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.

Previous Showing 21-28 of 28 results.

A218073 Number of profiles in domino tiling of a 2*n checkboard.

Original entry on oeis.org

0, 1, 2, 9, 12, 50, 60, 245, 280, 1134, 1260, 5082, 5544, 22308, 24024, 96525, 102960, 413270, 437580, 1755182, 1847560, 7407036, 7759752, 31097794, 32449872, 130007500, 135207800, 541574100, 561632400, 2249204040, 2326762800, 9316746045, 9617286240, 38504502630
Offset: 0

Views

Author

Michel Marcus, Oct 20 2012

Keywords

Links

Crossrefs

Cf. A005430 (bisection).

Programs

  • Maple
    a:= proc(n) option remember;
      `if`(n<3, n, (n*(5-7*n)*a(n-1) +4*(n-2)*(7*n+16)*a(n-3)
      +(24-12*n+172*n^2)*a(n-2))/ ((n+1)*(43*n-89)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Oct 20 2012
  • Mathematica
    a[n_] := n/2*Binomial[n + Mod[n, 2], (n + Mod[n, 2])/2]; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Feb 22 2013, after Joerg Arndt *)
  • Maxima
    a[0]:0$a[1]:1$a[2]:2$
a[n]:=(n*(5-7*n)*a[n-1] +4*(n-2)*(7*n+16)*a[n-3]+(24-12*n+172*n^2)*a[n-2])/ ((n+1)*(43*n-89))$
makelist(a[n] ,n,0,40); /* Martin Ettl, Oct 21 2012 */
  • PARI
    a(n) = n/2 * binomial(n+(n%2),(n+n%2)/2); /* Joerg Arndt, Oct 21 2012 */

Formula

If n is even, a(n) = binomial(n, n/2)*n/2.
If n is odd, a(n) = binomial(n + 1, (n + 1)/2)*n/2.

A287703 Triangle read by rows, numerators of T(n,k) = (-1)^n*binomial(n-1,k)*Bernoulli(n+k)/ (n+k) for n>=1, 0<=k<=n-1.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, -1, 0, 1, 0, 0, -1, 0, 1, 0, 1, 0, -1, 0, 5, 0, 0, 1, 0, -5, 0, 691, 0, -1, 0, 7, 0, -691, 0, 7, 0, 0, -2, 0, 691, 0, -14, 0, 3617, 0, 1, 0, -691, 0, 21, 0, -25319, 0, 43867, 0, 0, 691, 0, -10, 0, 75957, 0, -438670, 0, 174611, 0
Offset: 1

Views

Author

Peter Luschny, Jun 21 2017

Keywords

Comments

For the rational triangle the reciprocals of the row sums are the Apéry numbers A005430.

Examples

			The rational triangle starts (with row sums at the end of the line):
1: [1/2], 1/2
2: [1/12, 0], 1/12
3: [0, 1/60, 0], 1/60
4: [-1/120, 0, 1/84, 0], 1/280
5: [0, -1/63, 0, 1/60, 0], 1/1260
6: [1/252, 0, -1/24, 0, 5/132, 0], 1/5544
7: [0, 1/40, 0, -5/33, 0, 691/5460, 0], 1/24024
8: [-1/240, 0, 7/44, 0, -691/936, 0, 7/12, 0], 1/102960
9: [0, -2/33, 0, 691/585, 0, -14/3, 0, 3617/1020, 0], 1/437580
The numerators of the triangle are:
1: [ 1]
2: [ 1,  0]
3: [ 0,  1,  0]
4: [-1,  0,  1,   0]
5: [ 0, -1,  0,   1,    0]
6: [ 1,  0, -1,   0,    5,   0]
7: [ 0,  1,  0,  -5,    0, 691, 0]
8: [-1,  0,  7,   0, -691,   0, 7,   0]
9: [ 0, -2,  0, 691,    0, -14, 0, 3617, 0]

Crossrefs

Cf. A005430 (Apéry), A287704 (denominators).

