A257619 - OEIS

%I #13 Mar 24 2022 03:30:50

%S 1,2,2,4,44,4,8,564,564,8,16,6436,22560,6436,16,32,71404,637844,

%T 637844,71404,32,64,786948,15470232,36994952,15470232,786948,64,128,

%U 8660012,346391196,1660722424,1660722424,346391196,8660012,128

%N 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.

%H G. C. Greubel, <a href="/A257619/b257619.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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.

%F Sum_{k=0..n} T(n, k) = A144829(n).

%F From _G. C. Greubel_, Mar 24 2022: (Start)

%F 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.

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

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

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

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

%e Triangle begins as:

%e     1;

%e     2,       2;

%e     4,      44,         4;

%e     8,     564,       564,          8;

%e    16,    6436,     22560,       6436,         16;

%e    32,   71404,    637844,     637844,      71404,        32;

%e    64,  786948,  15470232,   36994952,   15470232,    786948,      64;

%e   128, 8660012, 346391196, 1660722424, 1660722424, 346391196, 8660012, 128;

%t 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]]];

%t Table[T[n,k,9,2], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 24 2022 *)

%o (PARI) f(x) = 9*x + 2;

%o 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)));

%o tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", "); ); print(); ); } \\ _Michel Marcus_, May 23 2015

%o (Sage)

%o def T(n,k,a,b): # A257619

%o     if (k<0 or k>n): return 0

%o     elif (n==0): return 1

%o     else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)

%o flatten([[T(n,k,9,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 24 2022

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

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

%Y Similar sequences listed in A256890.

%K nonn,tabl

%O 0,2

%A _Dale Gerdemann_, May 09 2015