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 1-3 of 3 results.

A123162 Triangle read by rows: T(n,k) = binomial(2*n - 1, 2*k - 1) for 0 < k <= n and T(n,0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 10, 1, 1, 7, 35, 21, 1, 1, 9, 84, 126, 36, 1, 1, 11, 165, 462, 330, 55, 1, 1, 13, 286, 1287, 1716, 715, 78, 1, 1, 15, 455, 3003, 6435, 5005, 1365, 105, 1, 1, 17, 680, 6188, 19448, 24310, 12376, 2380, 136, 1, 1, 19, 969, 11628, 50388, 92378, 75582, 27132, 3876, 171, 1
Offset: 0

Views

Author

Roger L. Bagula, Oct 02 2006

Keywords

Examples

			Triangle begins:
     1;
     1,  1;
     1,  3,   1;
     1,  5,  10,    1;
     1,  7,  35,   21,    1;
     1,  9,  84,  126,   36,    1;
     1, 11, 165,  462,  330,   55,    1;
     1, 13, 286, 1287, 1716,  715,   78,  1;
     1, 15, 455, 3003, 6435, 5005, 1365, 105, 1;
     ...

Links

Crossrefs

Cf. A002458, A007318, A034867, A103327, A122366, A123166, A259557.

Programs

  • GAP
    Flat(Concatenation([1],List([1..10],n->Concatenation([1],List([1..n],m->Binomial(2*n-1,2*m-1)))))); # Muniru A Asiru, Oct 11 2018
  • Magma
    A123162:= func< n,k | k eq 0 select 1 else Binomial(2*n-1, 2*k-1) >;
[A123162(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 18 2023
  • Mathematica
    T[n_, k_]= If [k==0, 1, Binomial[2*n-1, 2*k-1]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten
  • Maxima
    T(n, k) := if k = 0 then 1 else binomial(2*n - 1, 2*k - 1)$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, n))$ /* Franck Maminirina Ramaharo, Oct 10 2018 */
  • SageMath
    def A123162(n,k): return binomial(2*n-1, 2*k-1) + int(k==0)
flatten([[A123162(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 01 2022

Formula

From Paul Barry, May 26 2008: (Start)
T(n,k) = binomial(2*n - 1, 2*k - 1) + 0^k.
Column k has g.f. (x^k/(1 - x)^(2*k + 0^k))*Sum_{j=0..k} binomial(2*k, 2*j)*x^j. (End)
From Franck Maminirina Ramaharo, Oct 10 2018: (Start)
Row n = coefficients in the expansion of ((x + sqrt(x))*(sqrt(x) - 1)^(2*n) + (x - sqrt(x))*(sqrt(x) + 1)^(2*n) + 2*x - 2)/(2*x - 2).
G.f.: (1 - (2 + x)*y + (1 - 2*x)*y^2 - (x - x^2)*y^3)/(1 - (3 + 2*x)*y + (3 + x^2)*y^2 - (1 - 2*x + x^2)*y^3).
E.g.f.: ((x + sqrt(x))*exp(y*(sqrt(x) - 1)^2) + (x - sqrt(x))*exp(y*(sqrt(x) + 1)^2) + (2*x - 2)*exp(y) - 2*x)/(2*x - 2). (End)
From G. C. Greubel, Jul 18 2023: (Start)
Sum_{k=0..n} T(n,k) = A123166(n).
T(n, n-1) = (n-1)*T(n, 1), n >= 2.
T(2*n, n) = A259557(n).
T(2*n+1, n+1) = A002458(n). (End)

Extensions

Edited by N. J. A. Sloane, Oct 04 2006
Partially edited and offset corrected by Franck Maminirina Ramaharo, Oct 10 2018

A154221 A simple Pascal-like triangle.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 5, 1, 1, 9, 9, 9, 9, 1, 1, 17, 17, 17, 17, 17, 1, 1, 33, 33, 33, 33, 33, 33, 1, 1, 65, 65, 65, 65, 65, 65, 65, 1, 1, 129, 129, 129, 129, 129, 129, 129, 129, 1, 1, 257, 257, 257, 257, 257, 257, 257, 257, 257, 1
Offset: 0

Views

Author

Paul Barry, Jan 05 2009

Keywords

Comments

First column is A094373. Central coefficients are A123166.
Row sums are A154222. Diagonal sums are A154223.

Examples

			Triangle begins
1,
1, 1,
1, 2, 1,
1, 3, 3, 1,
1, 5, 5, 5, 1,
1, 9, 9, 9, 9, 1,
1, 17, 17, 17, 17, 17, 1,
1, 33, 33, 33, 33, 33, 33, 1,
1, 65, 65, 65, 65, 65, 65, 65, 1

Links

Programs

  • Magma
    /* As triangle */ [[1+(2^(k-1)-0^k/2)*(2^(n-k-1)-0^(n-k)/2): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 06 2016
  • Maple
    A154221 := proc(n,k)
    local f1,f2 ;
    f1 := 2^(k-1) ;
    if k = 0 then
        f1 := f1-1/2 ;
    end if;
    f2 := 2^(n-k-1) ;
    if n-k = 0 then
        f2 := f2-1/2 ;
    end if;
    1+f1*f2 ;
end proc:
seq(seq(A154221(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Feb 05 2015
  • Mathematica
    f[n_, k_] := 1 + (1/4)*(2^(k) - 0^k)*(2^(n - k) - 0^(n - k)); Table[f[n, i], {n, 0, 49}, {i, 0, n}] // Flatten (* G. C. Greubel, Sep 06 2016 *)

Formula

T(n,k)= 1 + (2^(k-1)-0^k/2)*(2^(n-k-1)-0^(n-k)/2).

A229135 n * (2 + 2^(2*n - 1)).

Original entry on oeis.org

0, 4, 20, 102, 520, 2570, 12300, 57358, 262160, 1179666, 5242900, 23068694, 100663320, 436207642, 1879048220, 8053063710, 34359738400, 146028888098, 618475290660, 2611340116006, 10995116277800, 46179488366634, 193514046488620, 809240558043182, 3377699720527920
Offset: 0

Views

Author

Paul Curtz, Sep 15 2013

Keywords

Comments

a(n) mod 9 is periodic: repeat 0, 4, 2, 3, 7, 5, 6, 1, 8.
b(n) = a(n) - n = 0, 3, 18, 99, 516, 2565, 12294,... = A215149(2n)/2.

Examples

			a(0)=0*(2+1/2)=0, a(1)=1*(2+2)=4, a(2)=2*(2+8)=20, a(3)=3*(2+32)=102, a(4)=4*(2+128)=520, a(5)=5*(2+512)=2570.

Links

Crossrefs

Cf. A005843, A052539, A081294, A215149, A228827, A000367, A002445, A123166.

Programs

Formula

a(n) = n + A215149(2n)/2.
a(n) = (A228827(n) - A000367(n))/A002445(n).
a(n) = 2*n*A123166(n).
G.f.: (34*x^3 - 20*x^2 + 4*x)/((1-x)^2*(1-4*x)^2). - Ralf Stephan, Sep 20 2013

Extensions

Better definition by Michel Marcus.
More terms from Vincenzo Librandi, Sep 20 2013
Showing 1-3 of 3 results.

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

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