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A257609 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 + 2.

%I A257609 #21 Jan 18 2025 03:09:56
%S A257609 1,2,2,4,16,4,8,88,88,8,16,416,1056,416,16,32,1824,9664,9664,1824,32,
%T A257609 64,7680,76224,154624,76224,7680,64,128,31616,549504,1999232,1999232,
%U A257609 549504,31616,128,256,128512,3739648,22587904,39984640,22587904,3739648,128512,256
%N A257609 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 + 2.
%H A257609 G. C. Greubel, <a href="/A257609/b257609.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257609 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 + 2.
%F A257609 Sum_{k=0..n} T(n, k) = A002866(n).
%F A257609 From _G. C. Greubel_, Mar 21 2022: (Start)
%F A257609 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 = 2.
%F A257609 T(n, n-k) = T(n, k).
%F A257609 T(n, 0) = A000079(n).
%F A257609 T(n, 1) = 2*A100575(n+1). (End)
%F A257609 T(n, k) = 2^n*A008292(n+1, k+1). - _G. C. Greubel_, Jan 17 2025
%e A257609 Triangle begins as:
%e A257609     1;
%e A257609     2,      2;
%e A257609     4,     16,       4;
%e A257609     8,     88,      88,        8;
%e A257609    16,    416,    1056,      416,       16;
%e A257609    32,   1824,    9664,     9664,     1824,       32;
%e A257609    64,   7680,   76224,   154624,    76224,     7680,      64;
%e A257609   128,  31616,  549504,  1999232,  1999232,   549504,   31616,    128;
%e A257609   256, 128512, 3739648, 22587904, 39984640, 22587904, 3739648, 128512, 256;
%t A257609 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 A257609 Table[T[n,k,2,2], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 21 2022 *)
%o A257609 (Magma)
%o A257609 function T(n,k,a,b)
%o A257609   if k lt 0 or k gt n then return 0;
%o A257609   elif k eq 0 or k eq n then return 1;
%o A257609   else return (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b);
%o A257609   end if; return T;
%o A257609 end function;
%o A257609 [T(n,k,2,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 21 2022
%o A257609 (Magma)
%o A257609 A257609:= func< n,k | 2^n*EulerianNumber(n+1, k) >;
%o A257609 [A257609(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Jan 17 2025
%o A257609 (Sage)
%o A257609 def T(n,k,a,b): # A257609
%o A257609     if (k<0 or k>n): return 0
%o A257609     elif (n==0): return 1
%o A257609     else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
%o A257609 flatten([[T(n,k,2,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 21 2022
%Y A257609 Cf. A000079, A002866 (row sums), A008292, A060187, A100575, A257611, A257613.
%Y A257609 Cf. A257615, A038208, A256890, A257610, A257612, A257614, A257616, A257617.
%Y A257609 Cf. A257618, A257619.
%Y A257609 Similar sequences listed in A256890.
%K A257609 nonn,tabl
%O A257609 0,2
%A A257609 _Dale Gerdemann_, May 03 2015