A001448 -id:A001448 - cp's OEIS frontend

A337350 a(n) is the number of lattice paths from (0,0) to (2n,2n) using only the steps (1,0) and (0,1) and which do not touch any other points of the form (2k,2k).

1, 6, 34, 300, 3146, 36244, 443156, 5646040, 74137050, 996217860, 13633173180, 189347631720, 2662142601924, 37815138677960, 541882155414376, 7823955368697776, 113712609033955834, 1662288563798703204, 24424940365489658540, 360537080085493670856

Offset: 0

Lucas A. Brown, Aug 24 2020

The terms of this sequence may be computed via a determinant; see Lemma 10.7.2 of the Krattenthaler reference for details.

Christian Krattenthaler, Lattice path enumeration, In: Handbook of Enumerative Combinatorics. Edited by Miklos Bona. CRC Press, 2015, pages 589-678.
R. J. Mathar, The Eggenberger-Polya urn process: Probabilities of revisited ball ratios, vixra:2502.0097 (2025) Table 4
Quy Nhan, A finite sum of binomial coefficients, MathStackExchange, 2025-06-21.

Cf. A000108, A001448, A337291, A337292, A337351, A337352.

seq(n)={Vec(2 - 1/(O(x*x^n) + sum(k=0, n, binomial(4*k,2*k)*x^k)))} \\ Andrew Howroyd, Aug 25 2020

G.f.: 2 - 1 / (Sum_{n>=0} binomial(4*n,2*n) * x^n).
a(n) = binomial(4*n,2*n) * (8*n+1) / (8*n^2 + 2*n - 1) for n >= 1.  For proof, see the Quy Nhan link.
D-finite with recurrence n*(2*n+1)*(8*n-7)*a(n) -2*(4*n-5)*(4*n-3)*(8*n+1)*a(n-1)=0. - R. J. Mathar, Jan 26 2025
From Lucas A. Brown, Jul 13 2025: (Start)
G.f.: 2 - sqrt(2-32*x) / sqrt(1+sqrt(1-16*x)).
a(n) = A000108(2*n) + 4 * A000108(2*n-1). (End)

A337396 Expansion of sqrt((1-8x+sqrt(1+64x^2)) / (2 * (1+64*x^2))).

Original entry on oeis.org

1, -2, -26, 76, 1222, -3772, -64676, 203992, 3607622, -11510636, -207302156, 666187432, 12142184476, -39211413464, -720760216328, 2335857124016, 43208062233158, -140406756766796, -2609918906614652, 8498967890177416, 158596941629422132, -517334728427373704, -9684521991498517112

Offset: 0

Seiichi Manyama, Aug 26 2020

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=4 of A337419.

Cf. A001448, A193619, A337397.

a[n_] := Sum[(-4)^(n - k) * Binomial[2*k, k] * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 23, 0] (* Amiram Eldar, Aug 26 2020 *)

N=40; x='x+O('x^N); Vec(sqrt((1-8*x+sqrt(1+64*x^2))/(2*(1+64*x^2))))

{a(n) = sum(k=0, n, (-4)^(n-k)*binomial(2*k, k)*binomial(2*n, 2*k))}

a(n) = Sum_{k=0..n} (-4)^(n-k) * binomial(2*k,k) * binomial(2*n,2*k).

a(0) = 1, a(1) = -2 and n * (2*n-1) * (4*n-5) * a(n) = (4*n-3) * 2 * a(n-1) - 64 * (n-1) * (2*n-3) * (4*n-1) * a(n-2) for n > 1. - Seiichi Manyama, Aug 28 2020

A370100 a(n) = Sum_{k=0..n} binomial(4n,k) binomial(2*n-k-1,n-k).

Original entry on oeis.org

1, 5, 47, 500, 5615, 65005, 767396, 9183144, 110995695, 1351922495, 16566597047, 204010570296, 2522556212228, 31298015910140, 389458822888280, 4858487926378000, 60742838865326319, 760901358321592611, 9547848458062427405, 119990407515367475700

Offset: 0

Seiichi Manyama, Feb 10 2024

Cf. A001448, A352373, A370101, A370102.

Cf. A365754.

