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%I #11 Mar 06 2021 03:33:01
%S 2,11,11,2,238,2,2,1329,1329,2,2,1353,5276,1353,2,2,1377,21248,21248,
%T 1377,2,2,1401,37508,100532,37508,1401,2,2,1425,54056,371768,371768,
%U 54056,1425,2,2,1449,70892,838412,1849388,838412,70892,1449,2,2,1473,88016,1503920,6777248,6777248,1503920,88016,1473,2
%N Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + 11*T(n-2, k-1), read by rows.
%H G. C. Greubel, <a href="/A153521/b153521.txt">Rows n = 1..50 of the triangle, flattened</a>
%F T(n, k)= T(n-1, k) + T(n-1, k-1) + 11*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) = (0,1,5).
%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)=(0,1,5) = A151617(n).
%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 for j=5. (End)
%e Triangle begins as:
%e 2;
%e 11, 11;
%e 2, 238, 2;
%e 2, 1329, 1329, 2;
%e 2, 1353, 5276, 1353, 2;
%e 2, 1377, 21248, 21248, 1377, 2;
%e 2, 1401, 37508, 100532, 37508, 1401, 2;
%e 2, 1425, 54056, 371768, 371768, 54056, 1425, 2;
%e 2, 1449, 70892, 838412, 1849388, 838412, 70892, 1449, 2;
%e 2, 1473, 88016, 1503920, 6777248, 6777248, 1503920, 88016, 1473, 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,0,1,5], {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,0,1,5) 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,0,1,5): 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), A153520 (0,1,4), this sequence (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. A151617 (row sums).
%K nonn,tabl,easy,less
%O 1,1
%A _Roger L. Bagula_, Dec 28 2008
%E Edited by _G. C. Greubel_, Mar 04 2021
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