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

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• nice

Comments

T(n,1) = T(n,floor(n/2)+1) = A002083(n+2). - Reinhard Zumkeller, Nov 18 2012

Examples

			Triangle:
  1;
  1,  2;
  2,  3,  4;
  3,  5,  6,  7;
  6,  9, 11, 12, 13;
  ...
		

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)

Crossrefs

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

Cf. A005255, A062178.

Programs

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

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 *)

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

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.

Extensions

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