Table[Sum[Binomial[4*n, k]*Binomial[2*n - k - 1, n - k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 12 2024 *)

a(n) = sum(k=0, n, binomial(4*n, k)*binomial(2*n-k-1, n-k));

a(n) = [x^n] ( (1+x)^4/(1-x) )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1-x)/(1+x)^4 ). See A365754.
From Peter Bala, Jun 08 2024: (Start)
2*n*(n - 1)*(2*n - 1)*(51*n^2 - 144*n + 100)*a(n) = -(n - 1)*(5457*n^4 - 20865*n^3 + 26366*n^2 - 12172*n + 1560)*a(n-1) + 64*(2*n - 3)*(4*n - 5)*(4*n - 7)*(51*n^2 - 42*n + 7)*a(n-2) with a(0) = 1 and a(1) = 5.
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and r. See A352373 for a more general conjecture. (End)
a(n) ~ sqrt(3 + 5/sqrt(17)) * (51*sqrt(17) - 107)^n / (sqrt(Pi*n) * 2^(3*n + 3/2)). - Vaclav Kotesovec, Jun 12 2024
From Seiichi Manyama, Aug 09 2025: (Start)
a(n) = [x^n] 1/((1-x)^(2*n+1) * (1-2*x)^n).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n,k) * binomial(3*n-k,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k-1,k) * binomial(3*n-k,n-k). (End)

A006934 A series for Pi.

Original entry on oeis.org

1, 1, 21, 671, 180323, 20898423, 7426362705, 1874409467055, 5099063967524835, 2246777786836681835, 2490122296790918386363, 1694873049836486741425113, 5559749161756484280905626951, 5406810236613380495234085140851, 12304442295910538475633061651918089

Offset: 0

Simon Plouffe and N. J. A. Sloane

nonn

Formula (21) in Luke (see ref.): Let y = 4*n+1. Then for n -> oo
Pi ~ 4*(n!)^4*2^(4*n)/(y*((2*n)!)^2)*(sum_{k>=0}((-1)^k*y^(-2*k)* A006934(k)/A123854(k)))^2. (Luke does not reference the sequences in this form.) - Peter Luschny, Mar 23 2014
This might be related to the numerators of eq. (18) in N. Elezovic' "Asymptotic Expansions of Central Binomial...", J. Int. Seq. 17 (2014) # 14.2.1. - R. J. Mathar, Mar 23 2014
Several references give an erroneous value of 1874409465055 instead of a(7) in the formula for pi. - M. F. Hasler, Mar 23 2014

Y. L. Luke, The Special Functions and their Approximation, Vol. 1, Academic Press, NY, 1969, see p. 36.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. L. Fields, A note on the asymptotic expansion of a ratio of gamma functions, Proc. Edinburgh Math. Soc. (15), 43-45, 1966.
A. Gil, J. Segura, N. M. Temme, Fast and accurate computation of the Weber parabolic cylinder function W(a,x), IMA J. Num. Anal. 31 (2011), 1194-1216, eq (3.8).
A. Lupas, Re: Pi Calculation ?, on mathforum.org, Feb 15 2003.
C. Mortici, On some accurate estimates of pi, Bull. Math. Anal. Appl. 2(4) (2010) 137-139. (Formula (1.5), same typo as in Luke)
Index entries for sequences related to the number Pi

Cf. A088802, A123854, A220412.

A006934_list := proc(n) local k, f, bp;
bp := proc(n,x) option remember; local k; if n = 0 then 1 else -x*add(binomial(n-1,2*k+1)*bernoulli(2*k+2)/(k+1)*bp(n-2*k-2,x), k=0..n/2-1) fi end:
f := n -> 2^(3*n-add(i, i=convert(n,base,2)));
add(bp(2*k,1/4)*binomial(4*k,2*k)*x^(2*k), k=0..n-1);
seq((-1)^k*f(k)*coeff(%,x,2*k), k=0..n-1) end:
A006934_list(15);  # Peter Luschny, Mar 23 2014
# Second solution, without using Nörlund's generalized Bernoulli polynomials, based on Euler numbers:
A006934_list := proc(n) local a,c,j;
c := n -> 4^n/2^add(i, i=convert(n,base,2));
a := [seq((-4)^j*euler(2*j)/(4*j), j=1..n)];
expand(exp(add(a[j]*x^(-j), j=1..n))); taylor(%, x=infinity, n+2);
subs(x=1/x, convert(%,polynom)): seq(c(iquo(j,2))*coeff(%,x,j), j=0..n) end:
A006934_list(14); # Peter Luschny, Apr 08 2014

