A257610 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 + 2.

1, 2, 2, 4, 20, 4, 8, 132, 132, 8, 16, 748, 2112, 748, 16, 32, 3964, 25124, 25124, 3964, 32, 64, 20364, 256488, 552728, 256488, 20364, 64, 128, 103100, 2398092, 9670840, 9670840, 2398092, 103100, 128, 256, 518444, 21246736, 147146804, 270783520, 147146804, 21246736, 518444, 256

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) = 3*x + 2.

Sum_{k=0..n} T(n, k) = A007559(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 = 2. - G. C. Greubel, Mar 20 2022

Example

Triangle begins as:
    1;
    2,      2;
    4,     20,        4;
    8,    132,      132,         8;
   16,    748,     2112,       748,        16;
   32,   3964,    25124,     25124,      3964,        32;
   64,  20364,   256488,    552728,    256488,     20364,       64;
  128, 103100,  2398092,   9670840,   9670840,   2398092,   103100,    128;
  256, 518444, 21246736, 147146804, 270783520, 147146804, 21246736, 518444, 256;

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, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)

Prog

(Sage)
def T(n, k, a, b): # A257610
    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, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022

Crossrefs

Cf. A007559 (row sums), A038208, A142458, A257620, A257622, A257624, A257626.

Cf. A256890, A257609, A257610, A257612, A257614, A257616, A257617, A257618, A257619.

See similar sequences listed in A256890.

Sequence in context: A006853 A120417 A175185 * A062267 A128501 A288497

Adjacent sequences: A257607 A257608 A257609 * A257611 A257612 A257613

Keyword

nonn,tabl

Author

Dale Gerdemann, May 03 2015

Status

approved