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A257613 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.

%I A257613 #25 Mar 21 2022 03:05:30
%S A257613 1,4,4,16,48,16,64,416,416,64,256,3136,6656,3136,256,1024,21888,84608,
%T A257613 84608,21888,1024,4096,145664,939520,1692160,939520,145664,4096,16384,
%U A257613 939520,9555456,28195840,28195840,9555456,939520,16384,65536,5932032,91475968,415734784,676700160,415734784,91475968,5932032,65536
%N A257613 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 A257613 G. C. Greubel, <a href="/A257613/b257613.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257613 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 A257613 Sum_{k=0..n} T(n, k) = A051580(n).
%F A257613 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 A257613 Triangle begins as:
%e A257613       1;
%e A257613       4,      4;
%e A257613      16,     48,      16;
%e A257613      64,    416,     416,       64;
%e A257613     256,   3136,    6656,     3136,      256;
%e A257613    1024,  21888,   84608,    84608,    21888,    1024;
%e A257613    4096, 145664,  939520,  1692160,   939520,  145664,   4096;
%e A257613   16384, 939520, 9555456, 28195840, 28195840, 9555456, 939520, 16384;
%t A257613 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 A257613 Table[T[n,k,2,4], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 20 2022 *)
%o A257613 (PARI) f(x) = 2*x + 4;
%o A257613 T(n, k) = t(n-k, k);
%o A257613 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 A257613 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ");); print();); \\ _Michel Marcus_, May 06 2015
%o A257613 (Sage)
%o A257613 def T(n,k,a,b): # A257613
%o A257613     if (k<0 or k>n): return 0
%o A257613     elif (n==0): return 1
%o A257613     else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
%o A257613 flatten([[T(n,k,2,4) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 20 2022
%Y A257613 Cf. A051580 (row sums), A060187, A257609, A257611, A257615.
%Y A257613 Cf. A257606, A257622.
%Y A257613 Similar sequences listed in A256890.
%K A257613 nonn,tabl
%O A257613 0,2
%A A257613 _Dale Gerdemann_, May 06 2015