A002897 -id:A002897 - cp's OEIS frontend

A186416 a(n) = binomial(2n,n)^4/(n+1)^3.

1, 2, 48, 2500, 192080, 18670176, 2125170432, 270968717448, 37634544090000, 5588044012339360, 875419364366134016, 143310129125665075392, 24338673855047938317568, 4264316875814353400000000, 767401591466550107174400000, 141345980472409642279275210000, 26569505644587874058090478570000

Offset: 0

Emanuele Munarini, Feb 21 2011

Cf. A000108, A002894, A000888, A002897, A186414, A186415.

A186416 := proc(n) binomial(2*n,n)^4/(n+1)^3 ; end proc: # R. J. Mathar, Feb 23 2011

Table[Binomial[2n,n]^4/(n+1)^3,{n,0,40}]

makelist(binomial(2*n,n)^4/(n+1)^3,n,0,40);

G.f.: 4F3(1/2,1/2,1/2,1/2;2,2,2;256*x), where nFm(...;..;.) denotes a generalized hypergeometric series.

a(n) = (A000108(n))^3*A000984(n). - R. J. Mathar, Feb 23 2011

A186284 Self-convolution square equals A127776.

Original entry on oeis.org

1, 2, 48, 1704, 71490, 3291780, 160844160, 8189867280, 429832053840, 23088359467040, 1263134996327680, 70138971602098560, 3942799810867610280, 223942062435751452240, 12831882367225056387840, 740872398293620831990080

Offset: 0

Paul D. Hanna, Feb 16 2011

nonn

			G.f.: A(x) = 1 + 2*x + 48*x^2 + 1704*x^3 + 71490*x^4 + 3291780*x^5 +...
Related expansions.
The g.f. of A127776 equals A(x)^2:
A(x)^2 = 1 + 4*x + 100*x^2 + 3600*x^3 + 152100*x^4 + 7033104*x^5 +...+ A004981(n)^2*x^n +...
The g.f. of A002897 equals A(x)^4:
A(x)^4 = 1 + 8*x + 216*x^2 + 8000*x^3 + 343000*x^4 + 16003008*x^5 +...+ A000984(n)^3*x^n +...
The g.f. of A004981 begins:
1/(1-8*x)^(1/4) = 1 + 2*x + 10*x^2 + 60*x^3 + 390*x^4 + 2652*x^5 +...
where A004981(n) = (2^n/n!)*Product_{k=0..n-1} (4k + 1).
The g.f. of A000984 begins:
1/(1-4*x)^(1/2) = 1 + 2*x + 6*x^2 + 20*x^3 + 70*x^4 + 252*x^5 +...
where A000984(n) = (2n)!/(n!)^2 forms the central binomial coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A004981, A000984, A127776, A002897.

nmax = 20; CoefficientList[Series[Sqrt[Hypergeometric2F1[ 1/4, 1/4, 1, 64*x]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2018 *)

{a(n)=local(A004981=1/(1-8*x+x*O(x^n))^(1/4),A=sum(m=0,n,polcoeff(A004981,m)^2*x^m+x*O(x^n))^(1/2));polcoeff(A,n)}

{a(n)=local(A000984=1/(1-4*x+x*O(x^n))^(1/2),A=sum(m=0,n,polcoeff(A000984,m)^3*x^m+x*O(x^n))^(1/4));polcoeff(A,n)}

Self-convolution 4th power equals A002897.
G.f.: sqrt( K(k)/(Pi/2) ) in powers of (kk'/4)^2, where K(k) is complete elliptic integral of first kind evaluated at modulus k. [From a formula by Michael Somos in A002897]
G.f.: sqrt( 1/AGM(1, (1-16x)^(1/2)) ) in powers of x(1-16x) where AGM() is the arithmetic-geometric mean. [From a formula by Michael Somos in A004981]
a(n) ~ Pi^(3/4) * 2^(6*n - 1/2) / (Gamma(1/4)^3 * n^(3/2)). - Vaclav Kotesovec, Apr 10 2018

A224735 G.f.: exp( Sum_{n>=1} binomial(2n,n)^3 x^n/n ).

