This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A153520 #8 Mar 06 2021 07:14:42
%S A153520 2,7,7,2,94,2,2,341,341,2,2,357,1340,357,2,2,373,4084,4084,373,2,2,
%T A153520 389,6956,17548,6956,389,2,2,405,9956,53092,53092,9956,405,2,2,421,
%U A153520 13084,111740,229020,111740,13084,421,2,2,437,16340,194516,712404,712404,194516,16340,437,2
%N A153520 Triangle T(n,k) = T(n-1, k) + T(n-1, k-1) + 7*T(n-2, k-1), read by rows.
%H A153520 G. C. Greubel, <a href="/A153520/b153520.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A153520 T(n,k) = T(n-1, k) + T(n-1, k-1) + 7*T(n-2, k-1).
%F A153520 From _G. C. Greubel_, Mar 04 2021: (Start)
%F A153520 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 A153520 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 A153520 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 A153520 Triangle begins as:
%e A153520 2;
%e A153520 7, 7;
%e A153520 2, 94, 2;
%e A153520 2, 341, 341, 2;
%e A153520 2, 357, 1340, 357, 2;
%e A153520 2, 373, 4084, 4084, 373, 2;
%e A153520 2, 389, 6956, 17548, 6956, 389, 2;
%e A153520 2, 405, 9956, 53092, 53092, 9956, 405, 2;
%e A153520 2, 421, 13084, 111740, 229020, 111740, 13084, 421, 2;
%e A153520 2, 437, 16340, 194516, 712404, 712404, 194516, 16340, 437, 2;
%t A153520 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 A153520 Table[T[n,k,0,1,4], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 04 2021 *)
%o A153520 (Sage)
%o A153520 @CachedFunction
%o A153520 def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
%o A153520 def T(n,k,p,q,j):
%o A153520 if (n==2): return nth_prime(j)
%o A153520 elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j)
%o A153520 elif (k==1 or k==n): return 2
%o A153520 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 A153520 flatten([[T(n,k,0,1,4) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 04 2021
%o A153520 (Magma)
%o A153520 f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
%o A153520 function T(n,k,p,q,j)
%o A153520 if n eq 2 then return NthPrime(j);
%o A153520 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 A153520 elif (k eq 1 or k eq n) then return 2;
%o A153520 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 A153520 end if; return T;
%o A153520 end function;
%o A153520 [T(n,k,0,1,4): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 04 2021
%Y A153520 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 A153520 nonn,tabl,easy,less
%O A153520 1,1
%A A153520 _Roger L. Bagula_, Dec 28 2008
%E A153520 Edited by _G. C. Greubel_, Mar 04 2021