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A153657 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +7)*prime(j)*T(n-2, k-1) with j=10, read by rows.

%I A153657 #10 Apr 25 2024 09:08:23
%S A153657 2,29,29,2,1678,2,2,24387,24387,2,2,25955,1362648,25955,2,2,27523,
%T A153657 20483624,20483624,27523,2,2,29091,40833912,1107920632,40833912,29091,
%U A153657 2,2,30659,62413512,17187432136,17187432136,62413512,30659,2,2,32227,85222424,49222798744,901876719128,49222798744,85222424,32227,2
%N A153657 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +7)*prime(j)*T(n-2, k-1) with j=10, read by rows.
%H A153657 G. C. Greubel, <a href="/A153657/b153657.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A153657 T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +7)*prime(j)*T(n-2, k-1) with j=10.
%F A153657 From _G. C. Greubel_, Mar 06 2021: (Start)
%F A153657 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) = (2,7,10).
%F A153657 Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j)=(2,7,10), = 2*A009973(n-1). (End)
%e A153657 Triangle begins as:
%e A153657    2;
%e A153657   29,    29;
%e A153657    2,  1678,        2;
%e A153657    2, 24387,    24387,           2;
%e A153657    2, 25955,  1362648,       25955,            2;
%e A153657    2, 27523, 20483624,    20483624,        27523,           2;
%e A153657    2, 29091, 40833912,  1107920632,     40833912,       29091,       2;
%e A153657    2, 30659, 62413512, 17187432136,  17187432136,    62413512,    30659,     2;
%e A153657    2, 32227, 85222424, 49222798744, 901876719128, 49222798744, 85222424, 32227, 2;
%t A153657 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 A153657 Table[T[n,k,2,7,10], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 06 2021 *)
%o A153657 (Sage)
%o A153657 @CachedFunction
%o A153657 def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
%o A153657 def T(n,k,p,q,j):
%o A153657     if (n==2): return nth_prime(j)
%o A153657     elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
%o A153657     elif (k==1 or k==n): return 2
%o A153657     else: return T(n-1,k,2,7,10) + T(n-1,k-1,p,q,j) + (p*j+q)*nth_prime(j)*T(n-2,k-1,p,q,j)
%o A153657 flatten([[T(n,k,p,q,j) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 06 2021
%o A153657 (Magma)
%o A153657 f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
%o A153657 function T(n,k,p,q,j)
%o A153657   if n eq 2 then return NthPrime(j);
%o A153657   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 A153657   elif (k eq 1 or k eq n) then return 2;
%o A153657   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 A153657   end if; return T;
%o A153657 end function;
%o A153657 [T(n,k,2,7,10): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 06 2021
%Y A153657 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), A153656 (2,3,9), this sequence (2,7,10).
%Y A153657 Cf. A009973 (powers of 29).
%K A153657 nonn,tabl,easy,less
%O A153657 1,1
%A A153657 _Roger L. Bagula_, Dec 30 2008
%E A153657 Edited by _G. C. Greubel_, Mar 06 2021