A022147 -id:A022147 - 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.]

A109001 Triangle, read by rows, where g.f. of row n equals the product of (1-x)^n and the g.f. of the coordination sequence for root lattice B_n, for n >= 0.

Original entry on oeis.org

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, 1, 91, 1281, 4795, 6155, 2793, 371, 1, 1, 120, 2268, 12040, 23750, 18888, 5852, 568, 1, 1, 153, 3732, 26628, 74574, 91118, 49380, 11124, 825, 1, 1, 190, 5805, 53544, 201810, 350196, 291410, 114600, 19629, 1150, 1

Offset: 0

Paul D. Hanna, Jun 17 2005

Compare to triangle A108558, where row n equals the (n+1)-th differences of the crystal ball sequence for D_n lattice.

			G.f.s of initial rows of square array A108998 are:
  (1),
  (1 + x)/(1-x),
  (1 + 6*x + x^2)/(1-x)^2;
  (1 + 15*x + 23*x^2 + x^3)/(1-x)^3;
  (1 + 28*x + 102*x^2 + 60*x^3 + x^4)/(1-x)^4.
Triangle begins:
  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;
  1,  91, 1281,  4795,  6155,  2793,   371,     1;
  1, 120, 2268, 12040, 23750, 18888,  5852,   568,   1;
  1, 153, 3732, 26628, 74574, 91118, 49380, 11124, 825, 1;

Muniru A Asiru, Rows n=0..100 of triangle, 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. A108998, A108999, A109000, 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).

Flat(List([0..10],n->List([0..n],k->Binomial(2*n+1,2*k)-2*n*Binomial(n-1,k-1)))); # Muniru A Asiru, Dec 14 2018

T[n_, k_] := Binomial[2n+1, 2k] - 2n * Binomial[n-1, k-1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Dec 14 2018 *)

T(n,k)=binomial(2*n+1,2*k)-2*n*binomial(n-1,k-1)

T(n, k) = C(2*n+1, 2*k) - 2*n*C(n-1, k-1).
Row sums are 2^n*(2^n - n) for n >= 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.]

A103883 Square array A(n,k) read by antidiagonals: coordination sequence for lattice B_n.

Original entry on oeis.org

1, 1, 8, 1, 18, 16, 1, 32, 74, 24, 1, 50, 224, 170, 32, 1, 72, 530, 768, 306, 40, 1, 98, 1072, 2562, 1856, 482, 48, 1, 128, 1946, 6968, 8130, 3680, 698, 56, 1, 162, 3264, 16394, 28320, 20082, 6432, 954, 64, 1, 200, 5154, 34624, 83442, 85992, 42130, 10304, 1250, 72

Offset: 2

Ralf Stephan, Feb 20 2005

			Array, A(n, k), begins:
  1,   8,    16,     24,      32,       40,        48, ... A022144;
  1,  18,    74,    170,     306,      482,       698, ... A022145;
  1,  32,   224,    768,    1856,     3680,      6432, ... A022146;
  1,  50,   530,   2562,    8130,    20082,     42130, ... A022147;
  1,  72,  1072,   6968,   28320,    85992,    214864, ... A022148;
  1,  98,  1946,  16394,   83442,   307314,    907018, ... A022149;
  1, 128,  3264,  34624,  216448,   954880,   3301952, ... A022150;
  1, 162,  5154,  67266,  507906,  2653346,  10666146, ... A022151;
  1, 200,  7760, 122264, 1099040,  6728168,  31208560, ... A022152;
  1, 242, 11242, 210474, 2224178, 15804866,  83999962, ... A022153;
  1, 288, 15776, 346304, 4254912, 34792672, 210482016, ... A022154;
  ...
Antidiagonals, T(n, k), begin as:
  1;
  1,   8;
  1,  18,   16;
  1,  32,   74,    24;
  1,  50,  224,   170,    32;
  1,  72,  530,   768,   306,    40;
  1,  98, 1072,  2562,  1856,   482,   48;
  1, 128, 1946,  6968,  8130,  3680,  698,  56;
  1, 162, 3264, 16394, 28320, 20082, 6432, 954, 64;

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, A022145, A022146, A022147, A022148, A022149, A022150, A022151, A022152, A022153, A022154.
Columns include A001105.
Cf. A103881, A103884.

A103883:= func< n,k | (&+[Binomial(n-j-1,n-k-1)*(Binomial(2*n-2*k+1,2*j) - 2*j*Binomial(n-k,j)) : j in [0..k]]) >;
[A103883(n,k): k in [0..n-2], n in [2..14]]; // G. C. Greubel, May 24 2023

offset = 2;
T[n_, k_] := SeriesCoefficient[Sum[(Binomial[2n + 1, 2i] - 2i Binomial[n, i]) x^i, {i, 0, n}]/(1 - x)^n, {x, 0, k}];
Table[T[n - k, k], {n, offset, 11}, {k, 0, n - offset}] // Flatten (* Jean-François Alcover, Feb 13 2019 *)

def A103883(n,k): return sum(binomial(n-j-1,n-k-1)*(binomial(2*n-2*k+1,2*j) - 2*j*binomial(n-k,j)) for j in range(k+1))
flatten([[A103883(n,k) for k in range(n-1)] for n in range(2,15)]) # G. C. Greubel, May 24 2023

G.f. of n-th row: (Sum_{i=0..n} (C(2n+1, 2*i) - 2*i*C(n, i))*x^i)/(1-x)^n.
From G. C. Greubel, May 24 2023: (Start)
A(n, k) = Sum_{j=0..k} binomial(n+k-j-1, n-1)*(binomial(2*n+1, 2*j) - 2*j*binomial(n, j)) (array).
T(n, k) = Sum_{j=0..k} binomial(n-j-1, n-k-1)*(binomial(2*n-2*k+1, 2*j) - 2*j*binomial(n-k, j)) (antidiagonals). (End)

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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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A109001 Triangle, read by rows, where g.f. of row n equals the product of (1-x)^n and the g.f. of the coordination sequence for root lattice B_n, for n >= 0.

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A103883 Square array A(n,k) read by antidiagonals: coordination sequence for lattice B_n.

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