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A153651 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)*prime(j)*T(n-2, k-1) with j=6, read by rows.

%I A153651 #8 Mar 07 2021 03:06:00
%S A153651 2,13,13,2,334,2,2,2195,2195,2,2,2483,52152,2483,2,2,2771,368520,
%T A153651 368520,2771,2,2,3059,726360,8194776,726360,3059,2,2,3347,1125672,
%U A153651 61619496,61619496,1125672,3347,2,2,3635,1566456,166614648,1295091960,166614648,1566456,3635,2
%N A153651 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)*prime(j)*T(n-2, k-1) with j=6, read by rows.
%H A153651 G. C. Greubel, <a href="/A153651/b153651.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A153651 T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+5)*prime(j)*T(n-2, k-1) with j=6.
%F A153651 From _G. C. Greubel_, Mar 06 2021: (Start)
%F A153651 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) = (1,5,6).
%F A153651 Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1) for j=6, = 2*A001022(n-1). (End)
%e A153651 Triangle begins as:
%e A153651    2;
%e A153651   13,   13;
%e A153651    2,  334,       2;
%e A153651    2, 2195,    2195,         2;
%e A153651    2, 2483,   52152,      2483,          2;
%e A153651    2, 2771,  368520,    368520,       2771,         2;
%e A153651    2, 3059,  726360,   8194776,     726360,      3059,       2;
%e A153651    2, 3347, 1125672,  61619496,   61619496,   1125672,    3347,    2;
%e A153651    2, 3635, 1566456, 166614648, 1295091960, 166614648, 1566456, 3635, 2;
%t A153651 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 A153651 Table[T[n,k,1,5,6], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 06 2021 *)
%o A153651 (Sage)
%o A153651 @CachedFunction
%o A153651 def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
%o A153651 def T(n,k,p,q,j):
%o A153651     if (n==2): return nth_prime(j)
%o A153651     elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
%o A153651     elif (k==1 or k==n): return 2
%o A153651     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 A153651 flatten([[T(n,k,1,5,6) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 06 2021
%o A153651 (Magma)
%o A153651 f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
%o A153651 function T(n,k,p,q,j)
%o A153651   if n eq 2 then return NthPrime(j);
%o A153651   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 A153651   elif (k eq 1 or k eq n) then return 2;
%o A153651   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 A153651   end if; return T;
%o A153651 end function;
%o A153651 [T(n,k,1,5,6): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 06 2021
%Y A153651 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), this sequence (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 A153651 Cf. A001022 (powers of 13).
%K A153651 nonn,tabl,easy,less
%O A153651 1,1
%A A153651 _Roger L. Bagula_, Dec 30 2008
%E A153651 Edited by _G. C. Greubel_, Mar 06 2021