A006934List[n_] := Module[{c, a, s, sx}, c[k_] := 4^k/2^Total[ IntegerDigits[k, 2]]; a = Table[(-4)^j EulerE[2j]/(4j), {j, 1, n}]; s[x_] = Series[Exp[Sum[a[[j]] x^(-j), {j, 1, n}]], {x, Infinity, n+2}] // Normal; sx = s[1/x]; Table[c[Quotient[j, 2]] Coefficient[sx, x, j], {j, 0, n}]];
A006934List[14] (* Jean-François Alcover, Jun 02 2019, from second Maple program *)

@CachedFunction
def p(n):
    if n < 2: return 1
    return -add(binomial(n-1,k-1)*bernoulli(k)*p(n-k)/k for k in range(2,n+1,2))/2
def A006934(n): return (-1)^n*p(2*n)*binomial(4*n,2*n)*2^(3*n-sum(n.digits(2)))
[A006934(n) for n in (0..14)]  # Peter Luschny, Mar 24 2014

Let p(n,x) = sum(k=0..n, x^k*A220412(n,k))/A220411(n) then a(n) = (-1)^n*p(n,1/4)*A123854(n)*A001448(n). - Peter Luschny, Mar 23 2014

Pi = lim_{n->oo} 2^{4n+2}/((4n+1)*C(2n,n)^2)*(sum_{k=0..oo} (-1)^k*a(k)/(A123854(k)*(4n+1)^{2k}))^2. - M. F. Hasler, Mar 23 2014

a(7) corrected, a(8)-a(14) from Peter Luschny, Mar 23 2014

A249332 a(n) = Sum_{k=0..2n} binomial(2n, k)^4.

Original entry on oeis.org

1, 18, 1810, 263844, 44916498, 8345319268, 1640651321764, 335556407724360, 70666388112940818, 15220552520052960516, 3337324864503769353060, 742446552655157791828680, 167167472732516775004539300, 38021985442071415426063237704, 8723111727436784830252513497928

Offset: 0

Michael Somos, Oct 25 2014

nonn

G. C. Greubel, Table of n, a(n) for n = 0..100

Cf. A001448, A005260, A006480, A050984.

[(&+[Binomial(2*n,k)^4: k in [0..2*n]]): n in [0..30]]; // G. C. Greubel, Aug 04 2018

Table[Sum[Binomial[2*n, k]^4, {k, 0, 2*n}], {n, 0, 30}] (* G. C. Greubel, Aug 04 2018 *)

{a(n) = sum(k=0, 2*n, binomial( 2*n, k)^4)};

a(n) = A005260(2*n).

A259613 a(n) = binomial(6n,2n)/3, n>0, a(0)=1.

Original entry on oeis.org

1, 5, 165, 6188, 245157, 10015005, 417225900, 17620076360, 751616304549, 32308782859535, 1397281501935165, 60727722660586800, 2650087220696342700, 116043807643289338428, 5096278545356362962504, 224377658168860057076688

Offset: 0

Vladimir Kruchinin, Jun 30 2015

G. C. Greubel, Table of n, a(n) for n = 0..600
V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.

Cf. A001448, A001450, A182959.

[1] cat [Binomial(6*n,2*n)/3: n in [1..20]]; // Vincenzo Librandi, Jul 01 2015

Join[{1}, Table[Binomial[6 n, 2 n]/3, {n, 30}]] (* Vincenzo Librandi, Jul 01 2015 *)

vector(20,n, n--; if (n==0, 1, binomial(6*n,2*n)/3)) \\ Michel Marcus, Jul 01 2015

