A037254 Triangle read by rows: T(n,k) (n >= 1, 1 <= k< = n) gives number of non-distorting tie-avoiding integer vote weights.

1, 1, 2, 2, 3, 4, 3, 5, 6, 7, 6, 9, 11, 12, 13, 11, 17, 20, 22, 23, 24, 22, 33, 39, 42, 44, 45, 46, 42, 64, 75, 81, 84, 86, 87, 88, 84, 126, 148, 159, 165, 168, 170, 171, 172, 165, 249, 291, 313, 324, 330, 333, 335, 336, 337, 330, 495, 579, 621, 643, 654, 660, 663, 665

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

Row sums give A005254. A002083 is a column. See also A005318, A096858.

Cf. A005255, A062178.

Sequence in context: A106408 A143061 A096858 * A316939 A259478 A155706

Adjacent sequences: A037251 A037252 A037253 * A037255 A037256 A037257

Keyword

nonn,tabl,nice

Author

N. J. A. Sloane

Extensions

More terms from (and formula corrected by) James A. Sellers, Feb 04 2000

Status

approved