Original entry on oeis.org

1, 8, 140, 3616, 116542, 4316080, 175593800, 7640774080, 349626142909, 16632958651688, 816163494236860, 41069537125459360, 2110206360805542510, 110346590629125981872, 5857345961837113457864, 314962180518584299711424, 17128125582951726423704502, 940726748732537798295599280

Offset: 0

Paul D. Hanna, Apr 16 2013

nonn

			G.f.: A(x) = 1 + 8*x + 140*x^2 + 3616*x^3 + 116542*x^4 + 4316080*x^5 +...
where
log(A(x)) = 2^3*x + 6^3*x^2/2 + 20^3*x^3/3 + 70^3*x^4/4 + 252^3*x^5/5 + 924^3*x^6/6 + 3432^3*x^7/7 + 12870^3*x^8/8 +...+ A000984(n)^3*x^n/n +...

Cf. A224732, A224734, A224736, A002897, A000984.

CoefficientList[Series[Exp[8*x*HypergeometricPFQ[{1, 1, 3/2, 3/2, 3/2}, {2, 2, 2, 2}, 64*x]], {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 27 2025 *)

{a(n)=polcoeff(exp(sum(k=1,n,binomial(2*k,k)^3*x^k/k)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

Logarithmic derivative yields A002897.

A268554 Diagonal of the rational function 1/((1 - w - u v) * (1 - x y - x z - y z)).

Original entry on oeis.org

1, 36, 6300, 1552320, 445945500, 139815211536, 46384755633216, 16009450307136000, 5689533506261190300, 2067982222137781950000, 765185639177176836418800, 287266309673587605560908800, 109149488451384203661831720000

Offset: 0

N. J. A. Sloane, Feb 29 2016

Each second element (which is zero) is skipped. - R. J. Mathar, Mar 10 2016

Annihilating differential operator: (-x^2+432*x^4)*Dx^4 + (-5*x+4320*x^3)*Dx^3 + (-4+10644*x^2)*Dx^2 + 6012*x*Dx + 288.

Vaclav Kotesovec, Table of n, a(n) for n = 0..200
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.
R. Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A268545 - A268555.

A268554 := proc(n)
    1/(1-w-u*v)/(1-x*y-x*z-y*z) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,n) ;
    coeftayl(%,z=0,n) ;
    coeftayl(%,u=0,n) ;
    coeftayl(%,v=0,n) ;
    coeftayl(%,w=0,n) ;
end proc:
seq(A268554(2*n),n=0..40) ; # R. J. Mathar, Mar 10 2016

Table[(4*n)!*(3*n)!/((n!)^3*(2*n)!^2), {n, 0, 15}] (* Vaclav Kotesovec, Jul 01 2016 *)

my(x1='x1, x2='x2, x3='x3, y1='y1, y2='y2, y3='y3);
R = 1/((1 - y1 - y2*y3) * (1 - x1*x2 - x1*x3 - x2*x3));
diag(n, expr, var) = {
  my(a = vector(n));
  for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
  for (k = 1, n, a[k] = expr;
       for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
  return(a);
};
diag(11, R, [x1,x2,x3,y1,y2,y3]) \\ Gheorghe Coserea, Jun 30 2016