Programs

  • Maple
    T := (n,k) -> numer((-1)^n*binomial(n-1,k)*bernoulli(k+n)/(k+n)):
for n from 1 to 9 do seq(T(n,k), k=0..n-1) od;
  • Mathematica
    T[n_, k_]:=Numerator[(-1)^n*Binomial[n - 1, k] BernoulliB[k + n]/(k + n)]; Table[T[n, k], {n, 11}, {k, 0, n - 1}]//Flatten (* Indranil Ghosh, Jul 27 2017 *)
  • PARI
    T(n, k) = numerator((-1)^n*binomial(n-1,k)*bernfrac(k+n)/(k+n));
tabl(nn) = for (n=1, nn, for (k=0, n-1, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 28 2017

Formula

A005430(n) = 1 / (Sum_{k=0..n-1} T(n,k)) for n>=1.

A289718 The order of the semigroup of orientation-preserving or reserving full transformations of n elements.

Original entry on oeis.org

1, 4, 27, 180, 1015, 5028, 23051, 101272, 434835, 1843320, 7753471, 32440884, 135195307, 561615460, 2326740315, 9617256944, 39671268187, 163352387952, 671559953015, 2756930503420, 11303415274179, 46290177094172, 189368906605747, 773942488240920, 3160265160762575, 12893881856438128, 52567364492251191, 214163336821005012, 871950728486358835, 3547937446945462500
Offset: 1

Views

Author

R. J. Mathar, Sep 02 2017

Keywords

Links

Programs

  • Maple
    A289718 := proc(n)
     n*binomial(2*n,n)-n^2*(n^2-2*n+5)/2+n ;
end proc:

Formula

a(n) = A005430(n)-n*A006004(n).

A337994 T(n, k) = (k*(2*k+2)*(2*k+1)*(2*n-1)*binomial(2*(n-1),n-1))/(n*(n+1)*(n+2)) for n, k > 0 and T(0, 0) = 1. Triangle read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, 0, 3, 15, 0, 6, 30, 84, 0, 14, 70, 196, 420, 0, 36, 180, 504, 1080, 1980, 0, 99, 495, 1386, 2970, 5445, 9009, 0, 286, 1430, 4004, 8580, 15730, 26026, 40040, 0, 858, 4290, 12012, 25740, 47190, 78078, 120120, 175032
Offset: 0

Views

Author

Peter Luschny, Nov 01 2020

Keywords

Comments

T(n, k) is divisible by A099398(n) for all 0 <= k <= n.

Examples

			Triangle starts:
[0] 1
[1] 0, 2
[2] 0, 3,    15
[3] 0, 6,    30,    84
[4] 0, 14,   70,    196,   420
[5] 0, 36,   180,   504,   1080,  1980
[6] 0, 99,   495,   1386,  2970,  5445,   9009
[7] 0, 286,  1430,  4004,  8580,  15730,  26026,  40040
[8] 0, 858,  4290,  12012, 25740, 47190,  78078,  120120, 175032
[9] 0, 2652, 13260, 37128, 79560, 145860, 241332, 371280, 541008, 755820

Crossrefs

Cf. A119578 (row sums), (-1)^n*A005430 (alternating row sums), A002740 (main diagonal), A007054 (col 1), A099398 (universal divisor), A000108 (Catalan).

Programs

  • Maple
    T := proc(n, k) if n = 0 then 1 else
(k*(2*k+2)*(2*k+1)*(2*n-1)*binomial(2*(n-1), n-1))/(n*(n+1)*(n+2)) fi end:
# Recursive:
CatalanNumber := n -> binomial(2*n, n)/(n+1):
T := proc(n, k) option remember; if k=0 then k^n elif k=n then CatalanNumber(n+1)* binomial(n+1, 2) else (4 - 10/(n + 2))*T(n-1, k) fi end:
seq(seq(T(n, k), k=0..n), n=0..9);
  • Mathematica
    T[n_, k_] := If[n == 0, 1, (k (2k + 2)(2k + 1)(2n - 1) CatalanNumber[n-1])/((n + 1) (n + 2))]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

Formula

Let t(n) denote the triangular numbers and C(n) the Catalan numbers.
T(n, k) = k*(2*n - 1)*(t(2*k + 1)/t(n + 1))*C(n - 1) for n, k > 0.
T(n, k) = k^n if k = 0; if k = n then C(n+1)*t(n+1); else T(n-1, k)*(4-10/(n+2)).

