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A153657
Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +7)*prime(j)*T(n-2, k-1) with j=10, read by rows.
14
2, 29, 29, 2, 1678, 2, 2, 24387, 24387, 2, 2, 25955, 1362648, 25955, 2, 2, 27523, 20483624, 20483624, 27523, 2, 2, 29091, 40833912, 1107920632, 40833912, 29091, 2, 2, 30659, 62413512, 17187432136, 17187432136, 62413512, 30659, 2, 2, 32227, 85222424, 49222798744, 901876719128, 49222798744, 85222424, 32227, 2
OFFSET
1,1
FORMULA
T(n, k) = T(n-1, k) + T(n-1, k-1) + (2*j +7)*prime(j)*T(n-2, k-1) with j=10.
From G. C. Greubel, Mar 06 2021: (Start)
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) = (2,7,10).
Sum_{k=0..n} T(n,k,p,q,j) = 2*prime(j)^(n-1), for (p,q,j)=(2,7,10), = 2*A009973(n-1). (End)
EXAMPLE
Triangle begins as:
2;
29, 29;
2, 1678, 2;
2, 24387, 24387, 2;
2, 25955, 1362648, 25955, 2;
2, 27523, 20483624, 20483624, 27523, 2;
2, 29091, 40833912, 1107920632, 40833912, 29091, 2;
2, 30659, 62413512, 17187432136, 17187432136, 62413512, 30659, 2;
2, 32227, 85222424, 49222798744, 901876719128, 49222798744, 85222424, 32227, 2;
MATHEMATICA
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, 2, 7, 10], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 06 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, 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, 2, 7, 10) + 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, p, q, j) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 06 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, 2, 7, 10): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 06 2021
CROSSREFS
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), A153656 (2,3,9), this sequence (2,7,10).
Cf. A009973 (powers of 29).
Sequence in context: A370322 A225544 A153655 * A140152 A089536 A019379
KEYWORD
nonn,tabl,easy,less
AUTHOR
Roger L. Bagula, Dec 30 2008
EXTENSIONS
Edited by G. C. Greubel, Mar 06 2021
STATUS
approved