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A153655
Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 10, read by rows.
14
2, 29, 29, 2, 1678, 2, 2, 24387, 24387, 2, 2, 25607, 1070676, 25607, 2, 2, 26827, 15947966, 15947966, 26827, 2, 2, 28047, 31569456, 683937616, 31569456, 28047, 2, 2, 29267, 47935146, 10427818366, 10427818366, 47935146, 29267, 2, 2, 30487, 65045036, 29701552216, 437373644876, 29701552216, 65045036, 30487, 2
OFFSET
1,1
FORMULA
T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j), T(3, 2, j) = 2*prime(j)^2 - 4, T(4, 2, j) = T(4, 3, j) = prime(j)^2 - 2, T(n, 1, j) = T(n, n, j) = 2 and j = 10.
From G. C. Greubel, Mar 04 2021: (Start)
Sum_{k=0..n} T(n, k, 10) = -(76/147)*[n=0] - (116/7)*[n=1] + 1682*(i*sqrt(609))^(n-2)*(ChebyshevU(n-2, -i/sqrt(609)) - (27*i/sqrt(609))*ChebyshevU(n-3, -i/sqrt(609) )).
Row sums satisfy the recurrence S(n) = 2*S(n-1) + 609*S(n-2) for n>4 with S(0) = 2, S(1) = 58, S(2) = 1682, S(3) = 48778. (End)
EXAMPLE
Triangle begins as:
2;
29, 29;
2, 1678, 2;
2, 24387, 24387, 2;
2, 25607, 1070676, 25607, 2;
2, 26827, 15947966, 15947966, 26827, 2;
2, 28047, 31569456, 683937616, 31569456, 28047, 2;
2, 29267, 47935146, 10427818366, 10427818366, 47935146, 29267, 2;
2, 30487, 65045036, 29701552216, 437373644876, 29701552216, 65045036, 30487, 2;
MATHEMATICA
T[n_, k_, j_]:= T[n, k, 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, j] + T[n-1, k-1, j] + (2*j+1)*Prime[j]*T[n-2, k-1, j] ]]];
Table[T[n, k, 10], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 03 2021 *)
PROG
(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, 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, j) + T(n-1, k-1, j) + (2*j+1)*nth_prime(j)*T(n-2, k-1, j)
flatten([[T(n, k, 10) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 03 2021
(Magma)
f:= func< n, j | Round(((3-(-1)^n)/2)*NthPrime(j)^(n-1) - 2^((3-(-1)^n)/2)) >;
function T(n, k, 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, j) + T(n-1, k-1, j) + (2*j+1)*NthPrime(j)*T(n-2, k-1, j);
end if; return T;
end function;
[T(n, k, 10): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 03 2021
CROSSREFS
Cf. A153652 (j=7), A153653 (j=8), A153654 (j=9), this sequence (j=10).
Sequence in context: A175932 A370322 A225544 * A153657 A140152 A089536
KEYWORD
nonn,tabl,easy,less
AUTHOR
Roger L. Bagula, Dec 30 2008
EXTENSIONS
Edited by G. C. Greubel, Mar 03 2021
STATUS
approved