A257612 - OEIS

%I #33 Mar 21 2022 03:04:49

%S 1,2,2,4,24,4,8,184,184,8,16,1216,3680,1216,16,32,7584,53824,53824,

%T 7584,32,64,46208,674752,1507072,674752,46208,64,128,278912,7764096,

%U 33244544,33244544,7764096,278912,128,256,1677312,84892672,636233728,1196803584,636233728,84892672,1677312,256

%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) =  4*x + 2.

%C Corresponding entries in this triangle and in A060187 differ only by powers of 2. - _F. Chapoton_, Nov 04 2020

%H G. C. Greubel, <a href="/A257612/b257612.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) = 4*x + 2.

%F Sum_{k=0..n} T(n,k) = A047053(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 = 4, and b = 2. - _G. C. Greubel_, Mar 20 2022

%e Triangle begins as:

%e     1;

%e     2,      2;

%e     4,     24,       4;

%e     8,    184,     184,        8;

%e    16,   1216,    3680,     1216,       16;

%e    32,   7584,   53824,    53824,     7584,      32;

%e    64,  46208,  674752,  1507072,   674752,   46208,     64;

%e   128, 278912, 7764096, 33244544, 33244544, 7764096, 278912, 128;

%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,4,2], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 20 2022 *)

%o (PARI) f(x) = 4*x + 2;

%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): # A257612

%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,4,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 20 2022

%Y Cf. A047053 (row sums), A060187, A142459, A257621.

%Y Cf. A038208, A256890, A257609, A257610, A257614, A257616, A257617, A257618, A257619.

%Y See similar sequences listed in A256890.

%K nonn,tabl

%O 0,2

%A _Dale Gerdemann_, May 06 2015