A153520 Triangle T(n,k) = T(n-1, k) + T(n-1, k-1) + 7*T(n-2, k-1), read by rows.

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

Offset: 1

Author

Roger L. Bagula, Dec 28 2008

Keywords

• nonn
• tabl
• easy
• less

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

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).

Programs

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

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 *)

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

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)

Extensions

Edited by G. C. Greubel, Mar 04 2021