A108479 -id:A108479 - cp's OEIS frontend

A086645 Triangle read by rows: T(n, k) = binomial(2n, 2k).

1, 1, 1, 1, 6, 1, 1, 15, 15, 1, 1, 28, 70, 28, 1, 1, 45, 210, 210, 45, 1, 1, 66, 495, 924, 495, 66, 1, 1, 91, 1001, 3003, 3003, 1001, 91, 1, 1, 120, 1820, 8008, 12870, 8008, 1820, 120, 1, 1, 153, 3060, 18564, 43758, 43758, 18564, 3060, 153, 1, 1, 190, 4845, 38760

Offset: 0

Philippe Deléham, Jul 26 2003

Terms have the same parity as those of Pascal's triangle.
Coefficients of polynomials (1/2)*((1 + x^(1/2))^(2n) + (1 - x^(1/2))^(2n)).
Number of compositions of 2n having k parts greater than 1; example: T(3, 2) = 15 because we have 4+2, 2+4, 3+2+1, 3+1+2, 2+3+1, 2+1+3, 1+3+2, 1+2+3, 2+2+1+1, 2+1+2+1, 2+1+1+2, 1+2+2+1, 1+2+1+2, 1+1+2+2, 3+3. - Philippe Deléham, May 18 2005
Number of binary words of length 2n - 1 having k runs of consecutive 1's; example: T(3,2) = 15 because we have 00101, 01001, 01010, 01011, 01101, 10001, 10010, 10011, 10100, 10110, 10111, 11001, 11010, 11011, 11101. - Philippe Deléham, May 18 2005
Let M_n be the n X n matrix M_n(i, j) = T(i, j-1); then for n > 0, det(M_n) = A000364(n), Euler numbers; example: det([1, 1, 0, 0; 1, 6, 1, 0; 1, 15, 15, 1; 1, 28, 70, 28 ]) = 1385 = A000364(4). - Philippe Deléham, Sep 04 2005
Equals ConvOffsStoT transform of the hexagonal numbers, A000384: (1, 6, 15, 28, 45, ...); e.g., ConvOffs transform of (1, 6, 15, 28) = (1, 28, 70, 28, 1). - Gary W. Adamson, Apr 22 2008
From Peter Bala, Oct 23 2008: (Start)
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 this array are the h-vectors of a unimodular triangulation of P(C_n) [Ardila et al.]. See A127674 for (a signed version of) the corresponding array of f-vectors for these type C_n polytopes. See A008459 for the array of h-vectors for type A_n polytopes and A108558 for the array of h-vectors associated with type D_n polytopes.
The Hilbert transform of this triangle is A142992 (see A145905 for the definition of this term).
(End)
Diagonal sums: A108479. - Philippe Deléham, Sep 08 2009
Coefficients of Product_{k=1..n} (cot(k*Pi/(2n+1))^2 - x) = Sum_{k=0..n} (-1)^k*binomial(2n,2k)*x^k/(2n+1-2k). - David Ingerman (daviddavifree(AT)gmail.com), Mar 30 2010
Generalized Narayana triangle for 4^n (or cosh(2x)). - Paul Barry, Sep 28 2010
Coefficients of the matrix inverse appear to be T^(-1)(n,k) = (-1)^(n+k)*A086646(n,k). - R. J. Mathar, Mar 12 2013
Let E(y) = Sum_{n>=0} y^n/(2*n)! = cosh(sqrt(y)). Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence (2*n)! as defined in Wang and Wang. Cf. A103327. - Peter Bala, Aug 06 2013
Row 6, (1,66,495,924,495,66,1), plays a role in expansions of powers of the Dedekind eta function. See the Chan link, p. 534, and A034839. - Tom Copeland, Dec 12 2016

			From _Peter Bala_, Oct 23 2008: (Start)
The triangle begins
n\k|..0.....1.....2.....3.....4.....5.....6
===========================================
0..|..1
1..|..1.....1
2..|..1.....6.....1
3..|..1....15....15.....1
4..|..1....28....70....28.....1
5..|..1....45...210...210....45.....1
6..|..1....66...495...924...495....66.....1
...
(End)
From _Peter Bala_, Aug 06 2013: (Start)
Viewed as the generalized Riordan array (cosh(sqrt(y)), y) with respect to the sequence (2*n)! the column generating functions begin
1st col: cosh(sqrt(y)) = 1 + y/2! + y^2/4! + y^3/6! + y^4/8! + ....
2nd col: 1/2!*y*cosh(sqrt(y)) = y/2! + 6*y^2/4! + 15*y^3/6! + 28*y^4/8! + ....
3rd col: 1/4!*y^2*cosh(sqrt(y)) = y^2/4! + 15*y^3/6! + 70*y^4/8! + 210*y^5/10! + .... (End)

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224.

