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 A153516 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. 14
 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, 340, 49, 2, 2, 57, 512, 1616, 1616, 512, 57, 2, 2, 65, 716, 3148, 5260, 3148, 716, 65, 2, 2, 73, 952, 5400, 13256, 13256, 5400, 952, 73, 2 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS G. C. Greubel, Rows n = 1..50 of the triangle, flattened FORMULA 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). 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). EXAMPLE Triangle begins as:   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,   340,    49,    2;   2, 57, 512, 1616,  1616,   512,   57,   2;   2, 65, 716, 3148,  5260,  3148,  716,  65,  2;   2, 73, 952, 5400, 13256, 13256, 5400, 952, 73, 2; MATHEMATICA 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] ]]]; Table[T[n, k, 0, 1, 2], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 03 2021 *) PROG (Sage) @CachedFunction def f(n, j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2) def T(n, k, p, q, j):     if (n==2): return nth_prime(j)     elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n, j)     elif (k==1 or k==n): return 2     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) flatten([[T(n, k, 0, 1, 2) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 03 2021 (Magma) f:= func< n, j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >; function T(n, k, p, q, j)   if n eq 2 then return NthPrime(j);   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);   elif (k eq 1 or k eq n) then return 2;   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);   end if; return T; end function; [T(n, k, 0, 1, 2): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 03 2021 CROSSREFS 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). Cf. A000244. Sequence in context: A184829 A338307 A153290 * A153311 A153312 A153283 Adjacent sequences:  A153513 A153514 A153515 * A153517 A153518 A153519 KEYWORD nonn,tabl,easy,less AUTHOR Roger L. Bagula, Dec 28 2008 EXTENSIONS Edited by G. C. Greubel, Mar 03 2021 STATUS approved

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Last modified July 23 12:19 EDT 2021. Contains 346259 sequences. (Running on oeis4.)