Original entry on oeis.org
1, 2, 2, 6, 24, 6, 20, 180, 180, 20, 70, 1120, 2520, 1120, 70, 252, 6300, 25200, 25200, 6300, 252, 924, 33264, 207900, 369600, 207900, 33264, 924, 3432, 168168, 1513512, 4204200, 4204200, 1513512, 168168, 3432, 12870, 823680, 10090080, 40360320
Offset: 0
A002894
a(n) = binomial(2n, n)^2.
Original entry on oeis.org
1, 4, 36, 400, 4900, 63504, 853776, 11778624, 165636900, 2363904400, 34134779536, 497634306624, 7312459672336, 108172480360000, 1609341595560000, 24061445010950400, 361297635242552100, 5445717990022688400, 82358080713306090000, 1249287673091590440000
Offset: 0
G.f. = 1 + 4*x + 36*x^2 + 400*x^3 + 4900*x^4 + 63504*x^5 + 853776*x^6 + ... - _Michael Somos_, Aug 06 2014
From _Peter Bala_, Jan 26 2018: (Start)
a(2) = 36: The thirty six 3 x k arrays with columns belonging to the set of column vectors S = {[1,0,0], [0,1,0], [1,0,1], [0,1,1]} and having all row sums equal to 2 are the 6 distinct arrays obtained by permuting the columns of
/1 1 0 0\
|0 0 1 1|,
\0 0 1 1/
the 6 distinct arrays obtained by permuting the columns of
/0 0 1 1\
|1 1 0 0|
\0 0 1 1/
and the 24 arrays obtained by permuting the columns of
/1 0 1 0\
|0 1 0 1|. (End)
\0 0 1 1/
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 591,828.
- J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 8.
- Matthijs Coster, Over 6 families van krommen [On 6 families of curves], Master's Thesis (unpublished), Aug 26 1983.
- Leonard Lipshitz and A. van der Poorten. "Rational functions, diagonals, automata and arithmetic." In Number Theory, Richard A. Mollin, ed., Walter de Gruyter, Berlin (1990): 339-358.
- 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
- 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].
- R. Bacher, Meander algebras, Institut Fourier, 1999.
- E. Barcucci, A. Frosini and S. Rinaldi, On directed-convex polyominoes in a rectangle, Discr. Math., 298 (2005). 62-78.
- Arnaud Beauville, Les familles stables de courbes elliptiques sur P^1 admettant quatre fibres singulières, Comptes Rendus, Académie Sciences Paris, no. 294, May 24 1982, 657-660. MR0664643 (83h:14008)
- Alin Bostan, Armin Straub, and Sergey Yurkevich, On the representability of sequences as constant terms, arXiv:2212.10116 [math.NT], 2022.
- Tricia Muldoon Brown, The Problem of Pawns, The Electronic Journal of Combinatorics (2019) Vol. 26, Issue 3, #P3.21. Also arXiv:1811.09606, [math.CO], 2018.
- John Maxwell Campbell, New series involving harmonic numbers and squared central binomial coefficients, Rocky Mountain J. Math., 49 (2019), 2513-2544.
- C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
- Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
- Murray Elder, Cogrowth, 2011.
- M. Elder, A. Rechnitzer, E. J. Janse van Rensburg, and T. Wong, The cogrowth series for BS(N,N) is D-finite, arXiv:1309.4184 [math.GR], 2013.
- Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 90.
- Florian Fürnsinn and Sergey Yurkevich, Algebraicity of hypergeometric functions with arbitrary parameters, arXiv:2308.12855 [math.CA], 2023.
- Davidson Noby Joseph and Igor Boettcher, Walking on Archimedean Lattices: Insights from Bloch Band Theory, arXiv:2507.12662 [cond-mat.stat-mech], 2025. See p. 18.
- Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2010.
- Markus Kuba and Alois Panholzer, Lattice paths and the diagonal of the cube, arXiv:2411.03930 [math.CO], 2024. See p. 14.
- L. Lipshitz and A. J. van der Poorten, Rational functions, diagonals, automata and arithmetic
- Raul Prisacariu, Littlewood Polynomials of Degree n with Closed Lill Paths
- Eric M. Rains, High powers of random elements of compact Lie groups, Probability Theory and Related Fields 107 (1997), 219-241.
- Grzegorz Siudem and Agata Fronczak, Bell polynomials in the series expansions of the Ising model, arXiv:2007.16132 [math-ph], 2020.
- Eric Weisstein's World of Mathematics, Lattice Path.
- D. Zagier, Integral solutions of Apéry-like recurrence equations. See line G in sporadic solutions table of page 5.
Cf.
A000984,
A000515,
A010370,
A054474 (INVERTi transform),
A172390,
A000897,
A002897,
A006480,
A008977,
A186420,
A188662,
A000894,
A241530,
A002898 (walks hex lattice).
-
[Binomial(2*n, n)^2: n in [0..20]]; // Vincenzo Librandi, Aug 07 2014
-
A002894 := n-> binomial(2*n,n)^2.
-
CoefficientList[Series[Hypergeometric2F1[1/2, 1/2, 1, 16x], {x, 0, 20}], x]
Table[Binomial[2n,n]^2,{n,0,20}] (* Harvey P. Dale, Jul 06 2011 *)
a[ n_] := SeriesCoefficient[ EllipticK[16 x] / (Pi/2), {x, 0, n}]; (* Michael Somos, Aug 06 2014 *)
a[n_] := 16^n HypergeometricPFQ[{1/2, -2 n, 2 n + 1}, {1, 1}, 1];
Table[a[n], {n, 0, 19}] (* Peter Luschny, Mar 14 2018 *)
-
{a(n) = binomial(2*n, n)^2};
-
{a(n) = if( n<0, 0, polcoeff( polcoeff( polcoeff( 1 / (1 - x * (y + z + 1/y + 1/z)) + x * O(x^(2*n)), 2*n), 0), 0))}; /* Michael Somos, Jun 12 2004 */
-
[binomial(2*n, n)**2 for n in range(17)] # Zerinvary Lajos, Apr 21 2009
A282252
Exponential Riordan array [Bessel_I(0,2*x)^2, x].