Indranil Ghosh, Rows 0.. 120 of triangle, flattened
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.
H. Chan, S. Cooper and P. Toh, The 26th power of Dedekind's eta function Advances in Mathematics, 207 (2006) 532-543.
T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras, 2012.
T. Copeland, Skipping over Dimensions, Juggling Zeros in the Matrix, 2020.
E. Lucas, Théorie des fonctions numériques simplement périodiques, Baltimore, 1878. See page 145 equation (153).
W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Cf. A098158, A000384, A081294.
Cf. A008459, A108558, A127674, A142992. - Peter Bala, Oct 23 2008
Cf. A103327 (binomial(2n+1, 2k+1)), A103328 (binomial(2n, 2k+1)), A091042 (binomial(2n+1, 2k)). -Wolfdieter Lang, Jan 06 2013
Cf. A086646 (unsigned matrix inverse), A103327.
Cf. A034839.

/* As triangle: */ [[Binomial(2*n, 2*k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Dec 14 2016

A086645:=(n,k)->binomial(2*n,2*k): seq(seq(A086645(n,k),k=0..n),n=0..12);

Table[Binomial[2 n, 2 k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 13 2016 *)

create_list(binomial(2*n,2*k),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

PARI
```
{T(n, k) = binomial(2*n, 2*k)};
```

{T(n, k) = sum( i=0, min(k, n-k), 4^i * binomial(n, 2*i) * binomial(n - 2*i, k-i))}; /* Michael Somos, May 26 2005 */

T(n, k) = (2*n)!/((2*(n-k))!*(2*k)!) row sums = A081294. COLUMNS: A000012, A000384
Sum_{k>=0} T(n, k)*A000364(k) = A000795(n) = (2^n)*A005647(n).
Sum_{k>=0} T(n, k)*2^k = A001541(n). Sum_{k>=0} T(n, k)*3^k = 2^n*A001075(n). Sum_{k>=0} T(n, k)*4^k = A083884(n). - Philippe Deléham, Feb 29 2004
O.g.f.: (1 - z*(1+x))/(x^2*z^2 - 2*x*z*(1+z) + (1-z)^2) = 1 + (1 + x)*z +(1 + 6*x + x^2)*z^2 + ... . - Peter Bala, Oct 23 2008
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A081294(n), A001541(n), A090965(n), A083884(n), A099140(n), A099141(n), A099142(n), A165224(n), A026244(n) for x = 0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Sep 08 2009
Product_{k=1..n} (cot(k*Pi/(2n+1))^2 - x) = Sum_{k=0..n} (-1)^k*binomial(2n,2k)*x^k/(2n+1-2k). - David Ingerman (daviddavifree(AT)gmail.com), Mar 30 2010
From Paul Barry, Sep 28 2010: (Start)
G.f.: 1/(1-x-x*y-4*x^2*y/(1-x-x*y)) = (1-x*(1+y))/(1-2*x*(1+y)+x^2*(1-y)^2);
E.g.f.: exp((1+y)*x)*cosh(2*sqrt(y)*x);
T(n,k) = Sum_{j=0..n} C(n,j)*C(n-j,2*(k-j))*4^(k-j). (End)
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) + 2*T(n-2,k-1) - T(n-2,k) - T(n-2,k-2), with T(0,0)=T(1,0)=T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Nov 26 2013
From Peter Bala, Sep 22 2021: (Start)
n-th row polynomial R(n,x) = (1-x)^n*T(n,(1+x)/(1-x)), where T(n,x) is the n-th Chebyshev polynomial of the first kind. Cf. A008459.
R(n,x) = Sum_{k = 0..n} binomial(n,2*k)*(4*x)^k*(1 + x)^(n-2*k).
R(n,x) = n*Sum_{k = 0..n} (n+k-1)!/((n-k)!*(2*k)!)*(4*x)^k*(1-x)^(n-k) for n >= 1. (End)

A381421 a(n) = Sum_{k=0..n} (k+1) * binomial(2k,2n-2*k).

