A109001 -id:A109001 - cp's OEIS frontend

A108998 Square array, read by antidiagonals, where row n equals the coordination sequence of B_n lattice, for n >= 0.

1, 1, 0, 1, 2, 0, 1, 8, 2, 0, 1, 18, 16, 2, 0, 1, 32, 74, 24, 2, 0, 1, 50, 224, 170, 32, 2, 0, 1, 72, 530, 768, 306, 40, 2, 0, 1, 98, 1072, 2562, 1856, 482, 48, 2, 0, 1, 128, 1946, 6968, 8130, 3680, 698, 56, 2, 0, 1, 162, 3264, 16394, 28320, 20082, 6432, 954, 64, 2, 0

Offset: 0

Paul D. Hanna, Jun 17 2005

Compare with A108553, where row n equals the crystal ball sequence for D_n lattice.

			Square array begins:
  1,  0,    0,     0,     0,      0,      0,      0, ...
  1,  2,    2,     2,     2,      2,      2,      2, ...
  1,  8,   16,    24,    32,     40,     48,     56, ...
  1, 18,   74,   170,   306,    482,    698,    954, ...
  1, 32,  224,   768,  1856,   3680,   6432,  10304, ...
  1, 50,  530,  2562,  8130,  20082,  42130,  78850, ...
  1, 72, 1072,  6968, 28320,  85992, 214864, 467544, ...
  1, 98, 1946, 16394, 83442, 307314, 907018, ...
Product of the g.f. of row n and (1-x)^n generates the rows of triangle A109001:
  1;
  1,  1;
  1,  6,   1;
  1, 15,  23,    1;
  1, 28, 102,   60,    1;
  1, 45, 290,  402,  125,   1;
  1, 66, 655, 1596, 1167, 226, 1; ...

Muniru A Asiru, Rows n=0..110 of antidiagonals, flattened
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.

Cf. A108999 (main diagonal), A109000 (antidiagonal sums), A109001, A022144 (row 2), A022145 (row 3), A022146 (row 4), A022147 (row 5), A022148 (row 6), A022149 (row 7), A022150 (row 8), A022151 (row 9), A022152 (row 10), A022153 (row 11), A022154 (row 12).

T(n,k)=if(n<0 || k<0,0,sum(j=0,k, binomial(n+k-j-1,k-j)*(binomial(2*n+1,2*j)-2*n*binomial(n-1,j-1))))

T(n, k) = Sum_{j=0..k} C(n+k-j-1, k-j)*(C(2*n+1, 2*j)-2*n*C(n-1, j-1)) for n >= k >= 0.

G.f. for coordination sequence of B_n lattice: ((Sum_{i=0..n} binomial(2*n+1, 2*i)*z^i)-2*n*z*(1+z)^(n-1))/(1-z)^n. [Bacher et al.]

A108999 Main diagonal of square array A108998, in which row n equals the coordination sequence of B_n lattice.

Original entry on oeis.org

1, 2, 16, 170, 1856, 20082, 214864, 2282394, 24165120, 255708578, 2708805776, 28752157898, 305908697152, 3262741154194, 34882914424528, 373781033269306, 4013444615232512, 43174945822078530, 465247083731404048

Offset: 0

Paul D. Hanna, Jun 17 2005

Compare to diagonal A108554 of square array A108553, in which row n equals the crystal ball sequence for D_n lattice.

Muniru A Asiru, Table of n, a(n) for n = 0..900

Cf. A108998, A109000, A109001.

List([0..20],n->Sum([0..n],j->Binomial(2*n-j-1,n-j)*(Binomial(2*n+1,2*j)-2*n*Binomial(n-1,j-1)))); # Muniru A Asiru, Nov 21 2018

a[n_]:= Sum[Binomial[2*n-j-1, n-j]*(Binomial[2*n+1, 2*j] - 2*n*Binomial[n-1, j-1]), {j,0,n}]; Array[a, 20, 0] (* Stefano Spezia, Nov 21 2018 *)

{a(n)=sum(j=0,n, binomial(2*n-j-1,n-j)*(binomial(2*n+1,2*j)-2*n*binomial(n-1,j-1)))}

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

a(n) ~ phi^(5*n+1) / (2*5^(1/4)*sqrt(Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 31 2025

A109000 Antidiagonal sums of square array A108998, in which row n equals the coordination sequence of B_n lattice.

Original entry on oeis.org

1, 1, 3, 11, 37, 133, 479, 1719, 6121, 21609, 75675, 263171, 909899, 3130963, 10730891, 36639987, 124528283, 420319907, 1403656123, 4615627555, 14868713515, 46702912307, 142489152555, 421113970835, 1203581558011

Offset: 0

Paul D. Hanna, Jun 17 2005

nonn

Limit a(n+1)/a(n) ~ 3.3829757679..., real root of cubic (1+x+3*x^2-x^3). Compare to antidiagonal sums A108555 of square array A108553, in which row n equals the crystal ball sequence for D_n lattice.

Cf. A108998, A108999, A109001.

{a(n)=sum(k=0,n,sum(j=0,k, binomial(n-j-1,k-j)*(binomial(2*n-2*k+1,2*j)-2*(n-k)*binomial(n-k-1,j-1))))}

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

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A108998 Square array, read by antidiagonals, where row n equals the coordination sequence of B_n lattice, for n >= 0.

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A108999 Main diagonal of square array A108998, in which row n equals the coordination sequence of B_n lattice.

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A109000 Antidiagonal sums of square array A108998, in which row n equals the coordination sequence of B_n lattice.

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