A103881 -id:A103881 - cp's OEIS frontend

A156554 The number of integer sequences of length d = 2n+1 such that the sum of the terms is 0 and the sum of the absolute values of the terms is d-1.

1, 6, 110, 2562, 66222, 1815506, 51697802, 1511679210, 45076309166, 1364497268946, 41800229045610, 1292986222651646, 40317756506959050, 1265712901796074842, 39965073938276694002, 1268208750951634765562, 40419340092267053380782, 1293151592990764737265490

Offset: 0

W. Edwin Clark, Feb 09 2009

Let b(n) = S(d,n) be the coordination sequence of the lattice A_d. Then this sequence is a(n) = S(2n,n). See Conway-Sloane. The sequence is defined by Couveignes et al.

			The a(1) = 6 sequences are (1,-1,0), (-1,1,0), (1,0,-1), (-1,0,1), (0,1,-1) and (0,-1,1).

R. H. Hardin and Colin Barker, Table of n, a(n) for n = 0..300 (terms up to n=26 from R. H. Hardin)
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
J.-M. Couveignes, T. Ezome, and R. Lercier, Elliptic periods and primality proving, arXiv:0810.2853 [math.NT], 2008-2009.

a(n) = A103881(2n, n), A103882.

S:=proc(d,n) add(binomial(d,k)^2*binomial(n-k+d-1,d-1),k=0..d); end proc; a:=n->S(2*n,n);

Table[ Binomial[-1 + 3 n, -1 + 2 n] HypergeometricPFQ[{-2 n, -2 n, -n}, {1, 1 - 3 n}, 1], {n, 0, 10}]  (* Eric W. Weisstein, Feb 10 2009 *)

S(d, n) = sum(k=0, d, binomial(d,k)^2*binomial(n-k+d-1, d-1));
concat(1, vector(20, n, S(2*n,n))) \\ Colin Barker, Dec 24 2015

a(n) = S(2n,n) where S(d,n) = Sum_{k=0..d} C(d,k)^2*C(n-k+d-1,d-1) from formula (22) in Conway-Sloane.
a(n) ~ (1 + sqrt(2))^(4*n + 1/2) / (2^(5/4) * Pi * n). - Vaclav Kotesovec, Apr 10 2018
From Peter Bala, Dec 19 2020: (Start)
a(n) = Sum_{k = 0..n} C(2*n,n-k)^2 * C(2*n+k-1,k).
a(n) = Sum_{k = 1..n} C(2*n, k)*C(2*n+k, k)*C(n-1,k-1) for n >= 1.
a(n) = [x^n] P(2*n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. Cf. A103882.
a(n) = C(2*n,n)^2 * hypergeom([-n, -n, 2*n], [n+1, n+1], 1).
n^2*(2*n - 1)^2*(24*n^3 - 105*n^2 + 152*n - 73)*a(n) = (3264*n^7 - 20808*n^6 + 53900*n^5 - 73159*n^4 + 55963*n^3 - 24107*n^2 + 5436*n - 504)*a(n - 1) - (2*n - 1)*(2*n - 3)*(n - 2)^2*(24*n^3 - 33*n^2 + 14*n - 2)*a(n - 2).
Conjectural: for any prime p >= 5, a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for all positive integers n and k.
More generally, if r and s are positive integers, we conjecture that the same supercongruences hold for the sequence defined by [x^(r*n)] P(s*n,(1 + x)/(1 - x)). (End)
Even more generally, we conjecture that the same supercongruences hold for the sequence defined by [x^(r*n)] (1 + x)^(A*n) * (1 - x)^(B*n) * P(s*n,(1 + x)/(1 - x)), where A and B are integers. - Peter Bala, Mar 17 2023
a(n) = 2*Sum_{k = 0..2*n-1} (-1)^(k+1)*binomial(2*n-1, k)*binomial(n+k, k)* binomial(2*n+k-1, k) for n >= 1. - Peter Bala, Sep 25 2024

Formula incorrectly copied from A143699 removed by R. J. Mathar, Mar 11 2010

A103882 a(n) = Sum_{i=0..n} C(n+1,i)C(n-1,i-1)C(2n-i,n).

