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A154230 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30)*T(n-2, k-1), read by rows.

%I A154230 #12 Mar 04 2021 07:04:40
%S A154230 1,1,1,1,100,1,1,455,455,1,1,1435,98810,1435,1,1,3711,1135370,1135370,
%T A154230 3711,1,1,8388,7849141,464306300,7849141,8388,1,1,17161,40410421,
%U A154230 10431621081,10431621081,40410421,17161,1,1,32495,169040786,130822910455,7140071740062,130822910455,169040786,32495,1
%N A154230 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30)*T(n-2, k-1), read by rows.
%C A154230 Row sums are: {1, 2, 102, 912, 101682, 2278164, 480021360, 20944097328, ...}.
%C A154230 The row sums of this class of sequences (see cross references) is given by the following. Let S(n) be the row sum then S(n) = 2*S(n-1) + f(n)*S(n-2) for a given f(n). For this sequence f(n) = (n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30 = A000538(n+1). - _G. C. Greubel_, Mar 02 2021
%H A154230 G. C. Greubel, <a href="/A154230/b154230.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A154230 T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30)*T(n-2, k-1) with T(n, 0) = T(n, n) = 1.
%e A154230 Triangle begins as:
%e A154230   1;
%e A154230   1,     1;
%e A154230   1,   100,        1;
%e A154230   1,   455,      455,           1;
%e A154230   1,  1435,    98810,        1435,           1;
%e A154230   1,  3711,  1135370,     1135370,        3711,        1;
%e A154230   1,  8388,  7849141,   464306300,     7849141,     8388,     1;
%e A154230   1, 17161, 40410421, 10431621081, 10431621081, 40410421, 17161, 1;
%p A154230 T:= proc(n, k) option remember;
%p A154230       if k=0 or k=n then 1
%p A154230     else T(n-1, k) +T(n-1, k-1) +((n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30)*T(n-2, k-1)
%p A154230       fi; end:
%p A154230 seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Mar 02 2021
%t A154230 T[n_, k_]:= T[n,k]= If[k==0 || k==n, 1, T[n-1, k] + T[n-1, k-1] + ((n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30)*T[n-2, k-1] ];
%t A154230 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Mar 02 2021 *)
%o A154230 (Sage)
%o A154230 def f(n): return (n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30
%o A154230 def T(n,k):
%o A154230     if (k==0 or k==n): return 1
%o A154230     else: return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1)
%o A154230 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 02 2021
%o A154230 (Magma)
%o A154230 f:= func< n | (n+1)*(n+2)*(2*n+3)*(3*n^2+9*n+5)/30 >;
%o A154230 function T(n,k)
%o A154230   if k eq 0 or k eq n then return 1;
%o A154230   else return T(n-1, k) + T(n-1, k-1) + f(n)*T(n-2, k-1);
%o A154230   end if; return T;
%o A154230 end function;
%o A154230 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 02 2021
%Y A154230 Cf. A154227, A154228, A154229, A154231, A154233.
%Y A154230 Cf. A000538.
%K A154230 nonn,tabl,easy
%O A154230 0,5
%A A154230 _Roger L. Bagula_, Jan 05 2009
%E A154230 Edited by _G. C. Greubel_, Mar 02 2021