Conjecture: n^3*(2*n-1)*a(n) -6*(4*n-1)*(3*n-1)*(3*n-2)*(4*n-3)*a(n-1)=0. - R. J. Mathar, Mar 10 2016
From Vaclav Kotesovec, Jul 01 2016: (Start)
a(n) = (4*n)! * (3*n)! / ((n!)^3 * (2*n)!^2).
a(n) ~ 2^(4*n - 3/2) * 3^(3*n + 1/2) / (Pi^(3/2) * n^(3/2)).
(End)
0 = (-x^2+432*x^4)*y'''' + (-5*x+4320*x^3)*y''' + (-4+10644*x^2)*y'' + 6012*x*y' + 288*y, where y = 1 + 36*x^2 + 6300*x^4  + ... is the g.f. - Gheorghe Coserea, Jul 03 2016
From Peter Bala, Oct 16 2024: (Start)
a(n) = 4 * Sum_{k = 0..2*n-1} (-1)^(n+k) * binomial(2*n-1, k) * binomial(4*n+k-1, k) * A108625(2*n, 2*n-k) for n >= 1 (verified using the MulZeil procedure in Doron Zeilberger's MultiZeilberger package). Cf. A002897.
a(n) = binomial(4*n, 2*n)*binomial(3*n, n)*binomial(2*n, n).
The supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and positive integers n and k (apply Meštrović, Section 6, equation 39).
a(n) = [x^(2*n)] (1 + x)^(4*n) * [x^n] (1 + x)^(3*n) * [x^n] (1 + x)^(2*n) = [x^n] F(x)^(36*n), where F(x) = 1 + x + 52*x^2 + 6919*x^3 + 1266837*x^4 + 275133604*x^5 + 66468858333*x^6 + 17272069128056*x^7 + 4732687104502730*x^8 + 1350192483617697301*x^9 + 397617338885817524186*x^10 + ... appears to have integer coefficients (checked up to O(x^500)).
Let E(x) = exp(Sum_{n >= 1} (1/36) *a(n)*x^n/n). Then E(x) = 1 + x + 88*x^2 + 14461*x^3 + 3115089*x^4 + 781116715*x^5 + 215898182457*x^6 + 63857605571783*x^7 + 19853845202113934*x^8 + 6413541401057933731*x^9 + 2135530251738770328084*x^10 + ... appears to have integer coefficients (checked up to O(x^500)).
a(n) = 36 * [x^n] ( x/series_reversion(E(x)) )^n.
For integer r and positive integer s, define sequences {u(n) : n >= 0} and {v(n) : n >= 0} by setting u(n) = [x^(s*n)] F(x)^(r*n) and v(n) = [x^(s*n)] E(x)^(r*n). We conjecture that both u(n) and v(n) satisfy the above supercongruences. (End)

A380243 Number of rooted 2n-regular planar maps with 3 vertices.

Original entry on oeis.org

1, 54, 3000, 171500, 10001880, 591666768, 35371207872, 2131746903000, 129299660919000, 7883256659941520, 482689850761774656, 29661047546558142624, 1828220386252351000000, 112982297841774018000000, 6998159395715622920640000, 434337846995341921726638000, 27004842919501042631643927000

Offset: 1

Andrew Howroyd, Jan 22 2025

nonn

There are no odd valent regular planar maps with 3 vertices.

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row n=3 of A380241.

Cf. A000984, A002897, A380242.

PARI
```
a(n) = n*binomial(2*n-1, n)^3
```

a(n) = n*binomial(2*n-1, n)^3.

a(n) = n*A002897(n - 1).

A186418 a(n) = binomial(2*n,n)^4/(n + 1)^2.

Original entry on oeis.org

1, 4, 144, 10000, 960400, 112021056, 14876193024, 2167749739584, 338710896810000, 55880440123393600, 9629613008027474176, 1719721549507980904704, 316402760115623198128384, 59700436261400947600000000

Offset: 0

Emanuele Munarini, Feb 21 2011

Cf. A000108, A002894, A000888, A002897, A186414, A186415, A186416.

Table[Binomial[2n,n]^4/(n+1)^2,{n,0,40}]

makelist(binomial(2*n,n)^4/(n+1)^2,n,0,40);

G.f.: 4F3({1/2,1/2,1/2,1/2},{1,2,2},256x), where 4F3 is a hypergeometric series.

A367177 Triangle read by rows, T(n, k) = [x^k] hypergeom([1/2, -n, -n], [1, 1], 4*x).

