A005254 Number of weighted voting procedures.
1, 3, 9, 21, 51, 117, 271, 607, 1363, 3013, 6643, 14491, 31495, 67965, 146115, 312483, 666015, 1413915, 2992815, 6315135, 13292007, 27906585, 58464339, 122229123, 255072423, 531369483, 1105217223, 2295383319, 4760727375
Offset: 1
References
- M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 122-123.
- T. V. Narayana, Recent progress and unsolved problems in dominance theory, pp. 68-78 of Combinatorial mathematics (Canberra 1977), Lect. Notes Math. Vol. 686, 1978.
- T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, pp. 100-101.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Solution to Board of Directors Problem, J. Rec. Math., 9 (No. 3, 1977), 240.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
- 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 of A037254.
Programs
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Haskell
a005254 = sum . a037254_row -- Reinhard Zumkeller, Nov 18 2012
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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]; A005254 = Table[ Sum[ a[n, k], {k, 1, n}], {n, 1, 29}] (* Jean-François Alcover, Apr 03 2012, after recurrence of A037254 *)
Extensions
More terms from James Sellers, Feb 04 2000