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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.

%I A257610 #22 Mar 21 2022 03:06:04
%S A257610 1,2,2,4,20,4,8,132,132,8,16,748,2112,748,16,32,3964,25124,25124,3964,
%T A257610 32,64,20364,256488,552728,256488,20364,64,128,103100,2398092,9670840,
%U A257610 9670840,2398092,103100,128,256,518444,21246736,147146804,270783520,147146804,21246736,518444,256
%N 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.
%H A257610 G. C. Greubel, <a href="/A257610/b257610.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257610 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.
%F A257610 Sum_{k=0..n} T(n, k) = A007559(n).
%F A257610 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
%e A257610 Triangle begins as:
%e A257610     1;
%e A257610     2,      2;
%e A257610     4,     20,        4;
%e A257610     8,    132,      132,         8;
%e A257610    16,    748,     2112,       748,        16;
%e A257610    32,   3964,    25124,     25124,      3964,        32;
%e A257610    64,  20364,   256488,    552728,    256488,     20364,       64;
%e A257610   128, 103100,  2398092,   9670840,   9670840,   2398092,   103100,    128;
%e A257610   256, 518444, 21246736, 147146804, 270783520, 147146804, 21246736, 518444, 256;
%t A257610 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 A257610 Table[T[n,k,3,2], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 20 2022 *)
%o A257610 (Sage)
%o A257610 def T(n,k,a,b): # A257610
%o A257610     if (k<0 or k>n): return 0
%o A257610     elif (n==0): return 1
%o A257610     else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
%o A257610 flatten([[T(n,k,3,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 20 2022
%Y A257610 Cf. A007559 (row sums), A038208, A142458, A257620, A257622, A257624, A257626.
%Y A257610 Cf. A256890, A257609, A257610, A257612, A257614, A257616, A257617, A257618, A257619.
%Y A257610 See similar sequences listed in A256890.
%K A257610 nonn,tabl
%O A257610 0,2
%A A257610 _Dale Gerdemann_, May 03 2015