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A153654 Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 9, read by rows.

%I A153654 #18 Mar 04 2021 10:59:30
%S A153654 2,23,23,2,1054,2,2,12165,12165,2,2,13041,484928,13041,2,2,13917,
%T A153654 5814074,5814074,13917,2,2,14793,11526908,223541684,11526908,14793,2,
%U A153654 2,15669,17623430,2775818930,2775818930,17623430,15669,2,2,16545,24103640,7830701156,103239353768,7830701156,24103640,16545,2
%N A153654 Triangle T(n, k, j) =  T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 9, read by rows.
%H A153654 G. C. Greubel, <a href="/A153654/b153654.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A153654 T(n, k, j) =  T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j), T(3, 2, j) = 2*prime(j)^2 - 4, T(4, 2, j) = T(4, 3, j) = prime(j)^2 - 2, T(n, 1, j) = T(n, n, j) = 2 and j = 9.
%F A153654 From _G. C. Greubel_, Mar 04 2021: (Start)
%F A153654 Sum_{k=0..n} T(n, k, 10) = -(68/361)*[n=0] - (92/19)*[n=1] + 1058*(i*sqrt(437))^(n-2)*(ChebyshevU(n-2, -i/sqrt(437)) - (21*i/sqrt(437))*ChebyshevU(n-3, -i/sqrt(437) )).
%F A153654 Row sums satisfy the recurrence S(n) = 2*S(n-1) + 437*S(n-2) for n>4 with S(0) = 2, S(1) = 46, S(2) = 1058, S(3) = 24334. (End)
%e A153654 Triangle begins as:
%e A153654    2;
%e A153654   23,    23;
%e A153654    2,  1054,        2;
%e A153654    2, 12165,    12165,          2;
%e A153654    2, 13041,   484928,      13041,            2;
%e A153654    2, 13917,  5814074,    5814074,        13917,          2;
%e A153654    2, 14793, 11526908,  223541684,     11526908,      14793,        2;
%e A153654    2, 15669, 17623430, 2775818930,   2775818930,   17623430,    15669,     2;
%e A153654    2, 16545, 24103640, 7830701156, 103239353768, 7830701156, 24103640, 16545, 2;
%t A153654 T[n_, k_, j_]:= T[n,k,j]= If[n==2, Prime[j], If[n==3 && k==2 || n==4 && 2<=k<=3, ((3-(-1)^n)/2)*Prime[j]^(n-1) -2^((3-(-1)^n)/2), If[k==1 || k==n, 2, T[n-1,k,j] + T[n-1,k-1,j] + (2*j+1)*Prime[j]*T[n-2,k-1,j] ]]];
%t A153654 Table[T[n,k,9], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 03 2021 *)
%o A153654 (Sage)
%o A153654 @CachedFunction
%o A153654 def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
%o A153654 def T(n,k,j):
%o A153654     if (n==2): return nth_prime(j)
%o A153654     elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
%o A153654     elif (k==1 or k==n): return 2
%o A153654     else: return T(n-1,k,j) + T(n-1,k-1,j) + (2*j+1)*nth_prime(j)*T(n-2,k-1,j)
%o A153654 flatten([[T(n,k,9) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 03 2021
%o A153654 (Magma)
%o A153654 f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
%o A153654 function T(n,k,j)
%o A153654   if n eq 2 then return NthPrime(j);
%o A153654   elif (n eq 3 and k eq 2 or n eq 4 and k eq 2 or n eq 4 and k eq 3) then return f(n,j);
%o A153654   elif (k eq 1 or k eq n) then return 2;
%o A153654   else return T(n-1,k,j) + T(n-1,k-1,j) + (2*j+1)*NthPrime(j)*T(n-2,k-1,j);
%o A153654   end if; return T;
%o A153654 end function;
%o A153654 [T(n,k,9): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 03 2021
%Y A153654 Cf. A153652 (j=7), A153653 (j=8), this sequence (j=9), A153655 (j=10).
%Y A153654 Cf. A153516, A153518, A153520, A153521, A153648, A153649, A153650, A153651, A153656, A153657.
%K A153654 nonn,tabl,easy,less
%O A153654 1,1
%A A153654 _Roger L. Bagula_, Dec 30 2008
%E A153654 Edited by _G. C. Greubel_, Mar 03 2021