Original entry on oeis.org

1, 2, 5, 22, 68, 206, 631, 1870, 5467, 15836, 45416, 129260, 365565, 1028122, 2877697, 8021010, 22274476, 61653850, 170152275, 468347046, 1286055927, 3523777912, 9635982160, 26302324504, 71674754873, 195015074610, 529846108989, 1437657038030, 3896050721940

Offset: 0

Seiichi Manyama, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (4,-2,0,-11,0,-2,4,-1).

Cf. A108479, A382230, A382470, A382471, A382472, A382473, A382474.

Cf. A034839.

[&+[(k+1) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 23 2025

Table[Sum[(k+1)*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 23 2025 *)

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

my(N=1, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: ((1-x-x^2)^2 + 4*x^3) / ((1-x-x^2)^2 - 4*x^3)^2.

a(n) = 4*a(n-1) - 2*a(n-2) - 11*a(n-4) - 2*a(n-6) + 4*a(n-7) - a(n-8).

A376729 Expansion of (1 - x^2 - x^3)/((1 - x^2 - x^3)^2 - 4*x^5).

Original entry on oeis.org

1, 0, 1, 1, 1, 6, 2, 15, 16, 29, 71, 73, 212, 276, 541, 1016, 1497, 3189, 4825, 9162, 16022, 26763, 50424, 82869, 151851, 262705, 456520, 820328, 1401913, 2511824, 4361521, 7657481, 13528913, 23509678, 41633002, 72630919, 127709888, 224418509, 392539055, 691382201

Offset: 0

Seiichi Manyama, Oct 03 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,2,-1).

Cf. A108479, A376730, A376731.

Cf. A376723, A376726.

my(N=40, x='x+O('x^N)); Vec((1-x^2-x^3)/((1-x^2-x^3)^2-4*x^5))

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

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6).

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

A382230 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(2k,2n-2*k).

Original entry on oeis.org

1, 3, 9, 46, 171, 591, 2033, 6714, 21606, 68308, 212370, 651234, 1974113, 5924277, 17623671, 52025858, 152539077, 444530073, 1288396257, 3715833732, 10668907932, 30507914696, 86912853588, 246755125332, 698353551105, 1970673504951, 5545952371509, 15568330002486

Offset: 0

Seiichi Manyama, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,-9,2,-18,30,7,30,-18,2,-9,6,-1).

Cf. A108479, A381421, A382470, A382471, A382472, A382473, A382474.

Cf. A034839, A377145.

[&+[Binomial(k+2, 2) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 22 2025

Table[Sum[Binomial[k+2,2]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 22 2025 *)

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

my(N=2, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..1} 4^k * binomial(3,2*k) * (1-x-x^2)^(3-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^3.

a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3) - 18*a(n-4) + 30*a(n-5) + 7*a(n-6) + 30*a(n-7) - 18*a(n-8) + 2*a(n-9) - 9*a(n-10) + 6*a(n-11) - a(n-12).

A382470 a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(2k,2n-2*k).

Original entry on oeis.org

1, 4, 14, 80, 345, 1336, 5074, 18404, 64460, 220276, 736242, 2415128, 7798043, 24833160, 78131242, 243211412, 749926963, 2292771088, 6956262660, 20959406680, 62753991192, 186809711448, 553172044548, 1630068765840, 4781871397429, 13969460520764

Offset: 0

Seiichi Manyama, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (8,-20,16,-26,88,-48,24,-163,24,-48,88,-26,16,-20,8,-1).

Cf. A108479, A381421, A382230, A382471, A382472, A382473, A382474.

Cf. A034839, A377148.

