A153652 - OEIS

%I #22 Mar 05 2021 02:51:16

%S 2,17,17,2,574,2,2,4911,4911,2,2,5423,156192,5423,2,2,5935,1413920,

%T 1413920,5935,2,2,6447,2802720,42656800,2802720,6447,2,2,6959,4322592,

%U 406009120,406009120,4322592,6959,2,2,7471,5973536,1125025312,11689502240,1125025312,5973536,7471,2

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

%H G. C. Greubel, <a href="/A153652/b153652.txt">Rows n = 1..50 of the triangle, flattened</a>

%F T(n, k, j) =  T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j), T(3, 2, j) = 2*prime(j)^2 - 4, T(4, 2, j) = T(4, 3, j) = prime(j)^2 - 2, T(n, 1, j) = T(n, n, j) = 2 and j = 7.

%F Sum_{k=0..n} T(n, k, j) = 2*prime(j)^(n-1) for j=7 = 2*A001026(n-1).

%e Triangle begins as:

%e    2;

%e   17,   17;

%e    2,  574,       2;

%e    2, 4911,    4911,          2;

%e    2, 5423,  156192,       5423,           2;

%e    2, 5935, 1413920,    1413920,        5935,          2;

%e    2, 6447, 2802720,   42656800,     2802720,       6447,       2;

%e    2, 6959, 4322592,  406009120,   406009120,    4322592,    6959,    2;

%e    2, 7471, 5973536, 1125025312, 11689502240, 1125025312, 5973536, 7471, 2;

%t T[n_, k_, j_]:= T[n,k,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,j] + T[n-1,k-1,j] + (2*j+1)*Prime[j]*T[n-2,k-1,j] ]]];

%t Table[T[n,k,7], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 02 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,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,j) + T(n-1,k-1,j) + (2*j+1)*nth_prime(j)*T(n-2,k-1,j)

%o flatten([[T(n,k,7) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 02 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,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,j) + T(n-1,k-1,j) + (2*j+1)*NthPrime(j)*T(n-2,k-1,j);

%o   end if; return T;

%o end function;

%o [T(n,k,7): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 02 2021

%Y Cf. this sequence (j=7), A153653 (j=8), A153654 (j=9), A153655 (j=10).

%Y Cf. A153516, A153518, A153520, A153521, A153648, A153649, A153650, A153651, A153656, A153657.

%Y Cf. A001026 (powers of 17).

%K nonn,tabl,easy,less

%O 1,1

%A _Roger L. Bagula_, Dec 30 2008

%E Edited by _G. C. Greubel_, Mar 02 2021