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 41-45 of 45 results.

A129690 A129688 * A007318.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 3, 5, 3, 1, 5, 8, 8, 4, 1, 5, 13, 16, 12, 5, 1, 7, 18, 29, 28, 17, 6, 1, 7, 25, 47, 57, 45, 23, 7, 1, 9, 32, 72, 104, 102, 68, 30, 8, 1, 9, 41, 104, 176, 206, 170, 98, 38, 9, 1
Offset: 1

Views

Author

Gary W. Adamson, Apr 28 2007

Keywords

Comments

Row sums = A084170: (1, 2, 6, 12, 52, 106, 426, ...). A129689 = A007318 * A129688. Left column = A109613: (1, 1, 3, 3, 5, 5, ...).

Examples

			First few rows of the triangle:
  1;
  1,  1;
  3,  2,  1;
  3,  5,  3,  1;
  5,  8,  8,  4,  1;
  5, 13, 16, 12,  5,  1;
  7, 18, 29, 28, 17,  6,  1;
  ...

Crossrefs

Cf. A057711, A129689, A007318, A129688, A109613.

Formula

A129688 * A007318 as infinite lower triangular matrices.

A276418 Starting a random walk on Z at 0 triangle T(j,k) gives the number of paths of length 2*j returning to 0 exactly k times.

Original entry on oeis.org

1, 2, 2, 6, 6, 4, 20, 20, 16, 8, 70, 70, 60, 40, 16, 252, 252, 224, 168, 96, 32, 924, 924, 840, 672, 448, 224, 64, 3432, 3432, 3168, 2640, 1920, 1152, 512, 128, 12870, 12870, 12012, 10296, 7920, 5280, 2880, 1152, 256, 48620, 48620, 45760, 40040, 32032, 22880, 14080, 7040, 2560, 512, 184756, 184756, 175032, 155584, 128128, 96096, 64064, 36608, 16896, 5632, 1024
Offset: 0

Views

Author

Franz Vrabec, Sep 27 2016

Keywords

Comments

The number of paths of odd length 2*j+1 is the same as the number of even length 2*j (returning to 0 exactly k times).

Examples

			Triangle T(j,k) begins:
      1
      2,     2
      6,     6,     4
     20,    20,    16,     8
     70,    70,    60,    40,    16
    252,   252,   224,   168,    96,    32
    924,   924,   840,   672,   448,   224,    64
   3432,  3432,  3168,  2640,  1920,  1152,   512,  128
  12870, 12870, 12012, 10296,  7920,  5280,  2880, 1152,  256
  48620, 48620, 45760, 40040, 32032, 22880, 14080, 7040, 2560, 512

Links

Programs

  • GAP
    Flat(List([0..10],j->List([0..j],k->2^k*Binomial(2*j-k,j-k)))); # Muniru A Asiru, May 18 2018

Formula

T(j,k) = (2^k)*C(2*j-k,j-k).
T(j,0) = T(j,1) for j>0.
T(j,0) = A000984(j).
T(j,1) = A000984(j) for j>0.
T(j,2) = A128650(j+1).
T(j,j) = A000079(j).
T(j,j-1) = A057711(j+1) for j>0.

A287879 Irregular triangle read by rows: normalized dimensions of certain generalized quadratic residue codes of length n.

Original entry on oeis.org

2, 4, 2, 8, 6, 16, 16, 18, 32, 40, 50, 64, 96, 132, 146, 128, 224, 336, 406, 256, 512, 832, 1088, 1186, 512, 1152, 2016, 2832, 3330, 1024, 2560, 4800, 7200, 9060, 9762, 2048, 5632, 11264, 17952, 24024, 27654, 4096, 12288, 26112, 44032, 62352, 76176, 81330, 8192, 26624, 59904, 106496, 158912, 204984, 232050, 16384, 57344, 136192, 254464, 398720, 540736, 645540, 684210
Offset: 1

Views

Author

N. J. A. Sloane, Jun 18 2017

Keywords

Examples

			Triangle begins:
[2],
[4, 2],
[8, 6],
[16, 16, 18],
[32, 40, 50],
[64, 96, 132, 146],
[128, 224, 336, 406],
[256, 512, 832, 1088, 1186],
[512, 1152, 2016, 2832, 3330],
[1024, 2560, 4800, 7200, 9060, 9762],
[2048, 5632, 11264, 17952, 24024, 27654],
[4096, 12288, 26112, 44032, 62352, 76176, 81330],
[8192, 26624, 59904, 106496, 158912, 204984, 232050],
[16384, 57344, 136192, 254464, 398720, 540736, 645540, 684210],
...

Links

Crossrefs

The 0th column is A000079; column 1 is essentially the same as A057711 or A129952, and is also essentially twice A001792 or A049610.
Row sums are twice A287880.

Programs

  • Maple
    g:=proc(m,w) local k;
if w=0 then 2^m else
2^m*add( (m/(m-w))*binomial(w-1,w-k)*binomial(m-w,k)/4^k, k=1..w);
fi;
end;
for n from 1 to 14 do
lprint( [seq(g(n,w),w=0..floor(n/2))]);
od;

Formula

See Ward, pp. 99-100, or the Maple code below.

