A257620 -id:A257620 - cp's OEIS frontend

A154693 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*A008292(n+1, k+1).

2, 3, 3, 5, 16, 5, 9, 66, 66, 9, 17, 260, 528, 260, 17, 33, 1026, 3624, 3624, 1026, 33, 65, 4080, 23820, 38656, 23820, 4080, 65, 129, 16302, 154548, 374856, 374856, 154548, 16302, 129, 257, 65260, 993344, 3529360, 4998080, 3529360, 993344, 65260, 257

Offset: 0

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

From G. C. Greubel, Jan 17 2025: (Start)
A more general triangle of coefficients may be defined by T(n, k, p, q) = (p^(n-k)*q^k + p^k*q^(n-k))*A008292(n+1, k+1). When (p, q) = (2, 1) this sequence is obtained.
Some related triangles are:
  (p, q) = (1, 1) : 2*A008292(n,k).
  (p, q) = (2, 2) : 2*A257609(n,k).
  (p, q) = (3, 2) : A154694(n,k).
  (p, q) = (3, 3) : 2*A257620(n,k). (End)

			The triangle begins as:
    2;
    3,     3;
    5,    16,      5;
    9,    66,     66,       9;
   17,   260,    528,     260,      17;
   33,  1026,   3624,    3624,    1026,      33;
   65,  4080,  23820,   38656,   23820,    4080,     65;
  129, 16302, 154548,  374856,  374856,  154548,  16302,   129;
  257, 65260, 993344, 3529360, 4998080, 3529360, 993344, 65260,  257;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
A. Lakhtakia, R. Messier, V. K. Varadan, and V. V. Varadan, Use of combinatorial algebra for diffusion on fractals, Physical Review A, volume 34, Number 3 (1986) p. 2501, (FIG. 3)

Cf. A000629 (row sums), A008292, A154694, A257609, A257620.

A154693:= func< n,k | (2^(n-k) + 2^k)*EulerianNumber(n+1, k) >;
[A154693(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 17 2025

p=2; q=1;
A008292[n_,k_]:= A008292[n,k]= Sum[(-1)^j*(k-j)^n*Binomial[n+1,j], {j,0,k}];
T[n_, m_]:= (p^(n-m)*q^m + p^m*q^(n-m))*A008292[n+1,m+1];
Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten

from sage.combinat.combinat import eulerian_number
def A154693(n,k): return (2^(n-k) +2^k)*eulerian_number(n+1,k)
print(flatten([[A154693(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 17 2025

T(n, k) = (2^(n-k) + 2^k)*A008292(n+1, k+1)

Sum_{k=0..n} T(n, k) = A000629(n+1).

Definition simplified by the Assoc. Eds. of the OEIS - Aug 08 2010.

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

Dale Gerdemann, May 10 2015

			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;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A051609 (row sums), A142458, A257610, A257620, A257622, A257624.

See similar sequences listed in A256890.

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 *)

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

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

cp's OEIS Frontend

A154693 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*A008292(n+1, k+1).

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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.

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cp's OEIS Frontend

A154693 Triangle read by rows: T(n, k) = (2^(n-k) + 2^k)*A008292(n+1, k+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

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.