cp's OEIS Frontend

A153518 Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.

%I A153518 #16 Mar 04 2021 07:04:34
%S A153518 2,5,5,2,46,2,2,123,123,2,2,135,476,135,2,2,147,1226,1226,147,2,2,159,
%T A153518 2048,4832,2048,159,2,2,171,2942,13010,13010,2942,171,2,2,183,3908,
%U A153518 26192,50180,26192,3908,183,2,2,195,4946,44810,141422,141422,44810,4946,195,2
%N A153518 Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.
%H A153518 G. C. Greubel, <a href="/A153518/b153518.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A153518 T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1).
%F A153518 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 A153518 From _G. C. Greubel_, Mar 04 2021: (Start)
%F A153518 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 A153518 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 A153518 Triangle begins as:
%e A153518   2;
%e A153518   5,   5;
%e A153518   2,  46,    2;
%e A153518   2, 123,  123,     2;
%e A153518   2, 135,  476,   135,      2;
%e A153518   2, 147, 1226,  1226,    147,      2;
%e A153518   2, 159, 2048,  4832,   2048,    159,     2;
%e A153518   2, 171, 2942, 13010,  13010,   2942,   171,    2;
%e A153518   2, 183, 3908, 26192,  50180,  26192,  3908,  183,   2;
%e A153518   2, 195, 4946, 44810, 141422, 141422, 44810, 4946, 195, 2;
%p A153518 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 A153518 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 A153518 Table[T[n,k,0,1,3], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 04 2021 *)
%o A153518 (Sage)
%o A153518 @CachedFunction
%o A153518 def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
%o A153518 def T(n,k,p,q,j):
%o A153518     if (n==2): return nth_prime(j)
%o A153518     elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
%o A153518     elif (k==1 or k==n): return 2
%o A153518     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 A153518 flatten([[T(n,k,0,1,3) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 04 2021
%o A153518 (Magma)
%o A153518 f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
%o A153518 function T(n,k,p,q,j)
%o A153518   if n eq 2 then return NthPrime(j);
%o A153518   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 A153518   elif (k eq 1 or k eq n) then return 2;
%o A153518   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 A153518   end if; return T;
%o A153518 end function;
%o A153518 [T(n,k,0,1,3): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 04 2021
%Y A153518 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 A153518 Cf. A123011.
%K A153518 nonn,tabl,easy,less
%O A153518 1,1
%A A153518 _Roger L. Bagula_, Dec 28 2008
%E A153518 More terms from _R. J. Mathar_, Jan 22 2009
%E A153518 Edited by _G. C. Greubel_, Mar 04 2021