[&+[Binomial(k+3,3) * Binomial(2*k,2*n-2*k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Apr 10 2025

Table[Sum[Binomial[k+3,3]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 10 2025 *)

a(n) = sum(k=0, n, binomial(k+3, 3)*binomial(2*k, 2*n-2*k));

my(N=3, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..2} 4^k * binomial(4,2*k) * (1-x-x^2)^(4-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^4.

a(n) = 8*a(n-1) - 20*a(n-2) + 16*a(n-3) - 26*a(n-4) + 88*a(n-5) - 48*a(n-6) + 24*a(n-7) - 163*a(n-8) + 24*a(n-9) - 48*a(n-10) + 88*a(n-11) - 26*a(n-12) + 16*a(n-13) - 20*a(n-14) + 8*a(n-15) - a(n-16).

A382471 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(2k,2n-2*k).

Original entry on oeis.org

1, 5, 20, 125, 610, 2611, 10815, 42610, 161005, 590155, 2106362, 7348265, 25141430, 84569395, 280246795, 916465742, 2961805180, 9470735650, 29994694130, 94172180660, 293326457342, 907028460410, 2786036875580, 8505001839950, 25815678641935, 77945771624609

Offset: 0

Seiichi Manyama, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-55,172,-250,100,-365,510,-29,510,-365,100,-250,172,-55,50,-35,10,-1).

Cf. A108479, A381421, A382230, A382470, A382472, A382473, A382474.

Cf. A034839, A377152.

[&+[Binomial(k+4,4) * Binomial(2*k,2*n-2*k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Apr 10 2025

Table[Sum[Binomial[k+4,4]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 10 2025 *)

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

my(N=4, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..2} 4^k * binomial(5,2*k) * (1-x-x^2)^(5-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^5.

A382472 a(n) = Sum_{k=0..n} binomial(k+5,5) * binomial(2k,2n-2*k).

Original entry on oeis.org

1, 6, 27, 182, 987, 4620, 20678, 87732, 355095, 1387462, 5258967, 19416222, 70086803, 248046540, 862694058, 2954279732, 9977518122, 33278815920, 109749059308, 358231786128, 1158357919194, 3713416860580, 11810098024410, 37285901203740, 116917784689237

Offset: 0

Seiichi Manyama, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (12,-54,112,-141,312,-674,588,-714,1812,-1134,888,-2741,888,-1134,1812,-714,588,-674,312,-141,112,-54,12,-1).

Cf. A108479, A381421, A382230, A382470, A382471, A382473, A382474.

Cf. A034839, A377153.

[&+[Binomial(k+5, 5) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Apr 11 2025

Table[Sum[Binomial[k+5,5]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 11 2025 *)

a(n) = sum(k=0, n, binomial(k+5, 5)*binomial(2*k, 2*n-2*k));

my(N=5, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..3} 4^k * binomial(6,2*k) * (1-x-x^2)^(6-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^6.

A382473 a(n) = Sum_{k=0..n} binomial(k+6,6) * binomial(2k,2n-2*k).

Original entry on oeis.org

1, 7, 35, 252, 1498, 7602, 36498, 165600, 713769, 2957647, 11850223, 46111352, 174956250, 649284286, 2362771938, 8449241836, 29744151416, 103237104740, 353744829032, 1198001464940, 4013905507150, 13316690882670, 43780154987030, 142726581203640

Offset: 0

Seiichi Manyama, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (14, -77, 210, -336, 602, -1435, 2018, -1981, 4312, -5894, 3360, -7721, 9562, -3079, 9562, -7721, 3360, -5894, 4312, -1981, 2018, -1435, 602, -336, 210, -77, 14, -1).

Cf. A108479, A381421, A382230, A382470, A382471, A382472, A382474.

Cf. A034839, A377158.

[&+[Binomial(k+6, 6) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Apr 11 2025

Table[Sum[Binomial[k+6,6]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 11 2025 *)

a(n) = sum(k=0, n, binomial(k+6, 6)*binomial(2*k, 2*n-2*k));

my(N=6, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..3} 4^k * binomial(7,2*k) * (1-x-x^2)^(7-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^7.

A382474 a(n) = Sum_{k=0..n} binomial(k+7,7) * binomial(2k,2n-2*k).