Original entry on oeis.org

1, 2, 12, 92, 780, 7002, 65226, 623576, 6077196, 60110030, 601585512, 6078578508, 61908797418, 634756203018, 6545498596110, 67830161708592, 705951252118284, 7375213677918294, 77310179609631564, 812839595630249540, 8569327862277434280, 90562666977432643862

Offset: 0

Ralf Stephan, Feb 20 2005

Number of permutations of n copies of 1..3 with all adjacent differences <= 1 in absolute value. - R. H. Hardin, May 06 2010 [Cf. A177316. - Peter Bala, Jan 14 2020]

Alois P. Heinz, Table of n, a(n) for n = 0..950 (terms n=1..94 from R. H. Hardin)
A. Straub, Multivariate Apéry numbers and supercongruences of rational functions, arXiv:1401.0854 [math.NT] (2014).

Cf. A005258, A156554, A177316, A352654.
Equals A103881(n, n).
Row n=3 of A331562.

[1] cat [&+[Binomial(n+1, i)*Binomial(n-1, i-1) * Binomial(2*n-i, n): i in [0..n]]:n in  [1..21]]; // Marius A. Burtea, Jan 19 2020

[&+[Binomial(n, k)^2*Binomial(n+k-1, k): k in [0..n]]:n in  [0..21]]; // Marius A. Burtea, Jan 19 2020

a:= proc(n) option remember; `if`(n<2, n+1,
      ((n-1)*(55*n^3-143*n^2+102*n-24)*a(n-1)+
      n*(5*n-3)*(n-2)^2*a(n-2))/((n-1)*(5*n-8)*n^2))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 29 2015
# Alternative:
a := n -> hypergeom([-n, -n, n], [1, 1], 1):
seq(simplify(a(n)), n=0..21); # Peter Luschny, Jan 19 2020

Drop[Table[Sum[Sum[Multinomial[r, g, n + 1 - r - g] Binomial[n - 1,n - r] Binomial[n - 1, n - g], {g, 1, n}], {r, 1, n}], {n, 0, 18}], 1] (* Geoffrey Critzer, Jun 29 2015 *)
Table[Sum[Binomial[n+1,k]Binomial[n-1,k-1]Binomial[2n-k,n],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jun 19 2021 *)

a(n) = polcoef(pollegendre(n, (1 + x)/(1 - x)) + O(x^(n+1)), n); \\ Michel Marcus, Dec 20 2020

def A103882(n):
    if n == 0: return 1
    m, g = 1, 0
    for k in range(n+1):
        g += m*n//(n+k)
        m *= (n+k+1)*(n-k)**2
        m //= (k+1)**3
    return g # Chai Wah Wu, Oct 04 2022

def A103882(n): return hypergeometric([-n,-n,n], [1,1], 1).simplify()
[A103882(n) for n in range(31)] # G. C. Greubel, May 24 2023

a(n) = (A005258(n-1) + 3*A005258(n))/5 (Apéry numbers). - Mark van Hoeij, Jul 13 2010
n^2*(n-1)*(5*n-8)*a(n) = (n-1)*(55*n^3-143*n^2+102*n-24)*a(n-1) + n*(n-2)^2*(5*n-3)*a(n-2). - Alois P. Heinz, Jun 29 2015
a(n) ~ phi^(5*n + 3/2) / (2*Pi*5^(1/4)*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 21 2019
From Peter Bala, Jan 14 2020: (Start)
a(n) = Sum_{k = 0..n} C(n,k)^2*C(n+k-1,k). Cf. A005258.
For any prime p >= 5, a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for all positive integers n and k (follows from known supercongruences satisfied by the Apéry numbers A005258 - see Straub, Example 3.4). (End)
a(n) = hypergeometric([-n, -n, n], [1, 1], 1). - Peter Luschny, Jan 19 2020
From Peter Bala, Dec 19 2020: (Start)
a(n) = Sum_{k = 1..n} C(n,k)*C(n+k,k)*C(n-1,k-1) for n >= 1.
a(n) = [x^n] P(n, (1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. Cf. A156554. (End)
a(n) = Sum_{k = 0..n} binomial(2*n-k-1,n-k)*binomial(n,k)^2. Cf. A108628. - Peter Bala, Mar 24 2022
From Peter Bala, Apr 15 2022: (Start)
a(-n) = (-1)^n*A352654(n).
a(n) = [x^n*y^n*z^(n-1)] 1/(1 - x - y - z + x*z + y*z - x*y*z) for n >= 1.
a(n) = B(n,n,n-1) in the notation of Straub, see equation 24.
a(n) = [x^n*y^n*z^(n-1)] (x + y + z)^n*(x + y)^n*(y + z)^(n-1) for n >= 1. (End)
D-finite with recurrence 9*n^2*a(n) -3*(31*n^2-27*n+6)*a(n-1) -2*(37*n^2-138*n+108)*a(n-2) -(n-3)*(17*n-56)*a(n-3) -(n-4)^2*a(n-4) = 0. - R. J. Mathar, Aug 01 2022
a(n) = Sum_{k = 0..n} (-1)^(n+k) * binomial(n-1, n-k)*binomial(n+k, k)*binomial(n+k-1, k). - Peter Bala, Aug 13 2023
a(n) = Sum_{k = 0..n} (-1)^k * binomial(n+1, k)*binomial(2*n-k, n-k)^2. - Peter Bala, Oct 05 2024

