A153520 - OEIS

A153520

Triangle T(n,k) = T(n-1, k) + T(n-1, k-1) + 7*T(n-2, k-1), read by rows.

%I #8 Mar 06 2021 07:14:42

%S 2,7,7,2,94,2,2,341,341,2,2,357,1340,357,2,2,373,4084,4084,373,2,2,

%T 389,6956,17548,6956,389,2,2,405,9956,53092,53092,9956,405,2,2,421,

%U 13084,111740,229020,111740,13084,421,2,2,437,16340,194516,712404,712404,194516,16340,437,2

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

%H G. C. Greubel, <a href="/A153520/b153520.txt">Rows n = 1..50 of the triangle, flattened</a>

%F T(n,k) = T(n-1, k) + T(n-1, k-1) + 7*T(n-2, k-1).

%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) = (0,1,4).

%F Sum_{k=1..n} T(n,k,p,q,j) = 2*(prime(j)-3)*[n=1] -2*prime(j)*(prime(j)-3)*[n=2] +2*prime(j)^2*(i*sqrt(prime(j)))^(n-3)*(ChebyshevU(n-3, -i/Sqrt(prime(j))) -((prime(j) -2)*i/sqrt(prime(j)))*ChebyshevU(n-4, -i/sqrt(prime(j)))) for (p,q,j)=(0,1,4).

%F Row sums satisfy the recurrence relation S(n) = 2*S(n-1) + prime(j)*S(n-2), for n > 4, with S(1) = 2, S(2) = 2*prime(j), S(3) = 2*prime(j)^2, S(4) = 2*prime(j)^3 with j=4. (End)

%e Triangle begins as:

%e 2;

%e 7, 7;

%e 2, 94, 2;

%e 2, 341, 341, 2;

%e 2, 357, 1340, 357, 2;

%e 2, 373, 4084, 4084, 373, 2;

%e 2, 389, 6956, 17548, 6956, 389, 2;

%e 2, 405, 9956, 53092, 53092, 9956, 405, 2;

%e 2, 421, 13084, 111740, 229020, 111740, 13084, 421, 2;

%e 2, 437, 16340, 194516, 712404, 712404, 194516, 16340, 437, 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,0,1,4], {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,0,1,4) 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,0,1,4): 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), this sequence (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), A153657 (2,7,10).

%K nonn,tabl,easy,less

%O 1,1

%A _Roger L. Bagula_, Dec 28 2008

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