Original entry on oeis.org

1, 1, 2, 1, 8, 6, 1, 18, 54, 20, 1, 32, 216, 320, 70, 1, 50, 600, 2000, 1750, 252, 1, 72, 1350, 8000, 15750, 9072, 924, 1, 98, 2646, 24500, 85750, 111132, 45276, 3432, 1, 128, 4704, 62720, 343000, 790272, 724416, 219648, 12870

Offset: 0

Peter Luschny, Nov 07 2023

			Triangle T(n, k) starts:
  [0] 1;
  [1] 1,   2;
  [2] 1,   8,    6;
  [3] 1,  18,   54,     20;
  [4] 1,  32,  216,    320,      70;
  [5] 1,  50,  600,   2000,    1750,     252;
  [6] 1,  72, 1350,   8000,   15750,    9072,     924;
  [7] 1,  98, 2646,  24500,   85750,  111132,   45276,    3432;
  [8] 1, 128, 4704,  62720,  343000,  790272,  724416,  219648,   12870;
  [9] 1, 162, 7776, 141120, 1111320, 4000752, 6519744, 4447872, 1042470, 48620;

Cf. A002893 (row sum), A002897 (central column), A000984 (main diagonal).

Cf. A367022, A367023, A387024.

p := n -> hypergeom([1/2, -n, -n], [1, 1], 4*x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

T(n, k) = binomial(n, k)^2 * binomial(2*k, k).

A186419 a(n) = binomial(2*n,n)^4/(n + 1).

Original entry on oeis.org

1, 8, 432, 40000, 4802000, 672126336, 104133351168, 17341997916672, 3048398071290000, 558804401233936000, 105925743088302215936, 20636658594095770856448, 4113235881503101575668992, 835806107659613266400000000, 172665358079973774114240000000

Offset: 0

Emanuele Munarini, Feb 21 2011

Cf. A000108, A002894, A000888, A002897, A186414, A186415, A186416, A186418.

Mathematica
```
Table[Binomial[2n,n]^4/(n+1),{n,0,40}]
```

makelist(binomial(2*n,n)^4/(n+1),n,0,12);

G.f.: 4F3({1/2,1/2,1/2,1/2},{1,1,2},256x), where 4F3 is a hypergeometric series.

A248600 G.f.: Sum_{n>=0} R_n(x+xy) x^(2n)y^n / (1-x-xy)^(4n+1) = Sum_{n>=0} Sum_{k=0..n} C(n,k)^4 * x^ny^k, where R_n(x+xy) equals the n-th row polynomial R_n(z) = Sum_{k=0..2n} T(n,k)z^k at z = x+x*y.

Original entry on oeis.org

1, 14, 8, 2, 786, 1056, 576, 96, 6, 61340, 131760, 117900, 48320, 9540, 720, 20, 5562130, 16481920, 20917120, 13847680, 5118400, 1025920, 105280, 4480, 70, 549676764, 2079579600, 3444581700, 3165926400, 1755532800, 598123008, 123656400, 14716800, 926100, 25200, 252, 57440496036