a(0)=1 prepended by Alois P. Heinz, Jun 29 2015

A103884 Square array A(n,k) read by antidiagonals: row n gives coordination sequence for lattice C_n.

Original entry on oeis.org

1, 1, 8, 1, 18, 16, 1, 32, 66, 24, 1, 50, 192, 146, 32, 1, 72, 450, 608, 258, 40, 1, 98, 912, 1970, 1408, 402, 48, 1, 128, 1666, 5336, 5890, 2720, 578, 56, 1, 162, 2816, 12642, 20256, 14002, 4672, 786, 64, 1, 200, 4482, 27008, 59906, 58728, 28610, 7392, 1026, 72

Offset: 2

Ralf Stephan, Feb 20 2005

			Array begins:
  1,   8,    16,     24,      32,       40,        48, ... A022144;
  1,  18,    66,    146,     258,      402,       578, ... A010006;
  1,  32,   192,    608,    1408,     2720,      4672, ... A019560;
  1,  50,   450,   1970,    5890,    14002,     28610, ... A019561;
  1,  72,   912,   5336,   20256,    58728,    142000, ... A019562;
  1,  98,  1666,  12642,   59906,   209762,    596610, ... A019563;
  1, 128,  2816,  27008,  157184,   658048,   2187520, ... A019564;
  1, 162,  4482,  53154,  374274,  1854882,   7159170, ... A035746;
  1, 200,  6800,  97880,  822560,  4780008,  21278640, ... A035747;
  1, 242,  9922, 170610, 1690370, 11414898,  58227906, ... A035748;
  1, 288, 14016, 284000, 3281280, 25534368, 148321344, ... A035749;
  ...
Antidiagonals, T(n, k), begins as:
  1;
  1,   8;
  1,  18,   16;
  1,  32,   66,   24;
  1,  50,  192,  146,   32;
  1,  72,  450,  608,  258,   40;
  1,  98,  912, 1970, 1408,  402,  48;
  1, 128, 1666, 5336, 5890, 2720, 578, 56;

G. C. Greubel, Antidiagonals n = 2..50, flattened
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122 [cond-mat.stat-mech], 1997.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

Rows include A022144, A010006, A019560, A019561, A019562, A019563, A019564, A035746, A035747, A035748, A035749, A035750 - A035787.
Columns include A001105, A035598, A035600, A035602, A035604, A035606.
Main diagonal is in A103885.
Cf. A103881, A103993, A108998.

A103884:= func< n,k | k eq 0 select 1 else 2*(&+[2^j*Binomial(n-k,j+1)*Binomial(2*k-1,j) : j in [0..2*k-1]]) >;
[A103884(n,k): k in [0..n-2], n in [2..12]]; // G. C. Greubel, May 23 2023

nmin = 2; nmax = 11; t[n_, 0]= 1; t[n_, k_]:= 2n*Hypergeometric2F1[1-2k, 1-n, 2, 2]; tnk= Table[ t[n, k], {n, nmin, nmax}, {k, 0, nmax-nmin}]; Flatten[ Table[ tnk[[ n-k+1, k ]], {n, 1, nmax-nmin+1}, {k, 1, n} ] ] (* Jean-François Alcover, Jan 24 2012, after formula *)

def A103884(n,k): return 1 if k==0 else 2*sum(2^j*binomial(n-k,j+1)*binomial(2*k-1,j) for j in range(2*k))
flatten([[A103884(n,k) for k in range(n-1)] for n in range(2,13)]) # G. C. Greubel, May 23 2023

