A153649 - OEIS

%I #7 Mar 05 2021 10:16:37

%S 2,7,7,2,94,2,2,341,341,2,2,413,3972,413,2,2,485,16320,16320,485,2,2,

%T 557,31260,171660,31260,557,2,2,629,48792,774120,774120,48792,629,2,2,

%U 701,68916,1917012,7556340,1917012,68916,701,2,2,773,91632,3693648,36567552,36567552,3693648,91632,773,2

%N Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (j+1)*prime(j)*T(n-2, k-1) with j=4, read by rows.

%H G. C. Greubel, <a href="/A153649/b153649.txt">Rows n = 1..50 of the triangle, flattened</a>

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

%F From _G. C. Greubel_, Mar 04 2021: (Start)

%F 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) = (1,1,4).

%F Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j)=(1,1,4), = 2*A000420(n-1). (End)

%e Triangle begins as:

%e   2;

%e   7,   7;

%e   2,  94,     2;

%e   2, 341,   341,       2;

%e   2, 413,  3972,     413,        2;

%e   2, 485, 16320,   16320,      485,        2;

%e   2, 557, 31260,  171660,    31260,      557,       2;

%e   2, 629, 48792,  774120,   774120,    48792,     629,     2;

%e   2, 701, 68916, 1917012,  7556340,  1917012,   68916,   701,   2;

%e   2, 773, 91632, 3693648, 36567552, 36567552, 3693648, 91632, 773, 2;

%t 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 Table[T[n,k,1,1,4], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 04 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,p,q,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,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 flatten([[T(n,k,1,1,4) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 04 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,p,q,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,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   end if; return T;

%o end function;

%o [T(n,k,1,1,4): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 04 2021

%Y 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), this sequence (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 Cf. A000420 (powers of 7).

%K nonn,tabl

%O 1,1

%A _Roger L. Bagula_, Dec 30 2008

%E Edited by _G. C. Greubel_, Mar 04 2021