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 A153656 #10 Mar 07 2021 03:06:26 %S A153656 2,23,23,2,1054,2,2,12165,12165,2,2,13133,533412,13133,2,2,14101, %T A153656 6422240,6422240,14101,2,2,15069,12779580,270482476,12779580,15069,2, %U A153656 2,16037,19605432,3385203976,3385203976,19605432,16037,2,2,17005,26899796,9577346548,137413443860,9577346548,26899796,17005,2 %N A153656 Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +3)*prime(j)*T(n-2, k-1) with j=9, read by rows. %H A153656 G. C. Greubel, <a href="/A153656/b153656.txt">Rows n = 1..50 of the triangle, flattened</a> %F A153656 T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +3)*prime(j)*T(n-2, k-1) with j=9. %F A153656 From _G. C. Greubel_, Mar 06 2021: (Start) %F A153656 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) = (2,3,9). %F A153656 Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j)=(2,3,9), = 2*A009967(n-1). (End) %e A153656 Triangle begins as: %e A153656 2; %e A153656 23, 23; %e A153656 2, 1054, 2; %e A153656 2, 12165, 12165, 2; %e A153656 2, 13133, 533412, 13133, 2; %e A153656 2, 14101, 6422240, 6422240, 14101, 2; %e A153656 2, 15069, 12779580, 270482476, 12779580, 15069, 2; %e A153656 2, 16037, 19605432, 3385203976, 3385203976, 19605432, 16037, 2; %e A153656 2, 17005, 26899796, 9577346548, 137413443860, 9577346548, 26899796, 17005, 2; %t A153656 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 A153656 Table[T[n,k,2,3,9], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 06 2021 *) %o A153656 (Sage) %o A153656 @CachedFunction %o A153656 def f(n,j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2) %o A153656 def T(n,k,p,q,j): %o A153656 if (n==2): return nth_prime(j) %o A153656 elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n,j) %o A153656 elif (k==1 or k==n): return 2 %o A153656 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 A153656 flatten([[T(n,k,2,3,9) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Mar 06 2021 %o A153656 (Magma) %o A153656 f:= func< n,j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >; %o A153656 function T(n,k,p,q,j) %o A153656 if n eq 2 then return NthPrime(j); %o A153656 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 A153656 elif (k eq 1 or k eq n) then return 2; %o A153656 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 A153656 end if; return T; %o A153656 end function; %o A153656 [T(n,k,2,3,9): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 06 2021 %Y A153656 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), 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), this sequence (2,3,9), A153657 (2,7,10). %Y A153656 Cf. A009967 (powers of 23). %K A153656 nonn,tabl,easy,less %O A153656 1,1 %A A153656 _Roger L. Bagula_, Dec 30 2008 %E A153656 Edited by _G. C. Greubel_, Mar 06 2021