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 21-24 of 24 results.

A257607 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 5.

Original entry on oeis.org

1, 5, 5, 25, 60, 25, 125, 535, 535, 125, 625, 4210, 7490, 4210, 625, 3125, 30885, 86110, 86110, 30885, 3125, 15625, 216560, 880735, 1377760, 880735, 216560, 15625, 78125, 1471235, 8330745, 18948695, 18948695, 8330745, 1471235, 78125, 390625, 9764910, 74498800, 234897010, 341076510, 234897010, 74498800, 9764910, 390625
Offset: 0

Views

Author

Dale Gerdemann, May 03 2015

Keywords

Examples

			Triangle begins as:
      1;
      5,       5;
     25,      60,      25;
    125,     535,     535,      125;
    625,    4210,    7490,     4210,      625;
   3125,   30885,   86110,    86110,    30885,    3125;
  15625,  216560,  880735,  1377760,   880735,  216560,   15625;
  78125, 1471235, 8330745, 18948695, 18948695, 8330745, 1471235, 78125;

Links

Crossrefs

Cf. A000351, A008292, A049198 (row sums), A256890, A257180, A257606
Cf. A257615, A257624
Similar sequences listed in A256890.

Programs

  • Mathematica
    T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,1,5], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)
  • Sage
    def T(n,k,a,b): # A257607
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,1,5) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022

Formula

T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 5.
Sum_{k=0..n} T(n, k) = A049198(n).
From G. C. Greubel, Mar 24 2022: (Start)
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 1, and b = 5.
T(n, n-k) = T(n, k).
T(n, 0) = A000351(n).
T(n, 1) = 10*6^n - 5^n*(10 + n).
T(n, 2) = 55*7^n - 10*6^n*(n+10) + 5^n*binomial(n+10, 2). (End)

A257626 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.

Original entry on oeis.org

1, 6, 6, 36, 108, 36, 216, 1404, 1404, 216, 1296, 15876, 33696, 15876, 1296, 7776, 166212, 642492, 642492, 166212, 7776, 46656, 1659204, 10701720, 19274760, 10701720, 1659204, 46656, 279936, 16052580, 163263924, 481752360, 481752360, 163263924, 16052580, 279936
Offset: 0

Views

Author

Dale Gerdemann, May 10 2015

Keywords

Examples

			Triangle begins as:
       1;
       6,        6;
      36,      108,        36;
     216,     1404,      1404,       216;
    1296,    15876,     33696,     15876,      1296;
    7776,   166212,    642492,    642492,    166212,      7776;
   46656,  1659204,  10701720,  19274760,  10701720,   1659204,    46656;
  279936, 16052580, 163263924, 481752360, 481752360, 163263924, 16052580, 279936;

Links

Crossrefs

Cf. A051609 (row sums), A142458, A257610, A257620, A257622, A257624.
See similar sequences listed in A256890.

Programs

  • Mathematica
    T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,3,6], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
  • Sage
    def T(n,k,a,b): # A257626
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,3,6) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022

Formula

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.
Sum_{k=0..n} T(n, k) = A051609(n).
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 3, and b = 6. - G. C. Greubel, Mar 20 2022

A154694 Triangle read by rows: T(n,k) = ((3/2)^k*2^n + (2/3)^k*3^n)*A008292(n+1,k+1).

Original entry on oeis.org

2, 5, 5, 13, 48, 13, 35, 330, 330, 35, 97, 2028, 4752, 2028, 97, 275, 11970, 54360, 54360, 11970, 275, 793, 69840, 557388, 1043712, 557388, 69840, 793, 2315, 407550, 5409180, 16868520, 16868520, 5409180, 407550, 2315, 6817, 2388516, 51011136, 247761072, 404844480, 247761072, 51011136, 2388516, 6817
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Jan 14 2009

Keywords

Examples

			Triangle begins as:
     2;
     5,      5;
    13,     48,      13;
    35,    330,     330,       35;
    97,   2028,    4752,     2028,       97;
   275,  11970,   54360,    54360,    11970,     275;
   793,  69840,  557388,  1043712,   557388,   69840,    793;
  2315, 407550, 5409180, 16868520, 16868520, 5409180, 407550, 2315;

Links

Crossrefs

Cf. A004123 (row sums), A154693, A256890.

Programs

  • Magma
    A154694:= func< n,k | (2^(n-k)*3^k+2^k*3^(n-k))*EulerianNumber(n+1, k) >;
[A154694(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 18 2025
  • Maple
    A154694 := proc(n,m)
    (3^m*2^(n-m)+2^m*3^(n-m))*A008292(n+1,m+1) ;
end proc:
seq(seq( A154694(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Mar 11 2024
  • Mathematica
    T[n_, k_, p_, q_] := (p^(n - k)*q^k + p^k*q^(n - k))*Eulerian[n+1,k];
Table[T[n,k,2,3], {n,0,12}, {k,0,n}]//Flatten
  • Python
    from sage.all import *
from sage.combinat.combinat import eulerian_number
def A154694(n,k): return (pow(2,n-k)*pow(3,k)+pow(2,k)*pow(3,n-k))*eulerian_number(n+1,k)
print(flatten([[A154694(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 18 2025

Formula

Sum_{k=0..n} T(n, k) = A004123(n+2).

Extensions

Definition simplified by the Assoc. Eds. of the OEIS, Jun 07 2010

A257608 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 1.

Original entry on oeis.org

1, 1, 1, 1, 20, 1, 1, 219, 219, 1, 1, 2218, 8322, 2218, 1, 1, 22217, 220222, 220222, 22217, 1, 1, 222216, 5006247, 12332432, 5006247, 222216, 1, 1, 2222215, 105340629, 530539235, 530539235, 105340629, 2222215, 1, 1, 22222214, 2123693776, 19700767514, 39259903390, 19700767514, 2123693776, 22222214, 1
Offset: 0

Views

Author

Dale Gerdemann, May 03 2015

Keywords

Examples

			Triangle begins as:
  1;
  1,       1;
  1,      20,         1;
  1,     219,       219,         1;
  1,    2218,      8322,      2218,         1;
  1,   22217,    220222,    220222,     22217,         1;
  1,  222216,   5006247,  12332432,   5006247,    222216,       1;
  1, 2222215, 105340629, 530539235, 530539235, 105340629, 2222215, 1;

Links

Crossrefs

Cf. A084949 (row sums), A257619.
Cf. A007318, A008292, A060187, A142458, A142459, A142460, A142461, A142462, A144431, A167884
Similar sequences listed in A256890.

Programs

  • Mathematica
    T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,9,1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
  • Sage
    def T(n,k,a,b): # A257608
    if (k<0 or k>n): return 0
    elif (k==0 or k==n): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,9,1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022

Formula

T(n, k) = t(n-k, k), where t(n,k) = f(k)*t(n-1, k) + f(n)*t(n, k-1), and f(n) = 9*n + 1.
Sum_{k=0..n} T(n, k) = A084949(n).
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = T(n, n) = 1, a = 9, and b = 1. - G. C. Greubel, Mar 20 2022
Previous Showing 21-24 of 24 results.

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