Offset: 0

Paul D. Hanna, Nov 26 2013

nonn

			G.f.: A(x) = 1 + 3*x + 19*x^2 + 159*x^3 + 1546*x^4 + 16517*x^5 + 188246*x^6 +...
where the g.f. satisfies:
(A(x) + x*A'(x)) / (A(x) - x*A(x)^2) = 1 + 2^2*x + 6^2*x^2 + 20^2*x^3 + 70^2*x^4 + 252^2*x^5 +...+ A000984(n)^2*x^n +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A232606, A000984, A002894.

{a(n)=local(CB2=sum(k=0,n,binomial(2*k,k)^2*x^k)+x*O(x^n), A=1+x*O(x^n));
for(i=1,n,A = 1 + intformal( (CB2-1)*A/x - CB2*A^2));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

G.f.: A(x) = (1/x)*Series_Reversion(x/F(x)) where F(x) = A(x/F(x)) is the g.f. of A232606.

Limit n->infinity a(n)^(1/n) = 16. - Vaclav Kotesovec, Jul 05 2014

A264960 Half-convolution of the central binomial coefficients A000984 with itself.

Original entry on oeis.org

1, 2, 10, 32, 146, 512, 2248, 8192, 35218, 131072, 556040, 2097152, 8815496, 33554432, 140107040, 536870912, 2230302098, 8589934592, 35541690568, 137438953472, 566823203656, 2199023255552, 9044910175520, 35184372088832, 144393718191496

Offset: 0

Peter Bala, Nov 29 2015

The half-convolution of a sequence {s(n)}n>=0 with itself is defined by r(n) := Sum_{k = 0..floor(n/2)} s(k)*s(n-k). See A201204.

Muniru A Asiru, Table of n, a(n) for n = 0..1520

Cf. A002894, A013776, A112830.

List([0..24],n->Sum([0..Int(n/2)],k->Binomial(2*k,k)*Binomial(2*n-2*k,n-k))); # Muniru A Asiru, Nov 25 2018

[(&+[Binomial(2*k,k)*Binomial(2*n-2*k, n-k): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Nov 26 2018

A264960:= n-> add(binomial(2*k,k)*binomial(2*n - 2*k, n - k),k = 0..floor(n/2)):
seq(A264960(n),n = 0..24);

a[n_] := Sum[Binomial[2k, k]*Binomial[2n - 2k, n - k], {k, 0, Floor[n/2]}]; Array[a, 30, 0] (* Amiram Eldar, Nov 25 2018 *)

a(n) = sum(k = 0, n\2, binomial(2*k,k)*binomial(2*n - 2*k, n - k)); \\ Michel Marcus, Nov 30 2015

[sum(binomial(2*k,k)*binomial(2*n-2*k, n-k) for k in (0..floor(n/2))) for n in range(30)] # G. C. Greubel, Nov 26 2018

a(n) = Sum_{k = 0..floor(n/2)} binomial(2*k,k)*binomial(2*n - 2*k, n - k).
a(2*n + 1) = 2^(4*n + 1) = A013776(n).
a(2*n) = (1/2)*(binomial(2*n,n)^2 + 16^n) = A112830(2*n,n).
O.g.f.: (1/2)*( 2/Pi*EllipticK(4*x) + 1/(1 - 4*x) ).
E.g.f.: (1/2)*( cosh(4*x) + sinh(4*x) + (BesselI(0,2*x))^2 ).
D-finite with recurrence: - (2*n-3)*n^2*a(n) + 4*(2*n-1)*(n-1)^2*a(n-1) + 16*(2*n-3)*(n-1)^2*a(n-2) - 64*(2*n-1)*(n-2)^2*a(n-3) = 0. - Georg Fischer, Nov 25 2022

A268150 A double binomial sum involving absolute values.