Offset: 0

Paul D. Hanna, Oct 11 2014

			Triangle begins:
[1],
[14, 8, 2],
[786, 1056, 576, 96, 6],
[61340, 131760, 117900, 48320, 9540, 720, 20],
[5562130, 16481920, 20917120, 13847680, 5118400, 1025920, 105280, 4480, 70],
[549676764, 2079579600, 3444581700, 3165926400, 1755532800, 598123008, 123656400, 14716800, 926100, 25200, 252],
[57440496036, 264565490112, 542687590368, 640299696960, 477284304420, 233110386432, 75243589344, 15835792896, 2103157980, 165802560, 7051968, 133056, 924],
[6242164112184, 33895475918304, 83073660613944, 119912994225024, 112698387745944, 72172565713248, 32111980788888, 9951304416768, 2124873478728, 305035899168, 28270554312, 1584815232, 48600552, 672672, 3432],
[698300344311570, 4368053451041280, 12465205610457600, 21305587665922560, 24216302627637120, 19255941998092800, 10989839486545920, 4550117424652800, 1366687981264320, 295074717949440, 44954858108160, 4691645038080, 320878958400, 13445752320, 311351040, 3294720, 12870], ...
where this triangle forms the coefficients in the series
B(x,y) = 1/(1-x-x*y) +
(14 + 8*(x+x*y) + 2*(x+x*y)^2) * x^2*y/(1-x-x*y)^5 +
(786 + 1056*(x+x*y) + 576*(x+x*y)^2 + 96*(x+x*y)^3 + 6*(x+x*y)^4) * x^4*y^2/(1-x-x*y)^9 +
(61340 + 131760*(x+x*y) + 117900*(x+x*y)^2 + 48320*(x+x*y)^3 + 9540*(x+x*y)^4 + 720*(x+x*y)^5 + 20*(x+x*y)^6) * x^6*y^3/(1-x-x*y)^13 +...
such that the sum may be expressed using binomial coefficients C(n,k)^4 like so:
B(x,y) =  1 +
x*(1 + y) +
x^2*(1 + 2^4*y + y^2) +
x^3*(1 + 3^4*y + 3^4*y^2 + y^3) +
x^4*(1 + 4^4*y + 6^4*y^2 + 4^4*y^3 + y^4) +
x^5*(1 + 5^4*y + 10^4*y^2 + 10^4*y^3 + 5^4*y^4 + y^5) +
x^6*(1 + 6^4*y + 15^4*y^2 + 20^4*y^3 + 15^4*y^4 + 6^4*y^5 + y^6) +...
The central terms of the rows begin:
[1, 8, 576, 48320, 5118400, 598123008, 75243589344, 9951304416768, 1366687981264320, ...].

Cf. A050983, A000984, A008977, A002897, A187056, A248706.

Leftmost border equals A050983, de Bruijn's S(4,n):
T(n,0) = Sum_{k=0..2*n} (-1)^(n+k) * C(2*n,k)^4.
Rightmost border equals A000984, the central binomial coefficients:
T(n,2*n) = Sum_{k=0..2*n} (-1)^(n+k)* C(2*n,k)^2 = (2*n)!/(n!)^2.
Row sums equal A008977(n) = (4*n)!/(n!)^4.
Sum_{k=0..n} (-1)^k * T(n,k) = A002897(n) = C(2*n,n)^3.

A307618 A Calabi-Yau period integral: a(n) = C(4n,2n)C(2n,n)^3.

Original entry on oeis.org

1, 48, 15120, 7392000, 4414410000, 2956651746048, 2133278987583744, 1621682968820428800, 1281351259836532170000, 1043032815185819858400000, 869343653096068540955685120, 738637974389826550020188712960, 637665137404661719206664998969600

Offset: 0

Bradley Klee, Jun 04 2019

nonn

Entry number six in the "Big Table" of Almkvist et al. (see links). The period T(x) = Sum_{n>=0} a(n)*x^(2*n) is also the first x-derivative of the 6-volume associated to the algebraic variety V6 = P1 & P2 & P3, with P1 : X1^2 + Y1^2 = X2^2 + Y2^2, P2 : X2^2 + Y2^2 = X3^2 + Y3^2, P3 : x=(X1^2 + X2^2 + X3^2 + Y1^2 + Y2^2 + Y3^2)^3*(1 - X1*X2*X3*Y1*Y2*Y3). The small x limit reduces V6 to a 6-ball with 6-volume proportional to x. Similar constructions are known to exist for a few other geometries on Almkvist's list, most notably #3: A186420, and #16: A039699.

G. Almkvist et al., Tables of Calabi-Yau Equations, arXiv:math/0507430 [math.AG], 2005-2010, p. 10.
G. Almkvist et al., Raw Calabi-Yau Period Data, Johannes Gutenberg Universität Mainz, 2019, case 1.6.

