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.

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A168622 Triangle read by rows: T(n, k) = [x^k]( 7*(1+x)^n - 6*(1+x^n) ) with T(0, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 14, 1, 1, 21, 21, 1, 1, 28, 42, 28, 1, 1, 35, 70, 70, 35, 1, 1, 42, 105, 140, 105, 42, 1, 1, 49, 147, 245, 245, 147, 49, 1, 1, 56, 196, 392, 490, 392, 196, 56, 1, 1, 63, 252, 588, 882, 882, 588, 252, 63, 1, 1, 70, 315, 840, 1470, 1764, 1470, 840, 315, 70, 1
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

Keywords

Examples

			Triangle begins as:
  1;
  1,  1;
  1, 14,   1;
  1, 21,  21,   1;
  1, 28,  42,  28,    1;
  1, 35,  70,  70,   35,    1;
  1, 42, 105, 140,  105,   42,    1;
  1, 49, 147, 245,  245,  147,   49,   1;
  1, 56, 196, 392,  490,  392,  196,  56,   1;
  1, 63, 252, 588,  882,  882,  588, 252,  63,  1;
  1, 70, 315, 840, 1470, 1764, 1470, 840, 315, 70, 1;

Links

Crossrefs

Cf. A022090, A131115, A132047, A168620, A168623.
Columns (essentially): A008589 (k=1), A024966 (k=2).

Programs

  • Magma
    A168622:= func< n,k | k eq 0 or k eq n select 1 else 7*Binomial(n,k) >;
[A168622(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 10 2025
  • Mathematica
    (* First program *)
p[x_, n_]:= With[{m=3}, If[n==0, 1, (2*m+1)(1+x)^n - 2*m*(1+x^n)]];
Table[CoefficientList[p[x,n], x], {n,0,12}]//Flatten
(* Second program *)
A168622[n_, k_]:= If[k==0 || k==n, 1, 7*Binomial[n,k]];
Table[A168622[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 10 2025 *)
  • SageMath
    def A168622(n,k):
    if k==0 or k==n: return 1
    else: return 7*binomial(n,k)
print(flatten([[A168622(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 10 2025

Formula

From G. C. Greubel, Apr 10 2025: (Start)
T(n, k) = 7*binomial(n, k), with T(n, 0) = T(n, n) = 1.
T(n, n-k) = T(n, k).
Sum_{k=0..n} T(n, k) = 2*A048489(n-1) + 6*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = -6*(1 + (-1)^n) + 13*[n=0].
Sum_{k=0..floor(n/2)} T(n-k, k) = A022090(n+1) - 3*(3 + (-1)^n) + 6*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (14/sqrt(3))*(-1)^n*cos((4*n+1)*Pi/6) - 6*(1 + (-1)^n*cos(n*Pi/2)) + 6*[n=0]. (End)

A168621 Triangle read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,k) = ((n - 1)! + 1)*binomial(n, k) for 1 <= k <= n - 1, n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 9, 9, 1, 1, 28, 42, 28, 1, 1, 125, 250, 250, 125, 1, 1, 726, 1815, 2420, 1815, 726, 1, 1, 5047, 15141, 25235, 25235, 15141, 5047, 1, 1, 40328, 141148, 282296, 352870, 282296, 141148, 40328, 1, 1, 362889, 1451556, 3386964, 5080446, 5080446, 3386964, 1451556, 362889, 1
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

Keywords

Comments

Row 0 is 1, and row n gives the coefficients in the expansion of (x + 1)^n + (n - 1)!*((x + 1)^n - x^n -1). - Franck Maminirina Ramaharo, Dec 22 2018

Examples

			Triangle begins:
  1;
  1,     1;
  1,     4,      1;
  1,     9,      9,      1;
  1,    28,     42,     28,      1;
  1,   125,    250,    250,    125,      1;
  1,   726,   1815,   2420,   1815,    726,      1;
  1,  5047,  15141,  25235,  25235,  15141,   5047,     1;
  1, 40328, 141148, 282296, 352870, 282296, 141148, 40328, 1;
  ...

Crossrefs

Row sums: A154252.
Cf. A007318, A132047, A155863, A155864, A155865 A168619, A219570.

Programs

  • Mathematica
    p[x_, n_] := If[n == 0, 1, (x + 1)^n + (n - 1)!*((x + 1)^n - x^n - 1)];
Table[CoefficientList[p[x, n], x], {n, 0, 12}] // Flatten (* Franck Maminirina Ramaharo, Dec 22 2018 *)
  • Maxima
    T(n, k) := if k = 0 or k = n then 1 else ((n - 1)! + 1)*binomial(n, k)$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 22 2018 */

Formula

From Franck Maminirina Ramaharo, Dec 22 2018: (Start)
T(n,k) = A007318(n,k) + A219570(n,k) for 1 <= k <= n - 1, n >= 2.
E.g.f.: exp((1 + x)*y) + log((1 - y)*(1 - x*y)/(1 - (1 + x)*y)). (End)

Extensions

Edited by Franck Maminirina Ramaharo, Dec 22 2018 (previous definition and examples were the same as A168620, but with different entries, as pointed out by R. J. Mathar, Oct 21 2012)
Showing 1-2 of 2 results.

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