Original entry on oeis.org
1, 0, 1, 4, 0, 1, 0, 12, 0, 1, 36, 0, 24, 0, 1, 0, 180, 0, 40, 0, 1, 400, 0, 540, 0, 60, 0, 1, 0, 2800, 0, 1260, 0, 84, 0, 1, 4900, 0, 11200, 0, 2520, 0, 112, 0, 1, 0, 44100, 0, 33600, 0, 4536, 0, 144, 0, 1, 63504, 0, 220500, 0, 84000, 0, 7560, 0, 180, 0, 1
Offset: 0
The triangle begins
1;
0, 1;
4, 0, 1;
0, 12, 0, 1;
36, 0, 24, 0, 1;
0, 180, 0, 40, 0, 1;
400, 0, 540, 0, 60, 0, 1;
...
T(3,1) = 12: on the square lattice, let L, R, U, D denote a left step, right step, up step and down step respectively. The 12 walks of length 3 containing a single loop are
loop L R, loop R L, loop U D, loop D U,
L loop R, R loop L, U loop D, D loop U,
L R loop, R L loop, U D loop, D U loop.
The infinitesimal generator of this array has integer entries and begins
0;
0, 0;
4, 0, 0;
0, 12, 0, 0;
-12, 0, 24, 0, 0;
0, -60, 0, 40, 0, 0;
160, 0, -180, 0, 60, 0, 0;
0, 1120, 0, -420, 0, 84, 0, 0;
-4620, 0, 4480, 0, -840, 0, 112, 0, 0;
...
It is the generalized exponential Riordan array [ 2*log(Bessel_I(0,2*x)), x ].
-
T := (n, k) -> (1/2)*binomial(n, k)*binomial(n-k, floor((1/2)*n-(1/2)*k))^2*(1+(-1)^(n-k)):
seq(seq(T(n, k), k = 0..n), n = 0..9);
-
Table[Binomial[n, k] Binomial[n - k, Floor[(n - k)/2]]^2*(1 + (-1)^(n - k))/2, {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 12 2017 *)
-
for(n=0,10, for(k=0,n, print1(binomial(n,k)*binomial(n-k,floor((n-k)/2))^2*(1 + (-1)^(n-k))/2, ", "))) \\ G. C. Greubel, Aug 16 2017
A320824
T(n, k) = (m*n)!/(k!*(n-k)!)^m with m = 3; triangle read by rows, 0 <= k <= n.
Original entry on oeis.org
1, 6, 6, 90, 720, 90, 1680, 45360, 45360, 1680, 34650, 2217600, 7484400, 2217600, 34650, 756756, 94594500, 756756000, 756756000, 94594500, 756756, 17153136, 3705077376, 57891834000, 137225088000, 57891834000, 3705077376, 17153136
Offset: 0
Triangle starts:
[0] 1;
[1] 6, 6;
[2] 90, 720, 90;
[3] 1680, 45360, 45360, 1680;
[4] 34650, 2217600, 7484400, 2217600, 34650;
[5] 756756, 94594500, 756756000, 756756000, 94594500, 756756;
-
Flat(List([0..6],n->List([0..n],k->Factorial(3*n)/(Factorial(k)*Factorial(n-k))^3))); # Muniru A Asiru, Oct 27 2018
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[[(Factorial(3*n)/(Factorial(n))^3)*Binomial(n, k)^3: k in [0..n]]: n in [0..15]]; // G. C. Greubel, Oct 27 2018
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T := (n, k, m) -> (m*n)!/(k!*(n-k)!)^m:
seq(seq(T(n, k, 3), k=0..n), n=0..7);
-
Table[((3*n)!/(n!)^3)*Binomial[n, k]^3, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 27 2018 *)
-
t(n, k) = (3*n)!/(k!*(n-k)!)^3
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 6 rows of triangle as follows: */
trianglerows(6) \\ Felix Fröhlich, Oct 21 2018
A068218
Triangle of numbers of square lattice walks that start and end at origin after 2k steps and contain exactly r steps to the east, not touching origin at intermediate stages.
Original entry on oeis.org
1, 2, 2, 2, 16, 2, 4, 84, 84, 4, 10, 400, 1056, 400, 10, 28, 1820, 9184, 9184, 1820, 28, 84, 8064, 66276, 126720, 66276, 8064, 84, 264, 35112, 426888, 1329768, 1329768, 426888, 35112, 264, 858, 151008, 2546544, 11737440, 19123776, 11737440
Offset: 0
T(3,1)=84 because there are 84 distinct lattice walks of length 2*3=6 starting and ending at the origin and containing exactly 1 step to the east and not touching origin at intermediate steps. Let E, W, S, N denote the 4 possible directions, then NNEWSS and NWSSNE are examples of such walks.
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A069466[k_, r_] := Binomial[2 k, k]*Binomial[k, r]^2; t[k_, r_] := t[k, r] = A069466[k, r] - Sum[Sum[t[i, j]*A069466[k - i, r - j], {j, 0, r}], {i, 1, k - 1}]; Table[t[k, r], {k, 0, 8}, {r, 0, k}] // Flatten (* Jean-François Alcover, Nov 21 2012, from formula *)
Showing 1-5 of 5 results.
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