A(n,k) = Sum_{i=1..2*k} 2^i*C(n, i)*C(2*k-1, i-1), A(n,0) = 1 (array).
G.f. of n-th row: (Sum_{i=0..n} C(2*n, 2*i)*x^i)/(1-x)^n.
T(n, k) = A(n-k, k) (antidiagonals).
T(n, n-2) = A022144(n-2).
T(n, k) = 2*(n-k)*Hypergeometric2F1([1+k-n, 1-2*k], [2], 2), T(n, 0) = 1, for n >= 2, 0 <= k <= n-2. - G. C. Greubel, May 23 2023
From  Peter Bala, Jul 09 2023: (Start)
T(n,k) = [x^k] Chebyshev_T(n, (1 + x)/(1 - x)), where Chebyshev_T(n, x) denotes the n-th Chebyshev polynomial of the first kind.
T(n+1,k) = T(n+1,k-1)  + 2*T(n,k) + 2*T(n,k-1) + T(n-1,k) - T(n-1,k-1). (End)

Definition clarified by N. J. A. Sloane, May 25 2023

A008387 Coordination sequence for A_6 lattice.

Original entry on oeis.org

1, 42, 462, 2562, 9492, 27174, 65226, 137886, 264936, 472626, 794598, 1272810, 1958460, 2912910, 4208610, 5930022, 8174544, 11053434, 14692734, 19234194, 24836196, 31674678, 39944058, 49858158, 61651128, 75578370, 91917462, 110969082

Offset: 0

N. J. A. Sloane and J. H. Conway

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Row 6 of A103881.

[n eq 0 select 1 else 7*n*(11*n^4+35*n^2+14)/10: n in [0..50]]; // G. C. Greubel, May 26 2023

1, seq(7*n*(11*n^4+35*n^2+14)/10, n=1..40);

LinearRecurrence[{6,-15,20,-15,6,-1}, {1,42,462,2562,9492,27174,65226}, 30] (* Jean-François Alcover, Jan 07 2019 *)

[7*n*(11*n^4 +35*n^2 +14)/10 +int(n==0) for n in range(51)] # G. C. Greubel, May 26 2023

a(n) = S(n,6) = 7*n*(11*n^4 + 35*n^2 + 14)/10, with S(n,m) = Sum_{k=0..m} binomial(m,k)^2 * binomial(n-k+m-1, m-1), for n > 0, and a(0) = 1.
G.f.: (1+36*x+225*x^2+400*x^3+225*x^4+36*x^5+x^6)/(1-x)^6 = 1 + 42*x*(1+5*x+10*x^2+5*x^3+x^4)/(1-x)^6. - Colin Barker, Sep 26 2012
E.g.f.: 1 + (1/10)*x*(420 + 1890*x + 2170*x^2 + 770*x^3 + 77*x^4)*exp(x). - G. C. Greubel, May 26 2023

A008395 Coordination sequence for A_10 lattice.

Original entry on oeis.org

1, 110, 3080, 40370, 322190, 1815506, 7925720, 28512110, 88206140, 241925530, 601585512, 1379301990, 2953859370, 5968878630, 11472968760, 21114177018, 37403270520, 64062783510, 106481351240, 172295622730, 272125000774, 420487598410

Offset: 0

N. J. A. Sloane and J. H. Conway

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Row 10 of A103881.

[1] cat [11*n*(4199*n^8 +72930*n^6 +327327*n^4 +406120*n^2 +96624)/90720: n in [1..40]]; // G. C. Greubel, May 27 2023

a:= n-> `if`(n=0, 1, 46189/90720*n^9+26741/3024*n^7+
         171457/4320*n^5+111683/2268*n^3+7381/630*n):
seq(a(n), n=0..25);

a[n_]:= If[n==0, 1, 11n(4199n^8 +72930n^6 +327327n^4 +406120n^2 +96624)/90720];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jan 07 2019 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1}, {1,110,3080, 40370,322190,1815506,7925720,28512110,88206140,241925530,601585512}, 30] (* Harvey P. Dale, Nov 27 2019 *)

