|
%I #18 Mar 06 2021 03:34:44
%S 2,19,19,2,718,2,2,6857,6857,2,2,7505,245628,7505,2,2,8153,2467944,
%T 2467944,8153,2,2,8801,4900212,84273732,4900212,8801,2,2,9449,7542432,
%U 886319856,886319856,7542432,9449,2,2,10097,10394604,2476630764,28993055148,2476630764,10394604,10097,2
%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 = 8, read by rows.
%H G. C. Greubel, <a href="/A153653/b153653.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 = 8.
%F Sum_{k=0..n} T(n, k, j) = 2*prime(j)^(n-1) for j=8 = 2*A001029(n-1).
%e Triangle begins as:
%e 2;
%e 19, 19;
%e 2, 718, 2;
%e 2, 6857, 6857, 2;
%e 2, 7505, 245628, 7505, 2;
%e 2, 8153, 2467944, 2467944, 8153, 2;
%e 2, 8801, 4900212, 84273732, 4900212, 8801, 2;
%e 2, 9449, 7542432, 886319856, 886319856, 7542432, 9449, 2;
%e 2, 10097, 10394604, 2476630764, 28993055148, 2476630764, 10394604, 10097, 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,8], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 03 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,8) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 03 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,8): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 03 2021
%Y Cf. A153652 (j=7), this sequence (j=8), A153654 (j=9), A153655 (j=10).
%Y Cf. A153516, A153518, A153520, A153521, A153648, A153649, A153650, A153651, A153656, A153657.
%Y Cf. A001029 (powers of 19).
%K nonn,tabl,easy,less
%O 1,1
%A _Roger L. Bagula_, Dec 30 2008
%E Edited by _G. C. Greubel_, Mar 03 2021
|