Original entry on oeis.org

0, 8, 2496, 177120, 7616000, 255780000, 7410154752, 194544814464, 4760448675840, 110493063252000, 2461297261280000, 53051182041906048, 1113060644163127296, 22833886572836393600, 459594580755139200000, 9100826722891800000000, 177680489488222659379200, 3426237501864596491802400

Offset: 0

Richard P. Brent, Jan 27 2016

A fast algorithm follows from Lemma 1 of Brent et al. article.

Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, Helmut Prodinger, Some binomial sums involving absolute values, arXiv:1411.1477v2 [math.CO], 2016.

Cf. A000984, A002894, A166337, A254408, A268148.

A268150 := proc(n)
    add( add( binomial(2*n,n+k)*binomial(2*n,n+l)*abs(k^2-l^2)^3,l=-n..n),k=-n..n) ;
end proc:
seq(A268150(n),n=0..10) ; # R. J. Mathar, Feb 27 2023

a(n) = sum(k=-n,n, sum(l=-n,n, binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k^2 - l^2)^3));

a(n) = Sum_{k=-n..n} (Sum_{l=-n..n} binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k^2 - l^2)^3).

Conjecture D-finite with recurrence -(4621*n-8921)*(n-1)^2*a(n) +4*(148256*n^3 -1055204*n^2 +2794799*n -2529792)*a(n-1) -64*(32443*n- 32400)*(2*n-3)*(2*n-5)*a(n-2)=0. - R. J. Mathar, Feb 27 2023

A268152 A double binomial sum involving absolute values.

Original entry on oeis.org

0, 8, 8832, 1228800, 79364096, 3562536960, 129276837888, 4079413624832, 116608362086400, 3096396542509056, 77661255048888320, 1861218099127123968, 42980384518787039232, 962362945373732864000, 20993511648589057622016, 447858123072052742062080, 9371462498278516088373248

Offset: 0

Richard P. Brent, Jan 27 2016

A fast algorithm follows from Theorem 5 of Brent et al. article.

Colin Barker, Table of n, a(n) for n = 0..800
Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, Helmut Prodinger, Some binomial sums involving absolute values, arXiv:1411.1477v2 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (80,-2560,40960,-327680,1048576).

Cf. A000984, A002894, A166337, A254408, A268148, A268150.

a(n) = sum(k=-n,n,sum(l=-n,n, binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k^2 - l^2)^4));

concat(0, Vec(8*x*(1+1024*x+67840*x^2+417792*x^3)/(1-16*x)^5 + O(x^20))) \\ Colin Barker, Feb 11 2016

a(n) = Sum_{k=-n..n} (Sum_{l=-n..n} binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k^2 - l^2)^4).
From Colin Barker, Feb 11 2016: (Start)
a(n) = 4^(2*n-1)*n*(36*n^3-84*n^2+67*n-17).
a(n) = 80*a(n-1)-2560*a(n-2)+40960*a(n-3)-327680*a(n-4)+1048576*a(n-5) for n>4.
G.f.: 8*x*(1+1024*x+67840*x^2+417792*x^3) / (1-16*x)^5.
(End)

A295870 a(n) = binomial(3n,n)*CQC(n), where CQC(n) = A005721(n) = A005190(2n) is a central quadrinomial coefficient.

Original entry on oeis.org

1, 12, 660, 48720, 4005540, 349260912, 31626298704, 2940502593600, 278788387440420, 26831860080682800, 2613367831568654160, 257012469788428710720, 25479526081439438845200, 2543092744417831625342400, 255292245777771431285140800, 25755871314484468746363582720