A350512 Triangle read by rows with T(n,0) = 1 for n >= 0 and T(n,k) = binomial(n-1,k-1)*(2*k*(n-k) + n)/k for 0 < k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 10, 18, 10, 1, 1, 13, 34, 34, 13, 1, 1, 16, 55, 80, 55, 16, 1, 1, 19, 81, 155, 155, 81, 19, 1, 1, 22, 112, 266, 350, 266, 112, 22, 1, 1, 25, 148, 420, 686, 686, 420, 148, 25, 1, 1, 28, 189, 624, 1218, 1512, 1218, 624, 189, 28, 1
Offset: 0

Views

Author

Werner Schulte, Jan 02 2022

Keywords

Comments

Depending on some fixed integer m there is a family of number triangles T(m; n,k) for 0 <= k <= n with entries: T(m; n,0) = 1 for n >= 0 and T(m; n,k) = binomial(n-1,k-1)*(m*k*(n-k) + n)/k for 0 < k <= n.
Special cases: m=0 (A007318), m=1 (A103450), and m=2 (this triangle).
Further properties: T(m; n,n) = 1 for n >= 0; T(m; n,k) = T(m; n,n-k) for 0 <= k <= n; T(m; 2*n,n) = A000108(n)*A086270(m,n+1) for n >= 0 and m > 0.
T(m; n,k) = T(m; n-1,k) + T(m; n-1,k-1) + m*binomial(n-2,k-1) for 0 < k < n.
G.f. of column k: (1 + m*k*x) * x^k / (1 - x)^(k+1).
G.f.: A(x, t) = (1 - (1+x)*t + m*x*t^2) / (1 - (1+x)*t)^2.
T(m; n,k) = [x^k] (1 + (m*n - m + 2)*x + x^2) * (1 + x)^(n-2) for n > 0.

Examples

			Triangle T(n, k) for 0 <= k <= n starts:
n\k :  0   1    2    3    4    5    6    7   8  9
=================================================
  0 :  1
  1 :  1   1
  2 :  1   4    1
  3 :  1   7    7    1
  4 :  1  10   18   10    1
  5 :  1  13   34   34   13    1
  6 :  1  16   55   80   55   16    1
  7 :  1  19   81  155  155   81   19    1
  8 :  1  22  112  266  350  266  112   22   1
  9 :  1  25  148  420  686  686  420  148  25  1
  etc.

Crossrefs

Row sums are A057711(n+1).
Cf. A000108, A007318, A086270, A103450.

Programs

  • Mathematica
    Flatten[Table[Join[{1},Table[Binomial[n-1,k-1](2*k*(n-k) + n)/k,{k,n}]],{n,0,10}]] (* Stefano Spezia, Jan 06 2022 *)

Formula

T(n, n) = 1; T(n, k) = T(n, n-k).
T(2*n, n) = (n+1)^2 * A000108(n).
T(n, k) = T(n-1, k) + T(n-1, k-1) + 2 * binomial(n-2,k-1) for 0 < k < n.
G.f. of column k: (1 + 2*k*x) * x^k / (1 - x)^(k+1).
G.f.: A(x,t) = (1 - (1 + x) * t + 2 * x * t^2) / (1 - (1 + x) * t)^2.
T(n,k) = [x^k] (1 + 2 * n * x + x^2) * (1 + x)^(n-2) for n > 0.

A140876 Triangle T(n,k) = A053120(n+2,k)-2*A053120(n+1,k)+A053120(n,k) read by rows, 0<=k

Original entry on oeis.org

2, 0, 6, -2, 2, 16, 0, -10, 10, 40, 2, -2, -36, 36, 96, 0, 14, -14, -112, 112, 224, -2, 2, 64, -64, -320, 320, 512, 0, -18, 18, 240, -240, -864, 864, 1152, 2, -2, -100, 100, 800, -800, -2240, 2240, 2560, 0, 22, -22, -440, 440, 2464, -2464, -5632, 5632, 5632, -2, 2, 144, -144, -1680, 1680, 7168, -7168, -13824, 13824
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Jul 21 2008

Keywords

Comments

Second differences downwards columns of the Chebyshev triangle A053120.
Row sums are 2, 6, 16, 40, 96, 224, 512, 1152, 2560, 5632, 12288,..., A057711.

Examples

			2;
0, 6;
-2, 2, 16;
0, -10, 10, 40;
2, -2, -36, 36, 96;
0, 14, -14, -112, 112, 224;
-2, 2, 64, -64, -320, 320, 512;
0, -18, 18, 240, -240, -864, 864, 1152;
2, -2, -100, 100, 800, -800, -2240, 2240, 2560;
0, 22, -22, -440, 440, 2464, -2464, -5632, 5632, 5632;
-2, 2, 144, -144, -1680, 1680, 7168, -7168, -13824, 13824, 12288;

Programs

  • Mathematica
    Clear[T, D2, x, n, m] T[n_, m_] := CoefficientList[ChebyshevT[n + 1, x], x][[m + 1]]; D2[n_, m_] := T[n + 2, m] - 2*T[n + 1, m] + T[n, m]; a = Table[Flatten[Table[D2[n, m], {m, 0, n}]], {n, 0, 10}]; Flatten[a]
Previous Showing 41-45 of 45 results.

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

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