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Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + j*prime(j)*T(n-2, k-1) with j=3, read by rows.
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%I #8 Mar 05 2021 02:51:30

%S 2,5,5,2,46,2,2,123,123,2,2,155,936,155,2,2,187,2936,2936,187,2,2,219,

%T 5448,19912,5448,219,2,2,251,8472,69400,69400,8472,251,2,2,283,12008,

%U 159592,437480,159592,12008,283,2,2,315,16056,298680,1638072,1638072,298680,16056,315,2

%N Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + j*prime(j)*T(n-2, k-1) with j=3, read by rows.

%H G. C. Greubel, <a href="/A153648/b153648.txt">Rows n = 1..50 of the triangle, flattened</a>

%F T(n, k) = T(n-1, k) + T(n-1, k-1) + j*prime(j)*T(n-2, k-1) with j=3.

%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) = (1,0,3).

%F Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j) = (1,0,3), = 2*A000351(n-1). (End)

%e Triangle begins as:

%e 2;

%e 5, 5;

%e 2, 46, 2;

%e 2, 123, 123, 2;

%e 2, 155, 936, 155, 2;

%e 2, 187, 2936, 2936, 187, 2;

%e 2, 219, 5448, 19912, 5448, 219, 2;

%e 2, 251, 8472, 69400, 69400, 8472, 251, 2;

%e 2, 283, 12008, 159592, 437480, 159592, 12008, 283, 2;

%e 2, 315, 16056, 298680, 1638072, 1638072, 298680, 16056, 315, 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,1,0,3], {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,1,0,3) 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,1,0,3): 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), this sequence (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. A000351 (powers of 5).

%K nonn,tabl

%O 1,1

%A _Roger L. Bagula_, Dec 30 2008

%E Edited by _G. C. Greubel_, Mar 04 2021