[11*n*(4199*n^8 +72930*n^6 +327327*n^4 +406120*n^2 +96624)//90720 +int(n==0) for n in range(41)] # G. C. Greubel, May 27 2023

a(n) = 46189/90720*n^9 +26741/3024*n^7 +171457/4320*n^5 +111683/2268*n^3 +7381/630*n for n >= 1.
Sum_{d=1}^10 C(11, d) C(m/2-1, d-1) C(10-d+m/2, m/2), where norm m is always even. (Serra-Sagrista)
G.f.: (1 +100*x +2025*x^2 +14400*x^3 +44100*x^4 +63504*x^5 +44100*x^6 +14400*x^7 +2025*x^8 +100*x^9 +x^10)/(1-x)^10. - Colin Barker, Sep 26 2012

A008389 Coordination sequence for A_7 lattice.

Original entry on oeis.org

1, 56, 812, 5768, 26474, 91112, 256508, 623576, 1356194, 2703512, 5025692, 8823080, 14768810, 23744840, 36881420, 55599992, 81659522, 117206264, 164826956, 227605448, 309182762, 413820584, 546468188, 712832792, 919453346

Offset: 0

N. J. A. Sloane and J. H. Conway

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Row 7 of A103881.

[1] cat [2 +n^2*(143*n^4 +770*n^2 +707)/30: n in [1..40]]; // G. C. Greubel, May 26 2023

1, seq(2 +n^2*(143*n^4 +770*n^2 +707)/30, n=1..50);

Table[n^2*(143*n^4 +770*n^2 +707)/30 +2 -Boole[n==0], {n,0,40}] (* G. C. Greubel, May 26 2023 *)

[2 +n^2*(143*n^4 +770*n^2 +707)/30 -int(n==0) for n in range(41)] # G. C. Greubel, May 26 2023

G.f.: (1+x)*(1+48*x+393*x^2+832*x^3+393*x^4+48*x^5+x^6)/(1-x)^7. - Colin Barker, Sep 26 2012
a(n) = 2 + n^2*(143*n^4 +770*n^2 +707)/30 with n>0, a(0)=1. - Bruno Berselli, Sep 26 2012
E.g.f.: -1 + (1/30)*(60 +1620*x +10530*x^2 +17490*x^3 +10065*x^4 +2145*x^5 +143*x^6)*exp(x). - G. C. Greubel, May 26 2023

A008391 Coordination sequence for A_8 lattice.

Original entry on oeis.org

1, 72, 1332, 11832, 66222, 271224, 889716, 2476296, 6077196, 13507416, 27717948, 53265960, 96900810, 168278760, 280819260, 452715672, 708113304, 1078467624, 1604095524, 2335932504, 3337508646, 4687156248, 6480461988, 8832976488, 11883194148, 15795816120

Offset: 0

N. J. A. Sloane and J. H. Conway

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Row 8 of A103881.

[1] cat [n*(715*n^6 + 6006*n^4 +10395*n^2 +3044)/280: n in [1..40]]; // G. C. Greubel, May 26 2023

1, seq(n*(715*n^6 + 6006*n^4 +10395*n^2 +3044)/280, n=1..40);

Join[{1},Table[143/56n^7+429/20n^5+297/8n^3+761/70n,{n,30}]] (* or *)
Join[{1},LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{72,1332,11832, 66222,271224,889716,2476296,6077196},30]] (* Harvey P. Dale, Mar 04 2012 *)

a[0]:1$
a[1]:72$
a[2]:1332$
a[3]:11832$
a[4]:66222$
a[5]:271224$
a[6]:889716$
a[7]:2476296$
a[8]:6077196$
a[n]:=8*a[n-1]-28*a[n-2]+ 56*a[n-3]- 70*a[n-4]+56*a[n-5]-28*a[n-6]+8*a[n-7]-a[n-8];
makelist(a[n],n,0,30); /* Martin Ettl, Oct 26 2012 */

[n*(715*n^6 + 6006*n^4 +10395*n^2 +3044)//280 +int(n==0) for n in range(41)] # G. C. Greubel, May 26 2023

a(n) = n*(715*n^6 + 6006*n^4 + 10395*n^2 + 3044)/280, a(0) = 1.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Harvey P. Dale, Mar 04 2012
G.f.: (1 + 64*x + 784*x^2 + 3136*x^3 + 4900*x^4 + 3136*x^5 + 784*x^6 + 64*x^7 + x^8)/(1-x)^8. - Colin Barker, Sep 26 2012

A008393 Coordination sequence for A_9 lattice.

