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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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.

FORMULA

T(1,1)=1; T(n,1)=T(n-1,[(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)

from sympy import floor

def T(n, k):

    if k==1:

        if n==1: return 1

        else: return T(n - 1, floor((n + 1)/2))

    return T(n, 1) + T(n - 1, k - 1)

for n in xrange(1, 12): print [T(n, k) for k in xrange(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 * A259478 A155706 A231227

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

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Last modified November 23 19:16 EST 2017. Contains 295128 sequences.