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.

A168641 Triangle read by rows: T(n,k) = [x^k] p(x,n), where p(x,n) = 3*(x + 1)^n - 2*(x^n + 1) - n*(x + x^(n - 1)) for n >= 2, p(x,0) = 1, and p(x,1) = x + 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 8, 18, 8, 1, 1, 10, 30, 30, 10, 1, 1, 12, 45, 60, 45, 12, 1, 1, 14, 63, 105, 105, 63, 14, 1, 1, 16, 84, 168, 210, 168, 84, 16, 1, 1, 18, 108, 252, 378, 378, 252, 108, 18, 1, 1, 20, 135, 360, 630, 756, 630, 360, 135, 20, 1
Offset: 0

Views

Author

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

Keywords

Examples

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  6,   6,   1;
  1,  8,  18,   8,   1;
  1, 10,  30,  30,  10,   1;
  1, 12,  45,  60,  45,  12,   1;
  1, 14,  63, 105, 105,  63,  14,   1;
  1, 16,  84, 168, 210, 168,  84,  16,  1;
  1, 18, 108, 252, 378, 378, 252, 108,  18,  1;
  1, 20, 135, 360, 630, 756, 630, 360, 135, 20, 1;
  ...

Links

Crossrefs

Cf. A095151, A132046, A168643, A168644, A168646.
Columns (essentially): A005843 (k=1), A045943 (k=2), A027480 (k=3), A050534 (k=4), A253942 (k=5), A253943 (k=6), A253944 (k=7).

Programs

  • Magma
    function f(n,k)
   if n le 2 then return 1;
   elif k eq 0 or k eq n then return 1;
   elif k eq 1 or k eq n-1 then return 2;
   else return 3;
   end if;
end function;
A168641:= func< n,k | Binomial(n,k)*f(n,k) >;
[A168641(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 24 2025
  • Mathematica
    p[x_, n_]:= If[n==0, 1, If[n==1, 1+x, 3*(1+x)^n -(1+x^n) -(1+n*x +n*x^(n-1) + x^n)]];
Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 10}]]
(* Second program *)
f[n_, k_]:= With[{b=Boole}, If[k<=n/2, b[k==0] +2*b[k==1] +3*b[2<=k<=n/2], f[n, n-k]]];
A168641[n_, k_]:= Binomial[n,k]*If[n<3,1,f[n,k]];
Table[A168641[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 24 2025 *)
  • Maxima
    T(n,k) := ratcoef(if n <= 2 then (1 + x)^n else 3*(x + 1)^n - (x^n + 1) - (x^n + n*x^(n - 1) + n*x + 1), x, k);
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Jan 02 2019 */
  • SageMath
    def f(n,k):
    if (k<=n/2): return int(k==0) + 2*int(k==1) + 3*int(1A168641(n,k):
    if (n<3): return binomial(n,k)
    else: return binomial(n,k)*f(n,k)
print(flatten([[A168641(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 24 2025

Formula

From G. C. Greubel, Mar 24 2025: (Start)
T(n, k) = 3*binomial(n, k), for n >= 4 and 2 <= k <= n-2, otherwise T(n, 0) = T(n, n) = 1, T(n, 1) = T(n, n-1) = 2*A065475(n-1).
T(n, n-k) = T(n, k).
T(n, 1) = A005843(n) - [n=1] - 2*[n=2].
Columns: T(n, k) = 3*binomial(n,k) - 2*[n=k] - (k+1)*[n=k+1], k >= 2.
Sum_{k=0..n} T(n, k) = 2*A095151(n-1) - 2*[n=0] - 2*[n=1].
Sum_{k=0..n} (-1)^k*T(n, k) = (1+(-1)^n)*(n-2) + 5*[n=0]. (End)

Extensions

Edited by Franck Maminirina Ramaharo, Jan 02 2019

A168644 Triangle read by rows: T(n, k) = [x^k] p(x,n), where p(x,0) = 1, p(x,n) = (7 - n)*(1+x)^n - (6-n)*(1 + x^n) for 1 <= n <= 5, and p(x,n) = 5*(1 + x)^n - Sum_{i=0..3} (Sum_{j=0..i} binomial(n, j)*(x^j + x^(n-j))) + (1/6)*n*(n - 1)*(n - 5)*x^(n-3) for n >= 6.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 12, 12, 1, 1, 12, 18, 12, 1, 1, 10, 20, 20, 10, 1, 1, 12, 45, 65, 45, 12, 1, 1, 14, 63, 140, 154, 63, 14, 1, 1, 16, 84, 224, 350, 252, 84, 16, 1, 1, 18, 108, 336, 630, 630, 384, 108, 18, 1, 1, 20, 135, 480, 1050, 1260, 1050, 555, 135, 20, 1
Offset: 0

