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A153655 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 = 10, read by rows.

%I A153655 #19 Mar 04 2021 10:59:46
%S A153655 2,29,29,2,1678,2,2,24387,24387,2,2,25607,1070676,25607,2,2,26827,
%T A153655 15947966,15947966,26827,2,2,28047,31569456,683937616,31569456,28047,
%U A153655 2,2,29267,47935146,10427818366,10427818366,47935146,29267,2,2,30487,65045036,29701552216,437373644876,29701552216,65045036,30487,2
%N A153655 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 = 10, read by rows.
%H A153655 G. C. Greubel, <a href="/A153655/b153655.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A153655 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 = 10.
%F A153655 From _G. C. Greubel_, Mar 04 2021: (Start)
%F A153655 Sum_{k=0..n} T(n, k, 10) = -(76/147)*[n=0] - (116/7)*[n=1] + 1682*(i*sqrt(609))^(n-2)*(ChebyshevU(n-2, -i/sqrt(609)) - (27*i/sqrt(609))*ChebyshevU(n-3, -i/sqrt(609) )).
%F A153655 Row sums satisfy the recurrence S(n) = 2*S(n-1) + 609*S(n-2) for n>4 with S(0) = 2, S(1) = 58, S(2) = 1682, S(3) = 48778. (End)
%e A153655 Triangle begins as:
%e A153655    2;
%e A153655   29,    29;
%e A153655    2,  1678,        2;
%e A153655    2, 24387,    24387,           2;
%e A153655    2, 25607,  1070676,       25607,            2;
%e A153655    2, 26827, 15947966,    15947966,        26827,           2;
%e A153655    2, 28047, 31569456,   683937616,     31569456,       28047,        2;
%e A153655    2, 29267, 47935146, 10427818366,  10427818366,    47935146,    29267,     2;
%e A153655    2, 30487, 65045036, 29701552216, 437373644876, 29701552216, 65045036, 30487, 2;
%t A153655 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 A153655 Table[T[n,k,10], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 03 2021 *)
%o A153655 (Sage)
%o A153655 @CachedFunction
%o A153655 def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
%o A153655 def T(n,k,j):
%o A153655     if (n==2): return nth_prime(j)
%o A153655     elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
%o A153655     elif (k==1 or k==n): return 2
%o A153655     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 A153655 flatten([[T(n,k,10) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 03 2021
%o A153655 (Magma)
%o A153655 f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
%o A153655 function T(n,k,j)
%o A153655   if n eq 2 then return NthPrime(j);
%o A153655   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 A153655   elif (k eq 1 or k eq n) then return 2;
%o A153655   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 A153655   end if; return T;
%o A153655 end function;
%o A153655 [T(n,k,10): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 03 2021
%Y A153655 Cf. A153652 (j=7), A153653 (j=8), A153654 (j=9), this sequence (j=10).
%Y A153655 Cf. A153516, A153518, A153520, A153521, A153648, A153649, A153650, A153651, A153656, A153657.
%K A153655 nonn,tabl,easy,less
%O A153655 1,1
%A A153655 _Roger L. Bagula_, Dec 30 2008
%E A153655 Edited by _G. C. Greubel_, Mar 03 2021