G.f.: A(x) = 1 + (x*B(x)')/(B(x)) where  B(x) = 2 * (1 + x*B(x)^2)^2 / (1 - 2*x*B(x)^2 + sqrt(1-8*x*B(x)^2)).
a(n) ~ 3^(6*n-1/2) / (sqrt(Pi*n) * 2^(4*n+3/2)). - Vaclav Kotesovec, Jul 01 2015
a(n) = A025174(2*n), n>0. - R. J. Mathar, Jun 07 2016
From Peter Bala, Jun 08 2024: (Start)
a(n) = (9/2)*(6*n-1)*(6*n-5)*(3*n-1)*(3*n-2)/((4*n-1)*(4*n-3)*(2*n-1)*n) * a(n-1) with a(0) = 1 and a(1) = 5.
Right-hand side of the identity (1/3)*Sum_{k = 0..2*n} (-1)^k*binomial(-n, k)* binomial(5*n-k, 2*n-k) = (1/3)*binomial(6*n, 2*n). Compare with the identity Sum_{k = 0..n} (-1)^k*binomial(n, k)*binomial(5*n-k, 2*n-k) = binomial(4*n, 2*n). (End)
From Karol A. Penson, Jan 26 2025: (Start)
G.f. for 3*a(n),a(0)=1, denoted A, expressible entirely by radicals: A = A1 + A2 with
A1 = ((4*sqrt(4 - 27*sqrt(z)) + 12*i*sqrt(3)*z^(1/4))^(1/3) + (4*sqrt(4 - 27*sqrt(z)) - 12*i*sqrt(3)*z^(1/4))^(1/3))/(4*sqrt(4 - 27*sqrt(z))), and
A2 = (1/(4*sqrt(4 + 27*sqrt(z)) + 12*sqrt(3)*z^(1/4))^(1/3) + 1/(4*sqrt(4 + 27*sqrt(z)) - 12*sqrt(3)*z^(1/4))^(1/3))/sqrt(4 + 27*sqrt(z)),
where i = sqrt(-1), the imaginary unit. (End)

A277860 a(n) = Sum_{k=0..n-1} binomial(4k, 2k+1)binomial(2k, k)48^(n-1-k).

Original entry on oeis.org

0, 8, 720, 50400, 3220000, 196885920, 11756961216, 692835631488, 40536961717824, 2363784447147552, 137716866109432896, 8030173585594013568, 469162781054378536320, 27486632292027996114560, 1615617140290621588826880, 95302760085090826490672640

Offset: 1

Zhi-Wei Sun, Nov 02 2016

nonn

Conjecture: For any prime p > 3 and positive integer n, we have
(Sum_{k=0..p*n-1} binomial(4k, 2k+1)*binomial(2k, k)/48^k - (p/3)*Sum_{r=0..n-1} binomial(4r, 2r+1)*binomial(2r, r)/48^r)*48^n/((p*n)^2*binomial(4n, 2n)*binomial(2n, n)) == (5/3)*B_{p-2}(1/3) (mod p), where (p/3) is the Legendre symbol and B_{p-2}(x) is the Bernoulli polynomial of degree p-2.
This conjecture with n = 1 gives the congruence a(p) == (5/12)*p^2*B_{p-2}(1/3) (mod p^3) for any prime p > 3.

			a(1) = 0 since binomial(4*0, 2*0+1)*binomial(2*0, 0)*48^(1-1-0) = 0.
a(2) = 8 since Sum_{k=0..1} binomial(4k, 2k+1)*binomial(2k, k)*48^(2-1-k) = binomial(4, 2+1)*binomial(2, 1)*48^0 = 8.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, Super congruences and Euler numbers, Sci. China Math. 54(2011), no.12, 2509--2535.
Zhi-Wei Sun, New series for some special values of L-functions, Nanjing Univ. J. Math. Biquarterly 32(2015), no.2, 189-218.
Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1610.03384 [math.NT], 2016.

Cf. A000984, A001448, A245089, A277640.

a[n_]:=Sum[Binomial[4k,2k+1]Binomial[2k,k]48^(n-1-k),{k,0,n-1}]
Table[a[n],{n,1,16}]

a(n) ~ 2^(6*n - 9/2) / (Pi*n). - Vaclav Kotesovec, Nov 06 2021

A342983 Number of tree-rooted planar maps with n+1 vertices and n+1 faces.

Original entry on oeis.org

1, 6, 280, 23100, 2522520, 325909584, 47117214144, 7383099180600, 1229149289511000, 214527522662653200, 38887279926227853120, 7271332144993605081120, 1395321310426879365566400, 273697641660657106322640000, 54708248601655917595233984000

Offset: 0

Andrew Howroyd, Apr 03 2021

nonn

The number of edges is 2*n.

