A173567 Triangle T(n, k) = (f(k, n-k+1) + f(n-k+1, k))/2 where f(n, k) = (1/2)*Sum_{j=1..2*n} k^j, read by rows.

2, 5, 5, 9, 30, 9, 14, 123, 123, 14, 20, 425, 1092, 425, 20, 27, 1413, 7650, 7650, 1413, 27, 35, 4872, 54051, 87380, 54051, 4872, 35, 44, 17783, 426573, 943190, 943190, 426573, 17783, 44, 54, 67875, 3655854, 12192579, 12207030, 12192579, 3655854, 67875, 54

Offset

1,1

Links

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

Formula

T(n, k) = (f(k, n-k+1) + f(n-k+1, k))/2 where f(n, k) = (1/2)*Sum_{j=1..2*n} k^j.

T(n, k) = (f(k, n-k+1) + f(n-k+1, k))/2 where f(n, k) = k*(1 - k^(2*n))/(1-k) with f(n, 1) = 2*n. - G. C. Greubel, Apr 25 2021

Example

Triangle begins as:
   2;
   5,     5;
   9,    30,       9;
  14,   123,     123,       14;
  20,   425,    1092,      425,       20;
  27,  1413,    7650,     7650,     1413,       27;
  35,  4872,   54051,    87380,    54051,     4872,      35;
  44, 17783,  426573,   943190,   943190,   426573,   17783,    44;
  54, 67875, 3655854, 12192579, 12207030, 12192579, 3655854, 67875, 54;

Mathematica

f[n_, k_]:= If[k==1, 2*n, k*(1-k^(2*n))/(1-k)];
T[n_, k_]:= (f[k, n-k+1] + f[n-k+1, k])/2;
Table[T[n, k], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 25 2021 *)

Prog

(Sage)
def f(n, k): return 2*n if k==1 else k*(1-k^(2*n))/(1-k)
def T(n, k): return (f(k, n-k+1) + f(n-k+1, k))/2
flatten([[T(n, k) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Apr 25 2021

Crossrefs

Sequence in context: A050175 A243333 A059797 * A288726 A344572 A265129

Adjacent sequences: A173564 A173565 A173566 * A173568 A173569 A173570

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 22 2010

Extensions

Edited by G. C. Greubel, Apr 25 2021

Status

approved