A119552 Binomial(binomial(2*n,n)*n,n).

Original entry on oeis.org

1, 2, 66, 34220, 250654530, 26255517781752, 40219012695854521452, 915602638690858051785326904, 313119156483427472599890586326722370, 1620758891081732168210880572699959520551984700
Offset: 0

Views

Author

Zerinvary Lajos, May 30 2006

Keywords

Crossrefs

Cf. A005430.

Programs

  • Maple
    [seq (binomial(binomial(2*n,n)*n,n),n=0..10)];

A131928 a(n) = phi(binomial(2*n,n)*n).

Original entry on oeis.org

0, 1, 4, 16, 96, 288, 1440, 5760, 23040, 92160, 552960, 1658880, 7299072, 36495360, 109486080, 510935040, 2043740160, 7664025600, 28740096000, 114960384000, 459841536000, 1839366144000, 8583708672000, 38626689024000
Offset: 0

Views

Author

Zerinvary Lajos, Oct 05 2007

Keywords

Crossrefs

Cf. A000010, A005430 Apery numbers: n*C(2n, n).

Programs

  • Maple
    with(numtheory):with(combinat):a:=n->phi(binomial(2*n,n)*n): seq(a(n), n=0..25);
  • Mathematica
    Table[EulerPhi[Binomial[2n,n]n],{n,0,30}] (* Harvey P. Dale, Apr 15 2012 *)

A157077 Triangle read by rows, coefficients of the Legendre polynomials P(n, x) times 2^n: T(n, k) = 2^n * [x^k] P(n, x), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, -2, 0, 6, 0, -12, 0, 20, 6, 0, -60, 0, 70, 0, 60, 0, -280, 0, 252, -20, 0, 420, 0, -1260, 0, 924, 0, -280, 0, 2520, 0, -5544, 0, 3432, 70, 0, -2520, 0, 13860, 0, -24024, 0, 12870, 0, 1260, 0, -18480, 0, 72072, 0, -102960, 0, 48620, -252, 0, 13860, 0, -120120, 0, 360360, 0, -437580, 0, 184756
Offset: 0

Views

Author

Roger L. Bagula, Feb 22 2009

Keywords

Examples

			The term order is Q(x) = a_0 + a_1*x + ... + a_n*x^n. The coefficients of the first few polynomials in this order are:
{1},
{0, 2},
{-2, 0, 6},
{0, -12, 0, 20},
{6, 0, -60, 0, 70},
{0, 60, 0, -280, 0, 252},
{-20, 0, 420, 0, -1260, 0, 924},
{0, -280, 0, 2520, 0, -5544, 0, 3432},
{70, 0, -2520, 0, 13860, 0, -24024, 0, 12870},
{0, 1260, 0, -18480, 0, 72072, 0, -102960, 0, 48620},
{-252, 0, 13860, 0, -120120, 0, 360360, 0, -437580, 0, 184756}.
.
From _Jon E. Schoenfield_, Jul 04 2022: (Start)
As a right-aligned triangle:
                                                             1;
                                                     0,      2;
                                                 -2, 0,      6;
                                         0,     -12, 0,     20;
                                      6, 0,     -60, 0,     70;
                              0,     60, 0,    -280, 0,    252;
                         -20, 0,    420, 0,   -1260, 0,    924;
                  0,    -280, 0,   2520, 0,   -5544, 0,   3432;
              70, 0,   -2520, 0,  13860, 0,  -24024, 0,  12870;
        0,  1260, 0,  -18480, 0,  72072, 0, -102960, 0,  48620;
  -252, 0, 13860, 0, -120120, 0, 360360, 0, -437580, 0, 184756. (End)

Links

Crossrefs

Cf. A100258, A126869, A000984, A005430, A000079, A069835, A084773, A098269, A098270.