Offset: 0

Bradley Klee, Feb 23 2018

nonn

Compare with EllipticK A002894 and the Ramanujan period-energy functions A113424, A006480, A000897. The series expansion "T(x) = 2*Pi*Sum_{n>=0} a_n*x^n" determines the real period T of elliptic curves in the family "x=p^2+q^2-4*(q^2-p^2)*q, 0 < x < 1/108". This sequence serves as a counterexample to the naive idea that elliptic integrals will always evaluate to a hypergeometric function such as 2F1(a,b;c;x).
A300058 is the complex period-energy function, after scaling energy and time dimensions such that all a(n) are integers and a(0)=1. The Picard-Fuchs equation is "(12-288*x+9216*x^2)*T(x) + (-1+232*x-8160*x^2+82944*x^3)*T'(x) + (-x+164*x^2-6432*x^3+41472*x^4)*T''(x)".
Although the sequence is not generated by a hypergeometric function, it can be formulated in terms of Hypergeometric numbers, specifically the binomial coefficients. Then Zeilberger's algorithm outputs a second order recurrence with polynomial coefficients.
The contour plot is nice to look at, with reflection symmetry, three critical points, and two separatrices dividing the phase plane into eight distinct regions.
Hyperbolic Critical points are located at (q,p) locations (1/6,0) and (-1/4,sqrt(5)/4) and (-1/4,-sqrt(5)/4). Is it possible to use chord-and-tangent addition rules to produce an exponentially-convergent Diophantine approximation to sqrt(5) that moves along the upper separatrix x=1/8?
Does there exist a period-preserving transformation that takes any one of the curves with 0 < x < 1/108 into a particular Weierstrass curve from the L-function and Modular Forms Database?

D. Husemöller, Elliptic Curves, 2nd ed., New York: Springer, 2004.
J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., New York: Springer, 2009.

J. Cremona, Elliptic Curves over Q, LMFDB 2017.
B. Klee, The Virtues of X_{n+1} = (4+3*X_{n})/(3+2*X_{n}), seqfans mailing list, 2017.
B. Klee, Geometric G.F. for Ramanujan Periods, seqfans mailing list, 2017.
Brad Klee, Deriving Hypergeometric Picard-Fuchs Equations, Wolfram Demonstrations Project (2018).
Bradley Klee, Phase Plane Geometry.
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22.
P. Paule and M. Schorn, FastZeil: the Paule/Schorn implementation of Gosper's and Zeilberger's algorithm, RISC 2017; Local copy, pdf file only, no active links
D. Zeilberger, The Method of Creative Telescoping, Journal of Symbolic Computation, 11.3 (1991), 195-204.

Cf. A000897, A002894, A006480, A113424.
Factors: A005190, A005809, A005721.
Complex Period: A300058.

b[NN_]:=Total/@Table[((-1)^k)*Binomial[3*n,n]*Binomial[2*n,k]*Binomial[5*n-4*k-1,3*n-4*k],{n,0,NN},{k,0,Floor[3*n/4]}];
c1=8*(-30+201*n-319*n^2+145*n^3);c2=-8640*(n-5/3)*(n-4/3)*(n-1/5);c3=10*(n-6/5)*n^2;a[0]=1;a[1]=12;a[n0_]:=ReplaceAll[(c1/c3)*a[n0-1]+(c2/c3)*a[n0-2],{n->n0}];
({#,SameQ[a/@Range[0, 15],#]}&@b[15])[[1]]

a(n) = A005809(n)*A005721(n).
a(n) = Sum_{k=0..floor(3n/4)} ((-1)^k)*binomial(3*n,n)*binomial(2 *n, k)*binomial(5*n - 4*k - 1, 3*n - 4*k).
c1 = 8 *(-30 + 201*n - 319*n^2 + 145*n^3); c2 = -8640*(n - 5/3)*(n - 4/3)*(n - 1/5); c3 = 10*(n - 6/5)*n^2; a(0)=1; a(1)=12; a(n) = (c1/c3)*a(n-1) + (c2/c3)*a(n-2).

A300058 a(n) = binomial(3n,n)/(2Pi)Integral_{x=0..2Pi} (12cos^2(x)sin(x) + 20sin^3(x))^(2n) dx.

