OFFSET
1,3
COMMENTS
T(n,1) = T(n,floor(n/2)+1) = A002083(n+2). - Reinhard Zumkeller, Nov 18 2012
REFERENCES
Author?, Solution to Board of Directors Problem, J. Rec. Math., 9 (No. 3, 1977), 240.
T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, pp. 100-101.
LINKS
Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 122-123.
G. Kreweras, Sur quelques problèmes relatifs au vote pondéré, [Some problems of weighted voting], Math. Sci. Humaines No. 84 (1983), 45-63.
B. E. Wynne & N. J. A. Sloane, Correspondence, 1976-84
B. E. Wynne & T. V. Narayana, Tournament configuration, weighted voting, and partitioned catalans, Preprint.
Bayard Edmund Wynne, and T. V. Narayana, Tournament configuration and weighted voting, Cahiers du bureau universitaire de recherche opérationnelle, 36 (1981): 75-78.
Solution to Board of Directors Problem, J. Rec. Math., 9 (No. 3, 1977), 240. (Annotated scanned copy)
FORMULA
T(1,1) = 1;
T(n,1) = T(n-1, floor((n+1)/2));
T(n,k) = T(n,1) + T(n-1,k-1) for k > 1.
EXAMPLE
Triangle:
1;
1, 2;
2, 3, 4;
3, 5, 6, 7;
6, 9, 11, 12, 13;
...
MATHEMATICA
a[1, 1] = 1; a[n_, 1] := a[n, 1] = a[n - 1, Floor[(n + 1)/2]]; a[n_, k_ /; k > 1] := a[n, k] = a[n, 1] + a[n - 1, k - 1]; A037254 = Flatten[ Table[ a[n, k], {n, 1, 11}, {k, 1, n}]] (* Jean-François Alcover, Apr 03 2012, after given recurrence *)
PROG
(Haskell)
a037254 n k = a037254_tabl !! (n-1) !! (k-1)
a037254_row n = a037254_tabl !! (n-1)
a037254_tabl = map fst $ iterate f ([1], drop 2 a002083_list) where
f (row, (x:xs)) = (map (+ x) (0 : row), xs)
-- Reinhard Zumkeller, Nov 18 2012
(Python)
def T(n, k):
if k==1:
if n==1: return 1
else: return T(n - 1, (n + 1)//2)
return T(n, 1) + T(n - 1, k - 1)
for n in range(1, 12): print([T(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, Jun 03 2017
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from (and formula corrected by) James A. Sellers, Feb 04 2000
STATUS
approved