A257611 - OEIS

%I #20 Mar 01 2022 01:23:34

%S 1,3,3,9,30,9,27,213,213,27,81,1308,2982,1308,81,243,7431,32646,32646,

%T 7431,243,729,40314,310263,587628,310263,40314,729,2187,212505,

%U 2695923,8701545,8701545,2695923,212505,2187,6561,1099704,22059036,113360904,191433990,113360904,22059036,1099704,6561

%N 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 G. C. Greubel, <a href="/A257611/b257611.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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 Sum_{k=0..n} T(n, k) = A051578(n).

%F From _G. C. Greubel_, Feb 28 2022: (Start)

%F t(k, n) = t(n, k).

%F T(n, n-k) = T(n, k).

%F t(0, n) = T(n, 0) = A000244(n). (End)

%e Array t(n,k) begins as:

%e     1,      3,        9,         27,          81,           243, ...;

%e     3,     30,      213,       1308,        7431,         40314, ...;

%e     9,    213,     2982,      32646,      310263,       2695923, ...;

%e    27,   1308,    32646,     587628,     8701545,     113360904, ...;

%e    81,   7431,   310263,    8701545,   191433990,    3579465642, ...;

%e   243,  40314,  2695923,  113360904,  3579465642,   93066106692, ...;

%e   729, 212505, 22059036, 1351133676, 59641127202, 2104476295026, ...;

%e Triangle T(n,k) begins as:

%e      1;

%e      3,      3;

%e      9,     30,       9;

%e     27,    213,     213,      27;

%e     81,   1308,    2982,    1308,      81;

%e    243,   7431,   32646,   32646,    7431,     243;

%e    729,  40314,  310263,  587628,  310263,   40314,    729;

%e   2187, 212505, 2695923, 8701545, 8701545, 2695923, 212505, 2187;

%t 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 T[n_, k_, p_, q_]= t[n-k, k, p, q];

%t Table[T[n,k,2,3], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 28 2022 *)

%o (PARI) f(x) = 2*x + 3;

%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 @CachedFunction

%o def t(n,k,p,q):

%o     if (n<0 or k<0): return 0

%o     elif (n==0 and k==0): return 1

%o     else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)

%o def A257611(n,k): return t(n-k,k,2,3)

%o flatten([[A257611(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 28 2022

%Y Cf. A000244, A051578 (row sums), A060187, A257609, A257613, A257615.

%Y Cf. A038221, A257180, A257611, A257620, A257621, A257623, A257625, A257627.

%Y Similar sequences listed in A256890.

%K nonn,tabl

%O 0,2

%A _Dale Gerdemann_, May 06 2015