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

%I A037254 #57 Jul 02 2025 16:01:56
%S A037254 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,
%T A037254 46,42,64,75,81,84,86,87,88,84,126,148,159,165,168,170,171,172,165,
%U A037254 249,291,313,324,330,333,335,336,337,330,495,579,621,643,654,660,663,665
%N A037254 Triangle read by rows: T(n,k) (n >= 1, 1 <= k< = n) gives number of non-distorting tie-avoiding integer vote weights.
%C A037254 T(n,1) = T(n,floor(n/2)+1) = A002083(n+2). - _Reinhard Zumkeller_, Nov 18 2012
%D A037254 Author?, Solution to Board of Directors Problem, J. Rec. Math., 9 (No. 3, 1977), 240.
%D A037254 T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, pp. 100-101.
%H A037254 Reinhard Zumkeller, <a href="/A037254/b037254.txt">Rows n = 1..150 of triangle, flattened</a>
%H A037254 M. Klamkin, ed., <a href="http://dx.doi.org/10.1137/1.9781611971729">Problems in Applied Mathematics: Selections from SIAM Review</a>, SIAM, 1990; see pp. 122-123.
%H A037254 G. Kreweras, <a href="http://www.numdam.org/item?id=MSH_1983__84__45_0">Sur quelques problèmes relatifs au vote pondéré</a>, [Some problems of weighted voting], Math. Sci. Humaines No. 84 (1983), 45-63.
%H A037254 B. E. Wynne & N. J. A. Sloane, <a href="/A002083/a002083_1.pdf">Correspondence, 1976-84</a>
%H A037254 B. E. Wynne & T. V. Narayana, <a href="/A002083/a002083_2.pdf">Tournament configuration, weighted voting, and partitioned catalans</a>, Preprint.
%H A037254 Bayard Edmund Wynne, and T. V. Narayana, <a href="http://www.numdam.org/item?id=BURO_1981__36__75_0">Tournament configuration and weighted voting</a>, Cahiers du bureau universitaire de recherche opérationnelle, 36 (1981): 75-78.
%H A037254 <a href="/A005254/a005254.pdf">Solution to Board of Directors Problem</a>, J. Rec. Math., 9 (No. 3, 1977), 240. (Annotated scanned copy)
%F A037254 T(1,1) = 1;
%F A037254 T(n,1) = T(n-1, floor((n+1)/2));
%F A037254 T(n,k) = T(n,1) + T(n-1,k-1) for k > 1.
%e A037254 Triangle:
%e A037254   1;
%e A037254   1,  2;
%e A037254   2,  3,  4;
%e A037254   3,  5,  6,  7;
%e A037254   6,  9, 11, 12, 13;
%e A037254   ...
%t A037254 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 *)
%o A037254 (Haskell)
%o A037254 a037254 n k = a037254_tabl !! (n-1) !! (k-1)
%o A037254 a037254_row n = a037254_tabl !! (n-1)
%o A037254 a037254_tabl = map fst $ iterate f ([1], drop 2 a002083_list) where
%o A037254    f (row, (x:xs)) = (map (+ x) (0 : row), xs)
%o A037254 -- _Reinhard Zumkeller_, Nov 18 2012
%o A037254 (Python)
%o A037254 def T(n, k):
%o A037254     if k==1:
%o A037254         if n==1: return 1
%o A037254         else: return T(n - 1, (n + 1)//2)
%o A037254     return T(n, 1) + T(n - 1, k - 1)
%o A037254 for n in range(1, 12): print([T(n, k) for k in range(1, n + 1)]) # _Indranil Ghosh_, Jun 03 2017
%Y A037254 Row sums give A005254. A002083 is a column. See also A005318, A096858.
%Y A037254 Cf. A005255, A062178.
%K A037254 nonn,tabl,nice
%O A037254 1,3
%A A037254 _N. J. A. Sloane_
%E A037254 More terms from (and formula corrected by) _James Sellers_, Feb 04 2000