A257619 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 + 2.

1, 2, 2, 4, 44, 4, 8, 564, 564, 8, 16, 6436, 22560, 6436, 16, 32, 71404, 637844, 637844, 71404, 32, 64, 786948, 15470232, 36994952, 15470232, 786948, 64, 128, 8660012, 346391196, 1660722424, 1660722424, 346391196, 8660012, 128

Offset

0,2

Links

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

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) = 9*x + 2.

Sum_{k=0..n} T(n, k) = A144829(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 = 9, and b = 2.

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

T(n, 0) = A000079(n).

T(n, 1) = (1/9)*(4*11^n - 2^n*(9*n + 4)).

T(n, 2) = (1/81)*(26*20^n - 4*(4+9*n)*11^n - 2^(n-1)*(20 + 9*n - 81*n^2)). (End)

Example

Triangle begins as:
    1;
    2,       2;
    4,      44,         4;
    8,     564,       564,          8;
   16,    6436,     22560,       6436,         16;
   32,   71404,    637844,     637844,      71404,        32;
   64,  786948,  15470232,   36994952,   15470232,    786948,      64;
  128, 8660012, 346391196, 1660722424, 1660722424, 346391196, 8660012, 128;

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, 9, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)

Prog

(PARI) f(x) = 9*x + 2;
t(n, m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, f(m)*t(n-1, m) + f(n)*t(n, m-1)));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", "); ); print(); ); } \\ Michel Marcus, May 23 2015
(Sage)
def T(n, k, a, b): # A257619
    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, 9, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022

Crossrefs

Cf. A000079, A144829 (row sums), A257608.

Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257616, A257617, A257618.

Similar sequences listed in A256890.

Sequence in context: A309344 A257618 A088895 * A075806 A295580 A177956

Adjacent sequences: A257616 A257617 A257618 * A257620 A257621 A257622

Keyword

nonn,tabl

Author

Dale Gerdemann, May 09 2015

Status

approved