Original entry on oeis.org

1, 492, 707220, 1298204880, 2654173160100, 5765723073622512, 13021894087331233104, 30217387890886676251200, 71532102917478013611243300, 171944976047709681477985038000, 418347201888204996027087975427920

Offset: 0

Bradley Klee, Feb 23 2018

nonn

Compare with A295870. The series expansion "T(x)=2*Pi*sqrt(3/5)*Sum_{n>=0} a_n*(x/25)^n" determines the period T of anharmonic oscillation along a contour of the Hamiltonian energy surface "x=2H=(5/3)*p^2+q^2+4*(p^2+q^2)*q,0

The period-energy function T(x) satisfies the Picard-Fuchs equation "(2460+28512*x+2239488*x^2)*T(x)-(125-24840*x-1423008*x^2-20155392*x^3)*T'(x)+(-125*x+1620*x^2+1189728*x^3+10077696 x^4)*T''(x)", also the P.F.Eq. of A295870 under transformation x->x'=1/108-x.

A300057 has a similar definition to A005721, with a couple of extra integers appearing in the integrand. This makes a nice analogy between real and complex periods A295870, A300058. Second-order recurrences with polynomial coefficients define both sequences.

Bradley Klee, Phase Plane Geometry.
Brad Klee, Deriving Hypergeometric Picard-Fuchs Equations, Wolfram Demonstrations Project (2018).
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22.

Cf. A002894, A113424, A006480, A000897. Factors: A005809, A300057. Real Period: A295870.

a := n -> 36^n*(3*n)!/n!^3*hypergeom([-2*n, n+1/2], [n+1], -2/3):
seq(simplify(a(n)), n=0..10); # Peter Luschny, Apr 19 2018

c1=12*(-230+2259*n-3933*n^2+1863*n^3);c2=5248800*(n-5/3)*(n-4/3)*(n-1/9);c3=9*n^2*(n-10/9);a[0]=1;a[1]=492;a[n0_]:=ReplaceAll[(c1/c3)*a[n0-1]+(c2/c3)*a[n0-2],n->n0];
b[NN_]:=Expand[Total[Flatten[#]]&/@Table[Binomial[3*n,n]*Binomial[2*n,k2]*Binomial[2*n,k1]*Binomial[2*n,3*n-k1-k2]*((4+Sqrt[15])^(2*n-k1))*((4-Sqrt[15])^(2*n-k2)),{n,0,NN},{k1,0,2*n},{k2,0,2*n}]]; ({#,SameQ[#,a/@Range[0,10]]}&@b[10])[[1]]
Table[Binomial[3*n, n] * SeriesCoefficient[(1 + 9*z + 9*z^2 + z^3)^(2*n), {z, 0, 3*n}], {n, 0, 15}] (* Vaclav Kotesovec, Apr 18 2018 *)

a(n) = A005809(n)*A300057(n).
a(n) = Sum_{k1=0..2n} Sum_{k2=0..2n} binomial(3*n,n)*binomial(2*n,k2)*binomial(2*n,k1)*binomial(2*n,3*n-k1-k2)*((4+sqrt(15))^(2*n-k1))*((4-sqrt(15))^(2*n-k2))
a(0) = 1; a(1) = 492; a(n):=(c1/c3)*a(n-1)+(c2/c3)*a(n-2); with
  c1 = 12*(-230+2259*n-3933*n^2+1863*n^3);
  c2 = 5248800*(n-5/3)*(n-4/3)*(n-1/9);
  c3 = 9*n^2*(n-10/9);
a(n) ~ 2^(2*n - 1) * 3^(3*n - 1/2) * 5^(2*n + 1/2) / (Pi*n). - Vaclav Kotesovec, Apr 18 2018

A303503 a(n) = (2n)! [x^(2n)] BesselI(0,2x)^n.

Original entry on oeis.org

1, 2, 36, 1860, 190120, 32232060, 8175770064, 2898980908824, 1369263687414480, 830988068906518380, 630109741730668410640, 583773362067938664133512, 648851848280206013365243776, 852146184628067383511375555000, 1305460597778526044143501996708800, 2307324514460203126471248458864413200

Offset: 0

Ilya Gutkovskiy, Apr 25 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..212

Main diagonal of A287318.