Views

Author

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

Keywords

Examples

			Triangle begins:
  1;
  1,  1;
  1, 10,   1;
  1, 12,  12,   1;
  1, 12,  18,  12,    1;
  1, 10,  20,  20,   10,    1;
  1, 12,  45,  65,   45,   12,    1;
  1, 14,  63, 140,  154,   63,   14,   1;
  1, 16,  84, 224,  350,  252,   84,  16,   1;
  1, 18, 108, 336,  630,  630,  384, 108,  18,  1;
  1, 20, 135, 480, 1050, 1260, 1050, 555, 135, 20, 1;
  ...

Links

Crossrefs

Cf. A132046, A168641, A168643, A168646.

Programs

  • Mathematica
    (* First program *)
p[x_, n_]:= If[n<2, (1+x)^n, 5*(1+x)^n -(1+x^n) - If[n>2, x^n +n*x^(n-1) + n*x +1, 1+x^n] - If[n>3, x^n +n*x^(n-1) +Binomial[n,2]*(x^2 +x^(n-2)) + n*x +1, 1+x^n] - If[n>4, x^n +n*x^(n-1) +Binomial[n,2]*(x^2 +x^(n-3) +x^(n-2)) + Binomial[n,3]*x^3 +n*x +1, 1+x^n]];
Flatten[Table[CoefficientList[p[x, n], x], {n, 0, 10}]]
(* Second program *)
f[n_, k_]:= With[{B=Boole}, If[k==0 || k==n, 1, If[1<=n<=5, (7-n) - (6-n)*(B[k==0] + B[k==n]), If[n==6, (k+1)*B[k<4] + (n-k+1)*B[k>3] - B[k==3], (k + 1)*B[k<4] + 5*B[3n-4]]]]];
A168644[n_, k_]:= Binomial[n,k]*f[n,k] + If[n>5, n*(n-1)*(n-5)*Boole[k==n-3]/6, 0];
Table[A168644[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 06 2025 *)
  • Maxima
    T(n, k) := if k = 0 or k = n then 1 else (if n <= 5 then (7 - n)*binomial(n, k) else ratcoef(5*(x + 1)^n - sum(sum(binomial(n, j)*(x^j + x^(n - j)), j, 1, i), i, 1, 3) + (1/6)*n*(n - 1)*(n - 5)*x^(n - 3), x, k))$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Jan 02 2019 */
  • SageMath
    def f(n, k):
    if k==0 or k==n: return 1
    elif 03) - int(k==3)
    else: return (k+1)*int(k<4) + 5*int(3n-4)
def A168644(n, k):
    if n<6: return binomial(n, k)*f(n, k)
    else: return binomial(n,k)*f(n,k) + n*(n-1)*(n-5)*int(k==n-3)//6
print(flatten([[A168644(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 06 2025

Extensions

Edited by Franck Maminirina Ramaharo, Jan 02 2019

A168646 Triangle read by rows: T(n,k) = [x^k] p(x,n), where p(x,0) = 1, p(x,n) = (8 - n)*(1+x)^n - (7 - n)*(1 + x^n) for 1 <= n <= 6, and p(x,n) = 6*(1+x)^n - Sum_{i=0..4} (Sum_{j=0..i} binomial(n, j)*(x^j + x^(n-j))) for n >= 7.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 15, 15, 1, 1, 16, 24, 16, 1, 1, 15, 30, 30, 15, 1, 1, 12, 30, 40, 30, 12, 1, 1, 14, 63, 315, 315, 63, 14, 1, 1, 16, 84, 224, 700, 224, 84, 16, 1, 1, 18, 108, 336, 630, 630, 336, 108, 18, 1, 1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1, 1, 22, 165, 660, 1650, 2772, 2772, 1650, 660, 165, 22, 1
Offset: 0

Views

Author

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

Keywords

Examples

			Triangle begins:
  1;
  1,  1;
  1, 12,   1;
  1, 15,  15,   1;
  1, 16,  24,  16,    1;
  1, 15,  30,  30,   15,    1;
  1, 12,  30,  40,   30,   12,    1;
  1, 14,  63, 315,  315,   63,   14,   1;
  1, 16,  84, 224,  700,  224,   84,  16,   1;
  1, 18, 108, 336,  630,  630,  336, 108,  18,  1;
  1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1;
  ...

