cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Paul D. Hanna, Jun 17 2005

Keywords

Comments

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

Links

Crossrefs

Cf. A108998, A109000, A109001.

Programs

  • GAP
    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
  • Mathematica
    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 *)
  • PARI
    {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)))}

Formula

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

Views

Author

Paul D. Hanna, Jun 17 2005

Keywords

Comments

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.

Crossrefs

Cf. A108998, A108999, A109001.

Programs

  • PARI
    {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))))}

Formula

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)).

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

Views

Author

Ralf Stephan, Feb 20 2005

Keywords

Examples

			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;

Links

Crossrefs

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.

Programs

  • Magma
    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
  • Mathematica
    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 *)
  • SageMath
    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

Formula

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)

Extensions

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

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

Views

Author

Paul D. Hanna, Jun 17 2005

Keywords

Comments

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

Examples

			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;

Links

Crossrefs

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).

Programs

  • GAP
    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
  • Mathematica
    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 *)
  • PARI
    T(n,k)=binomial(2*n+1,2*k)-2*n*binomial(n-1,k-1)

Formula

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.]
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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