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%I #10 Mar 07 2021 03:06:26
%S 2,23,23,2,1054,2,2,12165,12165,2,2,13133,533412,13133,2,2,14101,
%T 6422240,6422240,14101,2,2,15069,12779580,270482476,12779580,15069,2,
%U 2,16037,19605432,3385203976,3385203976,19605432,16037,2,2,17005,26899796,9577346548,137413443860,9577346548,26899796,17005,2
%N Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +3)*prime(j)*T(n-2, k-1) with j=9, read by rows.
%H G. C. Greubel, <a href="/A153656/b153656.txt">Rows n = 1..50 of the triangle, flattened</a>
%F T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +3)*prime(j)*T(n-2, k-1) with j=9.
%F From _G. C. Greubel_, Mar 06 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) = (2,3,9).
%F Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j)=(2,3,9), = 2*A009967(n-1). (End)
%e Triangle begins as:
%e 2;
%e 23, 23;
%e 2, 1054, 2;
%e 2, 12165, 12165, 2;
%e 2, 13133, 533412, 13133, 2;
%e 2, 14101, 6422240, 6422240, 14101, 2;
%e 2, 15069, 12779580, 270482476, 12779580, 15069, 2;
%e 2, 16037, 19605432, 3385203976, 3385203976, 19605432, 16037, 2;
%e 2, 17005, 26899796, 9577346548, 137413443860, 9577346548, 26899796, 17005, 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,2,3,9], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 06 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,2,3,9) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 06 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,2,3,9): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 06 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), A153650 (1,4,5), A153651 (1,5,6), A153652 (2,1,7), A153653 (2,1,8), A153654 (2,1,9), A153655 (2,1,10), this sequence (2,3,9), A153657 (2,7,10).
%Y Cf. A009967 (powers of 23).
%K nonn,tabl,easy,less
%O 1,1
%A _Roger L. Bagula_, Dec 30 2008
%E Edited by _G. C. Greubel_, Mar 06 2021
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