cp's OEIS Frontend

A257623 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.

%I A257623 #18 Feb 27 2022 02:06:50
%S A257623 1,3,3,9,48,9,27,501,501,27,81,4494,13026,4494,81,243,37815,250230,
%T A257623 250230,37815,243,729,309324,4122735,9008280,4122735,309324,729,2187,
%U A257623 2498649,62256627,256971945,256971945,62256627,2498649,2187
%N A257623 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 A257623 G. C. Greubel, <a href="/A257623/b257623.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257623 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 A257623 Sum_{k=0..n} T(n, k) = A008548(n).
%F A257623 From _G. C. Greubel_, Feb 27 2022: (Start)
%F A257623 t(k, n) = t(n, k).
%F A257623 T(n, n-k) = T(n, k).
%F A257623 t(0, n) = T(n, 0) = A000244(n). (End)
%e A257623 Array, t(n,k), begins as:
%e A257623     1,       3,         9,           27,             81, ... A000244;
%e A257623     3,      48,       501,         4494,          37815, ...;
%e A257623     9,     501,     13026,       250230,        4122735, ...;
%e A257623    27,    4494,    250230,      9008280,      256971945, ...;
%e A257623    81,   37815,   4122735,    256971945,    11820709470, ...;
%e A257623   243,  309324,  62256627,   6368680566,   450199373658, ...;
%e A257623   729, 2498649, 891791568, 144065371932, 15108742867890, ...;
%e A257623 Triangle, T(n,k), begins as:
%e A257623      1;
%e A257623      3,       3;
%e A257623      9,      48,        9;
%e A257623     27,     501,      501,        27;
%e A257623     81,    4494,    13026,      4494,        81;
%e A257623    243,   37815,   250230,    250230,     37815,      243;
%e A257623    729,  309324,  4122735,   9008280,   4122735,   309324,     729;
%e A257623   2187, 2498649, 62256627, 256971945, 256971945, 62256627, 2498649, 2187;
%t A257623 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 A257623 T[n_, k_, p_, q_]= t[n-k,k,p,q];
%t A257623 Table[T[n,k,5,3], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 27 2022 *)
%o A257623 (Sage)
%o A257623 @CachedFunction
%o A257623 def t(n,k,p,q):
%o A257623     if (n<0 or k<0): return 0
%o A257623     elif (n==0 and k==0): return 1
%o A257623     else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)
%o A257623 def A257623(n,k): return t(n-k,k,5,3)
%o A257623 flatten([[A257623(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 27 2022
%Y A257623 Cf. A000244, A008548, A142460, A257614.
%Y A257623 Cf. A038221, A257180, A257611, A257620, A257621, A257625, A257627.
%Y A257623 Similar sequences listed in A256890.
%K A257623 nonn,tabl
%O A257623 0,2
%A A257623 _Dale Gerdemann_, May 10 2015