A153518 Triangular T(n,k) = T(n-1, k) + T(n-1, k-1) + 5*T(n-2, k-1), read by rows.

2, 5, 5, 2, 46, 2, 2, 123, 123, 2, 2, 135, 476, 135, 2, 2, 147, 1226, 1226, 147, 2, 2, 159, 2048, 4832, 2048, 159, 2, 2, 171, 2942, 13010, 13010, 2942, 171, 2, 2, 183, 3908, 26192, 50180, 26192, 3908, 183, 2, 2, 195, 4946, 44810, 141422, 141422, 44810, 4946, 195, 2

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) + 5*T(n-2, k-1).

Recurrence row sums: s(n) = 2*s(n-1) + 5*s(n-2), n > 4, with s(1) = 2, s(2) = 10, s(3) = 50, s(4) = 250. - R. J. Mathar, Jan 22 2009

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) = (0,1,3).

Sum_{k=0..n} T(n,k,0,1,3) = 4*(-5)^n*[n<2] + 50*(i*sqrt(5))^(n-2)*(ChebyshevU(n-2, -i/sqrt(5)) - (3*i/sqrt(5))*ChebyshevU(n-3, -i/sqrt(5))) = 4*(-5)^n*[n<2] + 50*A123011(n-2). (End)

Example

Triangle begins as:
  2;
  5,   5;
  2,  46,    2;
  2, 123,  123,     2;
  2, 135,  476,   135,      2;
  2, 147, 1226,  1226,    147,      2;
  2, 159, 2048,  4832,   2048,    159,     2;
  2, 171, 2942, 13010,  13010,   2942,   171,    2;
  2, 183, 3908, 26192,  50180,  26192,  3908,  183,   2;
  2, 195, 4946, 44810, 141422, 141422, 44810, 4946, 195, 2;

Maple

A153518 := proc(n, k) option remember ; if n =1 then 2; elif n = 2 then 5; elif k=1 or k=n then 2; elif n = 3 then 46 ; elif n = 4 then 123 ; else procname(n-1, k-1)+procname(n-1, k)+5*procname(n-2, k-1) ; end: end: for n from 1 to 13 do for k from 1 to n do printf("%d, ", A153518(n, k)) ; od: od: # R. J. Mathar, Jan 22 2009

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, 3], {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, 3) 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, 3): 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), this sequences (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), A153657 (2,7,10).

Cf. A123011.

Sequence in context: A154953 A321310 A187340 * A153648 A153354 A153821

Adjacent sequences: A153515 A153516 A153517 * A153519 A153520 A153521

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Dec 28 2008

Extensions

More terms from R. J. Mathar, Jan 22 2009

Edited by G. C. Greubel, Mar 04 2021

Status

approved