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%I A153648 #8 Mar 05 2021 02:51:30
%S A153648 2,5,5,2,46,2,2,123,123,2,2,155,936,155,2,2,187,2936,2936,187,2,2,219,
%T A153648 5448,19912,5448,219,2,2,251,8472,69400,69400,8472,251,2,2,283,12008,
%U A153648 159592,437480,159592,12008,283,2,2,315,16056,298680,1638072,1638072,298680,16056,315,2
%N A153648 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 A153648 G. C. Greubel, <a href="/A153648/b153648.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A153648 T(n, k) = T(n-1, k) + T(n-1, k-1) + j*prime(j)*T(n-2, k-1) with j=3.
%F A153648 From _G. C. Greubel_, Mar 04 2021: (Start)
%F A153648 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 A153648 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 A153648 Triangle begins as:
%e A153648   2;
%e A153648   5,   5;
%e A153648   2,  46,     2;
%e A153648   2, 123,   123,      2;
%e A153648   2, 155,   936,    155,       2;
%e A153648   2, 187,  2936,   2936,     187,       2;
%e A153648   2, 219,  5448,  19912,    5448,     219,      2;
%e A153648   2, 251,  8472,  69400,   69400,    8472,    251,     2;
%e A153648   2, 283, 12008, 159592,  437480,  159592,  12008,   283,   2;
%e A153648   2, 315, 16056, 298680, 1638072, 1638072, 298680, 16056, 315, 2;
%t A153648 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 A153648 Table[T[n,k,1,0,3], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 04 2021 *)
%o A153648 (Sage)
%o A153648 @CachedFunction
%o A153648 def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
%o A153648 def T(n,k,p,q,j):
%o A153648     if (n==2): return nth_prime(j)
%o A153648     elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
%o A153648     elif (k==1 or k==n): return 2
%o A153648     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 A153648 flatten([[T(n,k,1,0,3) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 04 2021
%o A153648 (Magma)
%o A153648 f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
%o A153648 function T(n,k,p,q,j)
%o A153648   if n eq 2 then return NthPrime(j);
%o A153648   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 A153648   elif (k eq 1 or k eq n) then return 2;
%o A153648   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 A153648   end if; return T;
%o A153648 end function;
%o A153648 [T(n,k,1,0,3): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 04 2021
%Y A153648 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 A153648 Cf. A000351 (powers of 5).
%K A153648 nonn,tabl
%O A153648 1,1
%A A153648 _Roger L. Bagula_, Dec 30 2008
%E A153648 Edited by _G. C. Greubel_, Mar 04 2021

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