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%I #22 Mar 21 2022 02:14:23
%S 1,1,1,1,20,1,1,219,219,1,1,2218,8322,2218,1,1,22217,220222,220222,
%T 22217,1,1,222216,5006247,12332432,5006247,222216,1,1,2222215,
%U 105340629,530539235,530539235,105340629,2222215,1,1,22222214,2123693776,19700767514,39259903390,19700767514,2123693776,22222214,1
%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 + 1.
%H G. C. Greubel, <a href="/A257608/b257608.txt">Rows n = 0..50 of the triangle, flattened</a>
%H G. Strasser, <a href="http://dx.doi.org/10.1017/S0305004110000538">Generalisation of the Euler adic</a>, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_9(n,k).
%F 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.
%F Sum_{k=0..n} T(n, k) = A084949(n).
%F 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
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 20, 1;
%e 1, 219, 219, 1;
%e 1, 2218, 8322, 2218, 1;
%e 1, 22217, 220222, 220222, 22217, 1;
%e 1, 222216, 5006247, 12332432, 5006247, 222216, 1;
%e 1, 2222215, 105340629, 530539235, 530539235, 105340629, 2222215, 1;
%t 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]]];
%t Table[T[n,k,9,1], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 20 2022 *)
%o (Sage)
%o def T(n,k,a,b): # A257608
%o if (k<0 or k>n): return 0
%o elif (k==0 or k==n): 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,9,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 20 2022
%Y Cf. A084949 (row sums), A257619.
%Y Cf. A007318, A008292, A060187, A142458, A142459, A142460, A142461, A142462, A144431, A167884
%Y Similar sequences listed in A256890.
%K nonn,tabl
%O 0,5
%A _Dale Gerdemann_, May 03 2015