Also, a(n) is the number of discrete walks that start and stop at the origin, never pass below the x-axis nor to the left of the y-axis, and, in any order, have n steps that increment x, n steps that decrement x, n steps that increment y, and n steps that decrement y. It is in this sense a way to generalize the 2n-step one-dimensional walks counted by A000108 to a count in two dimensions. Proof: There are A001448(n) ways to interleave two length-2n Dyck words (A000108(n)^2) - Lee A. Newberg, Nov 17 2023

Andrew Howroyd, Table of n, a(n) for n = 0..200

Central coefficients of A342982.
Even bisection of A215288.
Cf. A000108, A001246, A001448.

PARI
```
a(n) = {(4*n)!/(n!*(n+1)!)^2}
```

a(n) = (4*n)!/(n!*(n+1)!)^2.

a(n) = A000108(n)^2 * A001448(n) = A001246(n) * A001448(n). - Alois P. Heinz, Aug 02 2023

A359647 a(n) = [x^n] hypergeom([1/4, 3/4], [2], 64*x). The central terms of the Motzkin triangle A359364 without zeros.

Original entry on oeis.org

1, 6, 140, 4620, 180180, 7759752, 356948592, 17210021400, 859544957700, 44123307828600, 2315270298060720, 123691561681243920, 6707888537328997200, 368417878127146461600, 20455964090297751153600, 1146556787261188952159280, 64797319609481605046295780

Offset: 0

Peter Luschny, Jan 09 2023

nonn

Number of Motzkin paths of length 4n with exactly 2n horizontal steps: a(1) = 6: UDHH, UHDH, UHHD, HUDH, HUHD, HHUD. - Alois P. Heinz, Aug 02 2023

Cf. A000108, A001006, A001448, A359364.

ser := series(hypergeom([1/4, 3/4], [2], 64*x), x, 20):
seq(coeff(ser, x, n), n = 0..16);

a(n) = A359364(4*n, 2*n).

a(n) = A000108(n) * A001448(n) = binomial(2*n,n)/(n+1)*binomial(4*n,2*n). - Alois P. Heinz, Aug 02 2023

A364518 Square array read by ascending antidiagonals: T(n,k) = [x^(2*k)] ( (1 + x)^(n+2)/(1 - x)^(n-2) )^k for n, k >= 0.

Original entry on oeis.org

1, 1, -2, 1, 0, 6, 1, 6, -10, -20, 1, 16, 70, 0, 70, 1, 30, 630, 924, 198, -252, 1, 48, 2310, 28672, 12870, 0, 924, 1, 70, 6006, 204204, 1385670, 184756, -4420, -3432, 1, 96, 12870, 860160, 19122246, 69206016, 2704156, 0, 12870, 1, 126, 24310, 2704156, 130378950, 1848483780, 3528923580, 40116600, 104006, -48620

Offset: 0

Peter Bala, Aug 07 2023

Compare with A364303 and A364519.
Given two sequences of integers c = (c_1, c_2, ..., c_K) and d = (d_1, d_2, ..., d_L), where c_1 + ... + c_K = d_1 + ... + d_L, we can define the factorial ratio sequence u_n(c, d) = (c_1*n)!*(c_2*n)!* ... *(c_K*n)!/ ( (d_1*n)!*(d_2*n)!* ... *(d_L*n)! ) and ask whether it is integral for all n >= 0. The integer L - K is called the height of the sequence. Bober completed the classification of integral factorial ratio sequences of height 1 (see A295431). Soundararajan gives many examples of two-parameter families of integral factorial ratio sequences of height 2.
Each row of the present table is an integral factorial ratio sequence of height 1. It is usually assumed that the c's and d's are integers but here some of the c's and d's are half-integers. See A276098 and the cross references there for further examples of this type.
It is known that the unsigned version of row 0 (the central binomial numbers A000984) and row 2 satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r. We conjecture that all the row sequences of the table satisfy the same supercongruences.