Cf. A000984, A002894, A002896, A033935, A039699, A287317, A361297.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1)*binomial(n, j)^2, j=0..n))
    end:
a:= n-> (2*n)!*b(n$2)/n!^2:
seq(a(n), n=0..17);  # Alois P. Heinz, Jan 29 2023

Table[(2 n)! SeriesCoefficient[BesselI[0, 2 x]^n, {x, 0, 2 n}], {n, 0, 15}]

a(n) = A287318(n,n).
a(n) ~ c * d^n * n^(2*n), where c = 1.72802011936236389522137050964080... and d = 1.1381284656425793765251319541847869000364101065484286935... - Vaclav Kotesovec, Apr 26 2018
a(n) = A000984(n)*A033935(n). - Alois P. Heinz, Jan 30 2023

A307468 Cogrowth sequence for the Heisenberg group.

Original entry on oeis.org

1, 4, 28, 232, 2156, 21944, 240280, 2787320, 33820044, 424925872, 5486681368, 72398776344, 972270849512, 13247921422480, 182729003683352, 2546778437385032, 35816909974343308, 507700854900783784, 7246857513425470288, 104083322583897779656

Offset: 0

Igor Pak, Apr 09 2019

This is the number of words of length 2n in the letters x,x^{-1},y,y^{-1} that equal the identity of the Heisenberg group H=.

Also, this is the number of closed walks of length 2n on the square lattice enclosing algebraic area 0 [Béguin et al.]. - Andrey Zabolotskiy, Sep 15 2021

			For n=1 the a(1)=4 words are x^{-1}x, xx^{-1}, y^{-1}y, yy^{-1}.

Jay Pantone, Table of n, a(n) for n = 0..200
Cédric Béguin, Alain Valette and Andrzej Zuk, On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper's operator, Journal of Geometry and Physics, 21 (1997), 337-356.
D. Lind and K, Schmidt, A survey of algebraic actions of the discrete Heisenberg group, arXiv:1502.06243 [math.DS], 2015; Russian Mathematical Surveys, 70:4 (2015), 77-142.

Related cogrowth sequences: Z A000984, Z^2 A002894, Z^3 A002896, (Z/kZ)^*2 for k = 2..5: A126869, A047098, A107026, A304979, Richard Thompson's group F A246877. The cogrowth sequences for BS(N,N) for N = 2..10 are A229644, A229645, A229646, A229647, A229648, A229649, A229650, A229651, A229652.

Cf. A352838, A178106.

Asymptotics: a(n) ~ (1/2) * 16^n * n^(-2).

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cp's OEIS Frontend

A364510 a(n) = binomial(4*n, n)^2.

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A071801 a(n) = binomial(2n, n) - binomial(n, floor(n/2))^2.

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A232607 G.f. A(x) satisfies: (A(x) + x*A'(x)) / (A(x) - x*A(x)^2) = Sum_{n>=0} binomial(2*n,n)^2*x^n.

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A264960 Half-convolution of the central binomial coefficients A000984 with itself.

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A268150 A double binomial sum involving absolute values.

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Maple

PARI

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A268152 A double binomial sum involving absolute values.

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PARI

PARI

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A295870 a(n) = binomial(3n,n)*CQC(n), where CQC(n) = A005721(n) = A005190(2n) is a central quadrinomial coefficient.

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A232607 G.f. A(x) satisfies: (A(x) + xA'(x)) / (A(x) - xA(x)^2) = Sum_{n>=0} binomial(2n,n)^2x^n.

A300058 a(n) = binomial(3n,n)/(2Pi)Integral_{x=0..2Pi} (12cos^2(x)sin(x) + 20sin^3(x))^(2n) dx.

A303503 a(n) = (2n)! [x^(2n)] BesselI(0,2x)^n.