Original entry on oeis.org

1, 8, 44, 336, 2166, 11832, 60576, 292248, 1334817, 5840296, 24637976, 100684376, 400255050, 1553016960, 5897388492, 21967711160, 80425346844, 289868771928, 1029979010972, 3612517052608, 12520285820362, 42919328903928, 145643017892472, 489606988741128

Offset: 0

Seiichi Manyama, Mar 28 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (16, -104, 352, -708, 1232, -2800, 5168, -6122, 9728, -18008, 16816, -19152, 37744, -27096, 24960, -50611, 24960, -27096, 37744, -19152, 16816, -18008, 9728, -6122, 5168, -2800, 1232, -708, 352, -104, 16, -1).

Cf. A108479, A381421, A382230, A382470, A382471, A382472, A382473.

Cf. A034839, A377159.

[&+[Binomial(k+7, 7) * Binomial(2*k, 2*n-2*k): k in [0..n]]: n in [0..29]]; // Vincenzo Librandi, Apr 22 2025

Table[Sum[Binomial[k+7,7]*Binomial[2*k,2*n-2*k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Apr 22 2025 *)

a(n) = sum(k=0, n, binomial(k+7, 7)*binomial(2*k, 2*n-2*k));

my(N=7, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, (N+1)\2, 4^k*binomial(N+1, 2*k)*X^(N+1-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1))

G.f.: (Sum_{k=0..4} 4^k * binomial(8,2*k) * (1-x-x^2)^(8-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^8.

A220602 T(n,k)=Number of ways to reciprocally link elements of an nXk array either to themselves or to exactly two horizontal, vertical and antidiagonal neighbors, without 3-loops.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 7, 7, 1, 1, 17, 65, 17, 1, 1, 44, 471, 471, 44, 1, 1, 117, 3469, 9047, 3469, 117, 1, 1, 305, 25747, 167614, 167614, 25747, 305, 1, 1, 798, 190425, 3174778, 7558495, 3174778, 190425, 798, 1, 1, 2091, 1408823, 60138423, 352586943

Offset: 1

R. H. Hardin Dec 16 2012

Table starts
.1.....1..........1...............1...................1
.1.....2..........7..............17..................44
.1.....7.........65.............471................3469
.1....17........471............9047..............167614
.1....44.......3469..........167614.............7558495
.1...117......25747.........3174778...........352586943
.1...305.....190425........60138423.........16505365334
.1...798....1408823......1137615308........770615995060
.1..2091...10424813.....21523535122......35978245868462
.1..5473...77135267....407249783145....1680090131446571
.1.14328..570740149...7705461280893...78453960278374893
.1.37513.4223043591.145792738613760.3663456651896335065

			Some solutions for n=3 k=4 0=self 2=n 3=ne 4=w 6=e 7=sw 8=s (reciprocal directions total 10)
.00.68.48.78...00.68.48.00...00.00.67.47...00.67.47.00...00.67.47.00
.68.24.23.28...68.24.28.00...00.38.36.48...36.34.67.47...38.37.68.48
.26.46.46.24...26.46.24.00...00.26.46.24...00.36.34.00...23.00.26.24

R. H. Hardin, Table of n, a(n) for n = 1..180

Column 2 is A108479

cp's OEIS Frontend

A086645 Triangle read by rows: T(n, k) = binomial(2n, 2k).

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A381421 a(n) = Sum_{k=0..n} (k+1) * binomial(2*k,2*n-2*k).

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A376729 Expansion of (1 - x^2 - x^3)/((1 - x^2 - x^3)^2 - 4*x^5).

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A382230 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(2*k,2*n-2*k).

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A382470 a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(2*k,2*n-2*k).

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A382471 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(2*k,2*n-2*k).

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A382472 a(n) = Sum_{k=0..n} binomial(k+5,5) * binomial(2*k,2*n-2*k).

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A381421 a(n) = Sum_{k=0..n} (k+1) * binomial(2k,2n-2*k).

A382230 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(2k,2n-2*k).

A382470 a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(2k,2n-2*k).

A382471 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(2k,2n-2*k).

A382472 a(n) = Sum_{k=0..n} binomial(k+5,5) * binomial(2k,2n-2*k).

A382473 a(n) = Sum_{k=0..n} binomial(k+6,6) * binomial(2k,2n-2*k).

A382474 a(n) = Sum_{k=0..n} binomial(k+7,7) * binomial(2k,2n-2*k).