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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 G. C. Greubel, Rows n = 1..50 of the triangle, flattened 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 Adjacent sequences: A153517 A153518 A153519 * A153521 A153522 A153523 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.)