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%I #7 Mar 05 2021 10:16:46
%S 2,11,11,2,238,2,2,1329,1329,2,2,1529,26220,1529,2,2,1729,159320,
%T 159320,1729,2,2,1929,312420,2914420,312420,1929,2,2,2129,485520,
%U 18999520,18999520,485520,2129,2,2,2329,678620,50414620,326526620,50414620,678620,2329,2
%N Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+4)*prime(j)*T(n-2, k-1) with j=5, read by rows.
%H G. C. Greubel, <a href="/A153650/b153650.txt">Rows n = 1..50 of the triangle, flattened</a>
%F T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+4)*prime(j)*T(n-2, k-1) with j=5.
%F From _G. C. Greubel_, Mar 04 2021: (Start)
%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) = (1,4,5).
%F Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1) for j=5 = 2*A001020(n-1). (End)
%e Triangle begins as:
%e 2;
%e 11, 11;
%e 2, 238, 2;
%e 2, 1329, 1329, 2;
%e 2, 1529, 26220, 1529, 2;
%e 2, 1729, 159320, 159320, 1729, 2;
%e 2, 1929, 312420, 2914420, 312420, 1929, 2;
%e 2, 2129, 485520, 18999520, 18999520, 485520, 2129, 2;
%e 2, 2329, 678620, 50414620, 326526620, 50414620, 678620, 2329, 2;
%e 2, 2529, 891720, 99159720, 2257893720, 2257893720, 99159720, 891720, 2529, 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,1,4,5], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 04 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,1,4,5) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 04 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,1,4,5): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 04 2021
%Y Sequences with variable (p,q,j): A153516 (0,1,2), A153518 (0,1,3), A153520 (0,1,4), A153521 (0,1,5), A153648 (1,0,3), A153649 (1,1,4), this sequence (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. A001020 (powers of 11).
%K nonn,tabl
%O 1,1
%A _Roger L. Bagula_, Dec 30 2008
%E Edited by _G. C. Greubel_, Mar 04 2021