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.

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

0,5

Links

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

G. Strasser, Generalisation of the Euler adic, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_9(n,k).

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

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A084949 (row sums), A257619.

Cf. A007318, A008292, A060187, A142458, A142459, A142460, A142461, A142462, A144431, A167884

Similar sequences listed in A256890.

Sequence in context: A155516 A174674 A144443 * A022183 A015146 A064033

Adjacent sequences: A257605 A257606 A257607 * A257609 A257610 A257611

Keyword

nonn,tabl

Author

Dale Gerdemann, May 03 2015

Status

approved