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%I #25 Mar 21 2022 03:05:30
%S 1,4,4,16,48,16,64,416,416,64,256,3136,6656,3136,256,1024,21888,84608,
%T 84608,21888,1024,4096,145664,939520,1692160,939520,145664,4096,16384,
%U 939520,9555456,28195840,28195840,9555456,939520,16384,65536,5932032,91475968,415734784,676700160,415734784,91475968,5932032,65536
%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) = 2*x + 4.
%H G. C. Greubel, <a href="/A257613/b257613.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) = 2*x + 4.
%F Sum_{k=0..n} T(n, k) = A051580(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) = 1, a = 2, and b = 4. - _G. C. Greubel_, Mar 20 2022
%e Triangle begins as:
%e 1;
%e 4, 4;
%e 16, 48, 16;
%e 64, 416, 416, 64;
%e 256, 3136, 6656, 3136, 256;
%e 1024, 21888, 84608, 84608, 21888, 1024;
%e 4096, 145664, 939520, 1692160, 939520, 145664, 4096;
%e 16384, 939520, 9555456, 28195840, 28195840, 9555456, 939520, 16384;
%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,2,4], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 20 2022 *)
%o (PARI) f(x) = 2*x + 4;
%o T(n, k) = t(n-k, k);
%o t(n, m) = if (!n && !m, 1, if (n < 0 || m < 0, 0, 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), ", ");); print();); \\ _Michel Marcus_, May 06 2015
%o (Sage)
%o def T(n,k,a,b): # A257613
%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,2,4) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 20 2022
%Y Cf. A051580 (row sums), A060187, A257609, A257611, A257615.
%Y Cf. A257606, A257622.
%Y Similar sequences listed in A256890.
%K nonn,tabl
%O 0,2
%A _Dale Gerdemann_, May 06 2015
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