Original entry on oeis.org

1, 90, 2070, 22530, 151560, 731502, 2777370, 8809110, 24314490, 60110030, 135916002, 285510150, 563873400, 1056789450, 1893408750, 3262336002, 5431848930, 8774904690, 13799638910, 21186110970, 31830097752, 46894786710

Offset: 0

N. J. A. Sloane and J. H. Conway

T. D. Noe, Table of n, a(n) for n = 0..1000
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Row 9 of A103881.

[1] cat [2 +11*n^2*(221*n^6 +2730*n^4 +7917*n^2 +5260)/2016: n in [1..40]]; // G. C. Greubel, May 27 2023

1, seq(2 +11*n^2*(221*n^6 +2730*n^4 +7917*n^2 +5260)/2016, n=1..40);

Table[11*n^2*(221*n^6 +2730*n^4 +7917*n^2 +5260)/2016 +2 -Boole[n==0], {n,0,40}] (* G. C. Greubel, May 27 2023 *)

[11*n^2*(221*n^6 +2730*n^4 +7917*n^2 +5260)//2016 +2 -int(n==0) for n in range(41)] # G. C. Greubel, May 27 2023

a(n) = 2 + 11*n^2*(221*n^6 + 2730*n^4 + 7917*n^2 + 5260)/2016, a(0) = 1.

G.f.: (1+x)*(1 + 80*x + 1216*x^2 + 5840*x^3 + 10036*x^4 + 5840*x^5 + 1216*x^6 + 80*x^7 + x^8)/(1-x)^9. - Colin Barker, Sep 26 2012

A157052 Number of integer sequences of length n+1 with sum zero and sum of absolute values 6.

Original entry on oeis.org

2, 18, 92, 340, 1010, 2562, 5768, 11832, 22530, 40370, 68772, 112268, 176722, 269570, 400080, 579632, 822018, 1143762, 1564460, 2107140, 2798642, 3670018, 4756952, 6100200, 7746050, 9746802, 12161268, 15055292, 18502290, 22583810, 27390112, 33020768, 39585282

Offset: 1

R. H. Hardin, Feb 22 2009

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A103881, A156554.

A157052:=n->n*(n + 1)*(n^4 + 2*n^3 + 11*n^2 + 10*n + 12)/36; seq(A157052(n), n=1..50); # Wesley Ivan Hurt, Feb 03 2014

Table[n(n+1)(n^4 +2n^3 +11n^2 +10n +12)/36, {n, 50}] (* Wesley Ivan Hurt, Feb 03 2014 *)

[n*(n+1)*(n^4 +2*n^3 +11*n^2 +10*n +12)/36 for n in (1..50)] # G. C. Greubel, Jan 23 2022

a(n) = T(n,3); T(n,k) = Sum_{i=1..n} binomial(n+1,i)*binomial(k-1,i-1)*binomial(n-i+k,k).
G.f.: 2*x*(1+2*x+4*x^2+2*x^3+x^4)/(1-x)^7. - Colin Barker, Mar 17 2012
a(n) = n*(n+1)*(n^4 +2*n^3 +11*n^2 +10*n +12)/36. - Bruno Berselli, Mar 17 2012
E.g.f.: (x/36)*(72 + 252*x + 264*x^2 + 108*x^3 + 18*x^4 + x^5)*exp(x). - G. C. Greubel, Jan 23 2022

A157068 Number of integer sequences of length n+1 with sum zero and sum of absolute values 38.

Original entry on oeis.org

2, 114, 3612, 80180, 1374690, 19234194, 227605448, 2335932504, 21186110970, 172295622730, 1271112537684, 8588601364668, 53573492643034, 310601807143530, 1683493452034320, 8573748834211984, 41210997268585158, 187693442844729174, 812839595630249540

Offset: 1

R. H. Hardin, Feb 22 2009

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (39,-741,9139,-82251,575757,-3262623,15380937, -61523748,211915132,-635745396,1676056044,-3910797436,8122425444,-15084504396, 25140840660,-37711260990,51021117810,-62359143990,68923264410,-68923264410, 62359143990,-51021117810,37711260990,-25140840660,15084504396,-8122425444, 3910797436,-1676056044,635745396,-211915132,61523748,-15380937,3262623,-575757, 82251,-9139,741,-39,1).

