%I #16 Mar 04 2021 07:04:34

%S 2,5,5,2,46,2,2,123,123,2,2,135,476,135,2,2,147,1226,1226,147,2,2,159,

%T 2048,4832,2048,159,2,2,171,2942,13010,13010,2942,171,2,2,183,3908,

%U 26192,50180,26192,3908,183,2,2,195,4946,44810,141422,141422,44810,4946,195,2

%N Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.

%H G. C. Greubel, <a href="/A153518/b153518.txt">Rows n = 1..50 of the triangle, flattened</a>

%F T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1).

%F Recurrence row sums: s(n) = 2*s(n-1) + 5*s(n-2), n > 4, with s(1) = 2, s(2) = 10, s(3) = 50, s(4) = 250. - _R. J. Mathar_, Jan 22 2009

%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) = (0,1,3).

%F Sum_{k=0..n} T(n,k,0,1,3) = 4*(-5)^n*[n<2] + 50*(i*sqrt(5))^(n-2)*(ChebyshevU(n-2, -i/sqrt(5)) - (3*i/sqrt(5))*ChebyshevU(n-3, -i/sqrt(5))) = 4*(-5)^n*[n<2] + 50*A123011(n-2). (End)

%e Triangle begins as:

%e 2;

%e 5, 5;

%e 2, 46, 2;

%e 2, 123, 123, 2;

%e 2, 135, 476, 135, 2;

%e 2, 147, 1226, 1226, 147, 2;

%e 2, 159, 2048, 4832, 2048, 159, 2;

%e 2, 171, 2942, 13010, 13010, 2942, 171, 2;

%e 2, 183, 3908, 26192, 50180, 26192, 3908, 183, 2;

%e 2, 195, 4946, 44810, 141422, 141422, 44810, 4946, 195, 2;

%p A153518 := proc(n,k) option remember ; if n =1 then 2; elif n = 2 then 5; elif k=1 or k=n then 2; elif n = 3 then 46 ; elif n = 4 then 123 ; else procname(n-1,k-1)+procname(n-1,k)+5*procname(n-2,k-1) ; end: end: for n from 1 to 13 do for k from 1 to n do printf("%d,",A153518(n,k)) ; od: od: # _R. J. Mathar_, Jan 22 2009

%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,0,1,3], {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,0,1,3) 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,0,1,3): 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), this sequences (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), A153657 (2,7,10).

%Y Cf. A123011.

%K nonn,tabl,easy,less

%O 1,1

%A _Roger L. Bagula_, Dec 28 2008

%E More terms from _R. J. Mathar_, Jan 22 2009

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