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A153521
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Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + 11*T(n-2, k-1), read by rows.
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15
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2, 11, 11, 2, 238, 2, 2, 1329, 1329, 2, 2, 1353, 5276, 1353, 2, 2, 1377, 21248, 21248, 1377, 2, 2, 1401, 37508, 100532, 37508, 1401, 2, 2, 1425, 54056, 371768, 371768, 54056, 1425, 2, 2, 1449, 70892, 838412, 1849388, 838412, 70892, 1449, 2, 2, 1473, 88016, 1503920, 6777248, 6777248, 1503920, 88016, 1473, 2
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OFFSET
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1,1
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LINKS
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FORMULA
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T(n, k)= T(n-1, k) + T(n-1, k-1) + 11*T(n-2, k-1).
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).
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).
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)
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EXAMPLE
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Triangle begins as:
2;
11, 11;
2, 238, 2;
2, 1329, 1329, 2;
2, 1353, 5276, 1353, 2;
2, 1377, 21248, 21248, 1377, 2;
2, 1401, 37508, 100532, 37508, 1401, 2;
2, 1425, 54056, 371768, 371768, 54056, 1425, 2;
2, 1449, 70892, 838412, 1849388, 838412, 70892, 1449, 2;
2, 1473, 88016, 1503920, 6777248, 6777248, 1503920, 88016, 1473, 2;
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MATHEMATICA
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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] ]]];
Table[T[n, k, 0, 1, 5], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 04 2021 *)
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PROG
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(Sage)
@CachedFunction
def f(n, j): return ((3-(-1)^n)/2)*nth_prime(j)^(n-1) - 2^((3-(-1)^n)/2)
def T(n, k, p, q, j):
if (n==2): return nth_prime(j)
elif (n==3 and k==2 or n==4 and 2<=k<=3): return f(n, j)
elif (k==1 or k==n): return 2
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)
flatten([[T(n, k, 0, 1, 5) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 04 2021
(Magma)
f:= func< n, j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
function T(n, k, p, q, j)
if n eq 2 then return NthPrime(j);
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);
elif (k eq 1 or k eq n) then return 2;
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);
end if; return T;
end function;
[T(n, k, 0, 1, 5): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 2021
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CROSSREFS
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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).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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