Cf. A103881, A156554.

A103881[n_, k_]:= (n+1)*Binomial[n+k-1,k]*HypergeometricPFQ[{1-n,-n,1-k}, {2, 1-n - k}, 1];
A157068[n_]:= A103881[n, 19];
Table[A157068[n], {n, 50}] (* G. C. Greubel, Jan 25 2022 *)

def A103881(n,k): return sum( binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k) for i in (0..n) )
def A157068(n): return A103881(n, 19)
[A157068(n) for n in (1..50)] # G. C. Greubel, Jan 25 2022

a(n) = T(n,19); T(n,k) = Sum_{i=1..n} binomial(n+1, i)*binomial(k-1, i-1)*binomial(n-i+k, k).
From G. C. Greubel, Jan 25 2022: (Start)
a(n) = (n+1)*binomial(n+18, 19)*Hypergeometric3F2([-18, -n, 1-n], [2, -n-18], 1).
a(n) = (35345263800/38!)*n*(n+1)*(778817392288148379660189696000000 + 1984223956005743569581323059200000*n + 3392214823876583668626122342400000*n^2 + 3227079634641025484578928197632000*n^3 + 2701114821085872776574503662387200*n^4 + 1477486663626167257723210367631360*n^5 + 794678697494482855499280703586304*n^6 + 289485264886342590944226501328896*n^7 + 112195641614805001937808853208064*n^8 + 29309532027252333838411983247872*n^9 + 8732100429652853130168723017472*n^10 + 1708566742801697011435174735872*n^11 + 408081704870580048838437092992*n^12 + 61460345467484307832839519168*n^13 + 12123027157132710911533327584*n^14 + 14298582910205269163512480328n^15 + 238150505845545646647030204*n^16 + 222226805381963345901159308n^17 + 3179819458407554816818235*n^18 + 235823049245552968253250*n^19 + 29394217444775030780985*n^20 + 17315150592375085755608n^21 + 190160234133314656140*n^22 + 8844512620448927880*n^23 + 864030358357843740*n^24 + 31339517913669420*n^25 + 2745580274521866*n^26 + 76036376515644*n^27 + 6015727425006*n^28 + 122857968168*n^29 + 8831668028*n^30 + 125358408*n^31 + 8231808*n^32 + 72522*n^33 + 4371*n^34 + 18*n^35 + n^36).
G.f.: 2*x*(1 + 18*x + 324*x^2 + 2754*x^3 + 23409*x^4 + 124848*x^5 + 665856*x^6 + 2496960*x^7 + 9363600*x^8 + 26218080*x^9 + 73410624*x^10 + 159056352*x^11 + 344622096*x^12 + 590780736*x^13 + 1012766976*x^14 + 1392554592*x^15 + 1914762564*x^16 + 2127513960*x^17 + 2363904400*x^18 + 2127513960*x^19 + 1914762564*x^20 + 1392554592*x^21 + 1012766976*x^22 + 590780736*x^23 + 344622096*x^24 + 159056352*x^25 + 73410624*x^26 + 26218080*x^27 + 9363600*x^28 + 2496960*x^29 + 665856*x^30 + 124848*x^31 + 23409*x^32 + 2754*x^33 + 324*x^34 + 18*x^35 + x^36)/(1-x)^39. (End)

cp's OEIS Frontend

A156554 The number of integer sequences of length d = 2n+1 such that the sum of the terms is 0 and the sum of the absolute values of the terms is d-1.

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A103882 a(n) = Sum_{i=0..n} C(n+1,i)*C(n-1,i-1)*C(2n-i,n).

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A103884 Square array A(n,k) read by antidiagonals: row n gives coordination sequence for lattice C_n.

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A008387 Coordination sequence for A_6 lattice.

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A008395 Coordination sequence for A_10 lattice.

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A008389 Coordination sequence for A_7 lattice.

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A103882 a(n) = Sum_{i=0..n} C(n+1,i)C(n-1,i-1)C(2n-i,n).