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%I #10 Mar 04 2021 07:04:28
%S 2,3,3,2,14,2,2,25,25,2,2,33,92,33,2,2,41,200,200,41,2,2,49,340,676,
%T 340,49,2,2,57,512,1616,1616,512,57,2,2,65,716,3148,5260,3148,716,65,
%U 2,2,73,952,5400,13256,13256,5400,952,73,2
%N Triangle T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j) and (p,q,j) = (0,1,2), read by rows.
%H G. C. Greubel, <a href="/A153516/b153516.txt">Rows n = 1..50 of the triangle, flattened</a>
%F T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p, q,j) with T(2,k,p,q,j) = prime(j), T(3,2,p,q,j) = 2*prime(j)^2 - 4, T(4,2,p,q,j) = T(4,3,p,q,j) = prime(j)^2 - 2, T(n,1,p,q,j) = T(n,n,p,q,j) = 2 and (p,q,j) = (0,1,2).
%F Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1) for (p,q,j) = (0,1,2) = 2*A000244(n-1).
%e Triangle begins as:
%e 2;
%e 3, 3;
%e 2, 14, 2;
%e 2, 25, 25, 2;
%e 2, 33, 92, 33, 2;
%e 2, 41, 200, 200, 41, 2;
%e 2, 49, 340, 676, 340, 49, 2;
%e 2, 57, 512, 1616, 1616, 512, 57, 2;
%e 2, 65, 716, 3148, 5260, 3148, 716, 65, 2;
%e 2, 73, 952, 5400, 13256, 13256, 5400, 952, 73, 2;
%t T[n_, k_, p_, q_, j_]:= T[n,k,p,q,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,p,q,j] + T[n-1,k-1,p,q,j] + (p*j+q)*Prime[j]*T[n-2,k-1,p,q,j] ]]];
%t Table[T[n,k,0,1,2], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 03 2021 *)
%o (Sage)
%o @CachedFunction
%o def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
%o def T(n,k,p,q,j):
%o if (n==2): return nth_prime(j)
%o elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
%o elif (k==1 or k==n): return 2
%o else: return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*nth_prime(j)*T(n-2,k-1,p,q,j)
%o flatten([[T(n,k,0,1,2) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 03 2021
%o (Magma)
%o f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
%o function T(n,k,p,q,j)
%o if n eq 2 then return NthPrime(j);
%o 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 elif (k eq 1 or k eq n) then return 2;
%o else return T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*NthPrime(j)*T(n-2,k-1,p,q,j);
%o end if; return T;
%o end function;
%o [T(n,k,0,1,2): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 03 2021
%Y Sequences with variable (p,q,j): this sequence (0,1,2), A153518 (0,1,3), A153520 (0,1,4), A153521 (0,1,5), A153648 (1,0,3), A153649 (1,1,4), A153650 (1,4,5), A153651 (1,5,6), A153652 (2,1,7), A153653 (2,1,8), A153654 (2,1,9), A153655 (2,1,10), A153656 (2,3,9), A153657 (2,7,10).
%Y Cf. A000244.
%K nonn,tabl,easy,less
%O 1,1
%A _Roger L. Bagula_, Dec 28 2008
%E Edited by _G. C. Greubel_, Mar 03 2021
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