%I #18 Feb 27 2022 02:06:50

%S 1,3,3,9,48,9,27,501,501,27,81,4494,13026,4494,81,243,37815,250230,

%T 250230,37815,243,729,309324,4122735,9008280,4122735,309324,729,2187,

%U 2498649,62256627,256971945,256971945,62256627,2498649,2187

%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) = 5*n + 3.

%H G. C. Greubel, <a href="/A257623/b257623.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) = 5*n + 3.

%F Sum_{k=0..n} T(n, k) = A008548(n).

%F From _G. C. Greubel_, Feb 27 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, ... A000244;

%e 3, 48, 501, 4494, 37815, ...;

%e 9, 501, 13026, 250230, 4122735, ...;

%e 27, 4494, 250230, 9008280, 256971945, ...;

%e 81, 37815, 4122735, 256971945, 11820709470, ...;

%e 243, 309324, 62256627, 6368680566, 450199373658, ...;

%e 729, 2498649, 891791568, 144065371932, 15108742867890, ...;

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

%e 1;

%e 3, 3;

%e 9, 48, 9;

%e 27, 501, 501, 27;

%e 81, 4494, 13026, 4494, 81;

%e 243, 37815, 250230, 250230, 37815, 243;

%e 729, 309324, 4122735, 9008280, 4122735, 309324, 729;

%e 2187, 2498649, 62256627, 256971945, 256971945, 62256627, 2498649, 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,5,3], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 27 2022 *)

%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 A257623(n,k): return t(n-k,k,5,3)

%o flatten([[A257623(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 27 2022

%Y Cf. A000244, A008548, A142460, A257614.

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

%Y Similar sequences listed in A256890.

%K nonn,tabl

%O 0,2

%A _Dale Gerdemann_, May 10 2015