Programs

  • Maple
    with(orthopoly):with(PolynomialTools): seq(print(CoefficientList (2^n*P(n, x), x,termorder=forward)),n=0..10); # Peter Luschny, Dec 18 2014
  • Mathematica
    Table[CoefficientList[2^n*LegendreP[n, x], x], {n, 0, 10}]; Flatten[%]
  • PARI
    tabl(nn) = for (n=0, nn, print(Vecrev(2^n*pollegendre(n)))); \\ Michel Marcus, Dec 18 2014
  • Sage
    def A157077_row(n):
    if n==0: return [1]
    T = [c[0] for c in (2^n*gen_legendre_P(n, 0, x)).coefficients()]
    return [0 if is_odd(n+k) else T[k//2] for k in (0..n)]
for n in range(9): print(A157077_row(n)) # Peter Luschny, Dec 19 2014

Formula

Row sums are 2^n.
From Peter Luschny, Dec 19 2014: (Start)
T(n,0) = A126869(n).
T(n,n) = A000984(n).
T(n,1) = (-1)^floor(n/2)*A005430(floor(n/2)+1) if n is odd else 0.
Let Q(n, x) = 2^n*P(n, x).
Q(n,0) = (-1)^floor(n/2)*A126869(floor(n/2)) if n is even else 0.
Q(n,1) = A000079(n).
Q(n,2) = A069835(n).
Q(n,3) = A084773(n).
Q(n,4) = A098269(n).
Q(n,5) = A098270(n). (End)
From Fabián Pereyra, Jun 30 2022: (Start)
n*T(n,k) = 2*(2*n-1)*T(n-1,k-1) - 4*(n-1)*T(n-2,k).
T(n,k) = (-1)^floor((n-k)/2)*binomial(n+k,k)*binomial(n,floor((n-k)/2))*(1+(-1)^(n-k))/2.
O.g.f.: A(x,t) = 1/sqrt(1-4*x*t+4*x^2) = 1 + (2*t)*x + (-2+6*t^2)*x^2 + (-12*t+20*t^3)*x^3 + (6-60*t^2+70*t^4)*x^4 + .... (End)

Extensions

Name clarified and edited by Peter Luschny, Dec 18 2014

A353596 Triangle read by rows, T(n, k) = [x^k] (-2)^n*GegenbauerC(n, -1/2, x).

Original entry on oeis.org

1, 0, 2, 2, 0, -2, 0, -4, 0, 4, -2, 0, 12, 0, -10, 0, 12, 0, -40, 0, 28, 4, 0, -60, 0, 140, 0, -84, 0, -40, 0, 280, 0, -504, 0, 264, -10, 0, 280, 0, -1260, 0, 1848, 0, -858, 0, 140, 0, -1680, 0, 5544, 0, -6864, 0, 2860, 28, 0, -1260, 0, 9240, 0, -24024, 0, 25740, 0, -9724
Offset: 0

Views

Author

Peter Luschny, May 06 2022

Keywords

Examples

			Triangle T(n, k) starts:
[0]   1;
[1]   0,   2;
[2]   2,   0,  -2;
[3]   0,  -4,   0,     4;
[4]  -2,   0,  12,     0,   -10;
[5]   0,  12,   0,   -40,     0,   28;
[6]   4,   0, -60,     0,   140,    0,  -84;
[7]   0, -40,   0,   280,     0, -504,    0,   264;
[8] -10,   0, 280,     0, -1260,    0, 1848,     0, -858;
[9]   0, 140,   0, -1680,     0, 5544,    0, -6864,    0, 2860;
.
Unsigned antidiagonals |T(n+k, n-k)|:
[0]  1;
[1]  2,   2;
[2]  2,   4,    2;
[3]  4,  12,   12,    4;
[4] 10,  40,   60,   40,   10;
[5] 28, 140,  280,  280,  140,  28;
[6] 84, 504, 1260, 1680, 1260, 504, 84;

Crossrefs

Diagonals (also divided by 2^k): A002420 (main), A028329 (main-2) (also A000984), A005430 (main-4) (also A002457), A002802 (main-6).

Programs

  • Maple
    g := n -> (-2)^n*GegenbauerC(n, -1/2, x):
seq(print(seq(coeff(simplify(g(n)), x, k), k = 0..n)), n = 0..9);
  • Mathematica
    s={}; For[n=0,n<11,n++,For[k=0,kDetlef Meya, Oct 03 2023 *)
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