Links

Crossrefs

Cf. A132046, A168641, A168643, A168644.

Programs

  • Mathematica
    (* First program *)
p[n_, x_]:= With[{B=Binomial}, If[n==0, 1, If[1<=n<=6, 1 + (8-n)*Sum[B[n,j]*x^j, {j, n -1}] +x^n, Sum[(j+1)*B[n,j]*x^j, {j,0,4}] +6*Sum[B[n,j]*x^j, {j,5,n-5}] + Sum[(n-j+ 1)*B[n,j]*x^j, {j,n-4,n}]]]];
Flatten[Table[CoefficientList[p[n,x], x], {n, 0, 12}]]
(* Second program *)
f[n_, k_]:= If[k==0||k==n,1,If[1<=n<= 6 && 1<=k<=n-1, 8-n, (k+1)*Boole[k<=4] + 6*Boole[5<=k<=n-5] +(n-k+1)*Boole[n-4<=k<=n]]];
A168646[n_, k_]:= Binomial[n,k]*f[n,k];
Table[A168646[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 05 2025 *)
  • Maxima
    T(n,k) := if k = 0 or k = n then 1 else (if n <= 6 then (8 - n)*binomial(n, k) else ratcoef(6*(x + 1)^n - sum(sum(binomial(n, j)*(x^j + x^(n - j)), j, 1, i), i, 1, 4), x, k))$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Jan 02 2019 */
  • SageMath
    def f(n,k):
    if k==0 or k==n: return 1
    elif 0n-5)
def A168646(n,k): return binomial(n,k)*f(n,k)
print(flatten([[A168646(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Apr 05 2025

Formula

T(n, n-k) = T(n, k). - G. C. Greubel, Apr 05 2025

Extensions

Edited by Franck Maminirina Ramaharo, Jan 02 2019
Data values T(7,3), T(7,4), T(8,4) corrected by G. C. Greubel, Apr 05 2025

A144480 T(n,k) = binomial(n, k)*min(k + 1, n - k + 1), triangle read by rows (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 6, 6, 1, 1, 8, 18, 8, 1, 1, 10, 30, 30, 10, 1, 1, 12, 45, 80, 45, 12, 1, 1, 14, 63, 140, 140, 63, 14, 1, 1, 16, 84, 224, 350, 224, 84, 16, 1, 1, 18, 108, 336, 630, 630, 336, 108, 18, 1, 1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1
Offset: 0

Views

Author

Roger L. Bagula, Oct 11 2008

Keywords

Examples

			Triangle begins:
  1;
  1,  1;
  1,  4,   1;
  1,  6,   6,   1;
  1,  8,  18,   8,    1;
  1, 10,  30,  30,   10,    1;
  1, 12,  45,  80,   45,   12,    1;
  1, 14,  63, 140,  140,   63,   14,   1;
  1, 16,  84, 224,  350,  224,   84,  16,   1;
  1, 18, 108, 336,  630,  630,  336, 108,  18,  1;
  1, 20, 135, 480, 1050, 1512, 1050, 480, 135, 20, 1;
  ...

Crossrefs

Row sums are in A245560.
Cf. A003983, A007318, A168643.

Programs

  • Mathematica
    Table[Table[Binomial[n, m]*If[m <= Floor[n/2], 1 + m, 1 + n - m], {m, 0, n}], {n, 0, 10}] // Flatten
  • Maxima
    create_list(binomial(n, k)*min(k + 1, n - k + 1), n, 0, 10, k, 0, n); /* Franck Maminirina Ramaharo, Dec 10 2018 */

Formula

If k <= floor(n/2), then T(n,k) = binomial(n, k)*(k + 1), otherwise T(n,k) = binomial(n, k)*(n - k - 1).
T(n,k) = A007318(n,k)*A003983(k+1,n-k+1), i.e., term-by term product of Pascal's triangle A007318 and A003983 as a triangle.

Extensions

Entry revised by N. J. A. Sloane, Aug 07 2014
Edited by Franck Maminirina Ramaharo, Dec 10 2018
Showing 1-4 of 4 results.

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

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