Hadamard Factors: A000984, A002894, A002897, A001448, A000897, A008977.

Calabi-Yau Periods: A008978, A186420, A268553, A039699.

Binomial[4*#,2*#]*Binomial[2*#,#]^3&/@Range[0,10]

G.f.: 4F3({1/4, 3/4, 1/2, 1/2}, {1, 1, 1}, 1024*x).
Define the period integral:
dt(x) = dz1*dz2*dz3/sqrt(1-32*x*cos(z1)*cos(z2)*cos(z3)).
T(x)=1/(2*Pi)^3*Integral_{0..2*Pi,0..2*Pi,0..2*Pi} dt(x),
the Picard-Fuchs coefficients:(c0,c1,c2,c3,c4)=
(768*x, 14592*x^2-1, x*(25344*x^2-7), 2*x^2*(5120*x^2-3), x^3*(32*x-1)*(32*x+1)),
and the certificate function:
G(z1,z2,z3)=(16*sin(z1)*(
       48*x*cos(z1)
     + cos(z2)*cos(z3)
     + 48*x*cos(z1)*(cos(z3)^2 + cos(z2)^2)
     + 2304*x^2*cos(z1)^2*cos(z2)*cos(z3)
     + 80*x*cos(z1)*cos(z2)^2*cos(z3)^2
     + 384*x^2*cos(z1)^2*(cos(z2)*cos(z3)^3 + cos(z2)^3*cos(z3))
     + 256*x^2*cos(z1)^2*cos(z2)^3*cos(z3)^3)
    )/(3*(1 - 32*x*cos(z1)*cos(z2)*cos(z3))^(7/2)),
Then: 0 = Sum_{n=0..4}cn*d^n/dx^n dt(x) + d/dz1 G(z1,z2,z3) + d/dz2 G(z2,z3,z1) + d/dz3 G(z3,z1,z2), thus: 0 = Sum_{n=0..4} cn*d^n/dx^n T(x).
Furthermore, let (a1,a2,a3)=(c1,c2,c3)/c0, then also: 0 = (1/2)*a2*a3 - (1/8)*a3^3 + d/dx(a2) - (3/4)*a3*d/dx(a3) - (1/2)*d^2/dx^2(a3) - a1.
D-finite with recurrence: n^4*a(n) -16*(4*n-1)*(4*n-3)*(-1+2*n)^2*a(n-1)=0. - R. J. Mathar, Jan 27 2020

cp's OEIS Frontend

A186416 a(n) = binomial(2n,n)^4/(n+1)^3.

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A186284 Self-convolution square equals A127776.

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A224735 G.f.: exp( Sum_{n>=1} binomial(2*n,n)^3 * x^n/n ).

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A268554 Diagonal of the rational function 1/((1 - w - u v) * (1 - x y - x z - y z)).

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A380243 Number of rooted 2n-regular planar maps with 3 vertices.

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A186418 a(n) = binomial(2*n,n)^4/(n + 1)^2.

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Maxima

Formula

A367177 Triangle read by rows, T(n, k) = [x^k] hypergeom([1/2, -n, -n], [1, 1], 4*x).

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Maple

Formula

A186419 a(n) = binomial(2*n,n)^4/(n + 1).

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Mathematica

A224735 G.f.: exp( Sum_{n>=1} binomial(2n,n)^3 x^n/n ).

A248600 G.f.: Sum_{n>=0} R_n(x+xy) x^(2n)y^n / (1-x-xy)^(4n+1) = Sum_{n>=0} Sum_{k=0..n} C(n,k)^4 * x^ny^k, where R_n(x+xy) equals the n-th row polynomial R_n(z) = Sum_{k=0..2n} T(n,k)z^k at z = x+x*y.

A307618 A Calabi-Yau period integral: a(n) = C(4n,2n)C(2n,n)^3.