			 Square array begins:
 n\k|  0   1      2        3           4             5
  - + - - - - - - - - - - - - - - - - - - - - - - - - -
  0 |  1  -2      6      -20          70          -252   ... see A000984
  1 |  1   0    -10        0         198             0   ... see A211419
  2 |  1   6     70      924       12870        184756   ... A001448
  3 |  1  16    630    28672     1385670      69206016   ... A091496
  4 |  1  30   2310   204204    19122246    1848483780   ... A061162
  5 |  1  48   6006   860160   130378950   20392706048   ... A276098
  6 |  1  70  12870  2704156   601080390  137846528820   ... A001448 bisected
  7 |  1  96  24310  7028736  2149374150  678057476096   ... A276099

Peter Bala, Some integer ratios of factorials
J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc. (2) 79 2009, 422-444.
K. Soundararajan, Integral factorial ratios: irreducible examples with height larger than 1, Phil. Trans. Royal Soc., A378: 2018044, 2019.
Wikipedia, Dixon's identity
Wikipedia, Hypergeometric function

Cf. A000984 (row 0 unsigned), A211419 (row 1 unsigned without 0's), A001448 (row 2), A091496 (row 3), A061162 (row 4), A276098 (row 5), A001448 bisected (row 6), A276099 (row 7).

Cf. A330843, A364303, A364506, A364509, A364513, A364519.

T(n,k) = add( binomial((n+2)*k, j)*binomial(n*k-j-1, 2*k-j), j = 0..2*k):
# display as a square array
seq(print(seq(T(n, k), k = 0..10)), n = 0..10);
# display as a sequence
seq(seq(T(n-k, k), k = 0..n), n = 0..10);

T(n,k) = sum(j = 0, 2*k, binomial((n+2)*k, j)*binomial(n*k-j-1, 2*k-j));
lista(nn) = for( n=0, nn, for (k=0, n, print1(T(n-k, k), ", "))); \\ Michel Marcus, Aug 13 2023

T(n,k) = Sum_{j = 0..2*k} binomial((n+2)*k, j)*binomial(n*k-j-1, 2*k-j).
T(2,k) = binomial(4*k,2*k).
For n >= 3, T(n,k) = binomial(n*k-1,2*k) * hypergeom([-(n+2)*k, -2*k], [1 - n*k], -1) except when (n,k) = (3,1).
For n >= 2, T(n,k) = ((n+2)*k)!*((n-2)*k/2)!/(((n+2)*k/2)!*((n-2)*k)!*(2*k)!) by Kummer's Theorem.
T(n,k) = [x^k] (1 - x)^(2*k) * Chebyshev_T(n*k, (1 + x)/(1 - x)).
T(n,k) = Sum_{j = 0..k} binomial(2*n*k, 2*j)*binomial((n-1)*k-j-1, k-j).
For n >= 3, T(n,k) = binomial((n-1)*k-1,k) * hypergeom([-n*k, -k, -n*k + 1/2], [1 - (n-1)*k, 1/2], 1).
The row generating functions are algebraic functions over the field of rational functions Q(x).

cp's OEIS Frontend

A337350 a(n) is the number of lattice paths from (0,0) to (2n,2n) using only the steps (1,0) and (0,1) and which do not touch any other points of the form (2k,2k).

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A337396 Expansion of sqrt((1-8*x+sqrt(1+64*x^2)) / (2 * (1+64*x^2))).

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A370100 a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(2*n-k-1,n-k).

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A006934 A series for Pi.

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A249332 a(n) = Sum_{k=0..2*n} binomial(2*n, k)^4.

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A259613 a(n) = binomial(6*n,2*n)/3, n>0, a(0)=1.

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A277860 a(n) = Sum_{k=0..n-1} binomial(4k, 2k+1)*binomial(2k, k)*48^(n-1-k).

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A342983 Number of tree-rooted planar maps with n+1 vertices and n+1 faces.

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A337396 Expansion of sqrt((1-8x+sqrt(1+64x^2)) / (2 * (1+64*x^2))).

A370100 a(n) = Sum_{k=0..n} binomial(4n,k) binomial(2*n-k-1,n-k).

A249332 a(n) = Sum_{k=0..2n} binomial(2n, k)^4.

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A277860 a(n) = Sum_{k=0..n-1} binomial(4k, 2k+1)binomial(2k, k)48^(n-1-k).