A144442 Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

1, 1, 1, 1, 17, 1, 1, 118, 118, 1, 1, 729, 2681, 729, 1, 1, 4400, 41745, 41745, 4400, 1, 1, 26431, 555240, 1349245, 555240, 26431, 1, 1, 158622, 6816846, 33456685, 33456685, 6816846, 158622, 1, 1, 951773, 80034743, 715321156, 1411926995, 715321156, 80034743, 951773, 1

Offset

1,5

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

Sum_{k=1..n} T(n, k) = s(n), where s(n) = (5*n-8)*s(n-1) + 5*s(n-2), with s(1) = 1, s(2) = 2.

From G. C. Greubel, Mar 03 2022: (Start)

T(n, n-k) = T(n, k).

T(n, 2) = (1/5)*(17*6^(n - 2) - (5*n + 2)).

T(n, 3) = (1/50)*(25*n^2 - 5*n - 31 - 34*6^(n - 3)*(30*n - 13) +

2489*11^(n - 3)). (End)

Example

Triangle begins as:
  1;
  1,      1;
  1,     17,        1;
  1,    118,      118,         1;
  1,    729,     2681,       729,          1;
  1,   4400,    41745,     41745,       4400,         1;
  1,  26431,   555240,   1349245,     555240,     26431,        1;
  1, 158622,  6816846,  33456685,   33456685,   6816846,   158622,      1;
  1, 951773, 80034743, 715321156, 1411926995, 715321156, 80034743, 951773, 1;

Mathematica

T[n_, k_, m_, j_]:= T[n, k, m, j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1, k-1, m, j] + (m*(k-1)+1)*T[n-1, k, m, j] + j*T[n-2, k-1, m, j]];
Table[T[n, k, 5, 5], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 03 2022 *)

Prog

(Sage)
def T(n, k, m, j):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1, k-1, m, j) + (m*(k-1)+1)*T(n-1, k, m, j) + j*T(n-2, k-1, m, j)
def A144442(n, k): return T(n, k, 5, 5)
flatten([[A144442(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 03 2022

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144441, A144443, A144444, A144445.

Sequence in context: A229956 A157274 A218115 * A157151 A176794 A176244

Adjacent sequences: A144439 A144440 A144441 * A144443 A144444 A144445

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Oct 05 2008

Status

approved