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A153520 Triangle T(n,k) = T(n-1, k) + T(n-1, k-1) + 7*T(n-2, k-1), read by rows. 14
2, 7, 7, 2, 94, 2, 2, 341, 341, 2, 2, 357, 1340, 357, 2, 2, 373, 4084, 4084, 373, 2, 2, 389, 6956, 17548, 6956, 389, 2, 2, 405, 9956, 53092, 53092, 9956, 405, 2, 2, 421, 13084, 111740, 229020, 111740, 13084, 421, 2, 2, 437, 16340, 194516, 712404, 712404, 194516, 16340, 437, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
FORMULA
T(n,k) = T(n-1, k) + T(n-1, k-1) + 7*T(n-2, k-1).
From G. C. Greubel, Mar 04 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) = (0,1,4).
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)=(0,1,4).
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 with j=4. (End)
EXAMPLE
Triangle begins as:
2;
7, 7;
2, 94, 2;
2, 341, 341, 2;
2, 357, 1340, 357, 2;
2, 373, 4084, 4084, 373, 2;
2, 389, 6956, 17548, 6956, 389, 2;
2, 405, 9956, 53092, 53092, 9956, 405, 2;
2, 421, 13084, 111740, 229020, 111740, 13084, 421, 2;
2, 437, 16340, 194516, 712404, 712404, 194516, 16340, 437, 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, 0, 1, 4], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 04 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, 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, 4) 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, 4): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 04 2021
CROSSREFS
Sequences with variable (p,q,j): A153516 (0,1,2), A153518 (0,1,3), this sequence (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), A153657 (2,7,10).
Sequence in context: A251809 A016639 A138341 * A153649 A020770 A177003
KEYWORD
nonn,tabl,easy,less
AUTHOR
Roger L. Bagula, Dec 28 2008
EXTENSIONS
Edited by G. C. Greubel, Mar 04 2021
STATUS
approved

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Last modified May 19 09:42 EDT 2024. Contains 372683 sequences. (Running on oeis4.)