cp's OEIS Frontend

A257611 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*n + 3.

%I A257611 #20 Mar 01 2022 01:23:34
%S A257611 1,3,3,9,30,9,27,213,213,27,81,1308,2982,1308,81,243,7431,32646,32646,
%T A257611 7431,243,729,40314,310263,587628,310263,40314,729,2187,212505,
%U A257611 2695923,8701545,8701545,2695923,212505,2187,6561,1099704,22059036,113360904,191433990,113360904,22059036,1099704,6561
%N A257611 Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*n + 3.
%H A257611 G. C. Greubel, <a href="/A257611/b257611.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257611 T(n, k) = t(n-k, k), where 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), and f(n) = 2*n + 3.
%F A257611 Sum_{k=0..n} T(n, k) = A051578(n).
%F A257611 From _G. C. Greubel_, Feb 28 2022: (Start)
%F A257611 t(k, n) = t(n, k).
%F A257611 T(n, n-k) = T(n, k).
%F A257611 t(0, n) = T(n, 0) = A000244(n). (End)
%e A257611 Array t(n,k) begins as:
%e A257611     1,      3,        9,         27,          81,           243, ...;
%e A257611     3,     30,      213,       1308,        7431,         40314, ...;
%e A257611     9,    213,     2982,      32646,      310263,       2695923, ...;
%e A257611    27,   1308,    32646,     587628,     8701545,     113360904, ...;
%e A257611    81,   7431,   310263,    8701545,   191433990,    3579465642, ...;
%e A257611   243,  40314,  2695923,  113360904,  3579465642,   93066106692, ...;
%e A257611   729, 212505, 22059036, 1351133676, 59641127202, 2104476295026, ...;
%e A257611 Triangle T(n,k) begins as:
%e A257611      1;
%e A257611      3,      3;
%e A257611      9,     30,       9;
%e A257611     27,    213,     213,      27;
%e A257611     81,   1308,    2982,    1308,      81;
%e A257611    243,   7431,   32646,   32646,    7431,     243;
%e A257611    729,  40314,  310263,  587628,  310263,   40314,    729;
%e A257611   2187, 212505, 2695923, 8701545, 8701545, 2695923, 212505, 2187;
%t A257611 t[n_, k_, p_, q_]:= t[n, k, p, q] = If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+q)*t[n-1,k,p,q] + (p*n+q)*t[n,k-1,p,q]]];
%t A257611 T[n_, k_, p_, q_]= t[n-k, k, p, q];
%t A257611 Table[T[n,k,2,3], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 28 2022 *)
%o A257611 (PARI) f(x) = 2*x + 3;
%o A257611 T(n, k) = t(n-k, k);
%o A257611 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 A257611 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ");); print();); \\ _Michel Marcus_, May 06 2015
%o A257611 (Sage)
%o A257611 @CachedFunction
%o A257611 def t(n,k,p,q):
%o A257611     if (n<0 or k<0): return 0
%o A257611     elif (n==0 and k==0): return 1
%o A257611     else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)
%o A257611 def A257611(n,k): return t(n-k,k,2,3)
%o A257611 flatten([[A257611(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 28 2022
%Y A257611 Cf. A000244, A051578 (row sums), A060187, A257609, A257613, A257615.
%Y A257611 Cf. A038221, A257180, A257611, A257620, A257621, A257623, A257625, A257627.
%Y A257611 Similar sequences listed in A256890.
%K A257611 nonn,tabl
%O A257611 0,2
%A A257611 _Dale Gerdemann_, May 06 2015