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A059233 Number of rows in which n appears in Pascal's triangle A007318. 9
1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,5
COMMENTS
Central binomial coefficients c = A000984(n) > 1 appear once in the middle column C(2n, n), and thereafter in one or more later rows to the left as C(r,k) and to the right as C(r, r-k), k < r/2; the last time in row r = c = C(c,1) = C(c,c-1). For these, a(n) = (A003016(n)+1)/2. For all other numbers n > 1, a(n) = A003016(n)/2. - M. F. Hasler, Mar 01 2023
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 93, #47.
C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 96.
LINKS
D. Singmaster, How often does an integer occur as a binomial coefficient?, Amer. Math. Monthly, 78 (1971), 385-386.
Eric Weisstein's World of Mathematics, Pascal's Triangle
FORMULA
a(A180058(n)) = n and a(m) < n for m < A180058(n); a(A182237(n)) = 2; a(A098565(n)) = 3. - Reinhard Zumkeller, Dec 24 2012
a(n) = ceiling(A003016(n)/2). - M. F. Hasler, Mar 01 2023
EXAMPLE
6 appears in both row 4 and row 6 in Pascal's triangle, therefore a(6) = 2.
MATHEMATICA
nmax = 101; A007318 = Table[Binomial[n, k], {n, 0, nmax}, {k, 0, n}]; a[n_] := Position[A007318, n][[All, 1]] // Union // Length; Table[a[n], {n, 2, nmax}] (* Jean-François Alcover, Sep 09 2013 *)
PROG
(Haskell)
a059233 n = length $ filter (n `elem`) $
take (fromInteger n) $ tail a007318_tabl
a059233_list = map a059233 [2..]
-- Reinhard Zumkeller, Dec 24 2012
(PARI) A059233(n)=A003016(n)\/2 \\ M. F. Hasler, Mar 01 2023
CROSSREFS
Sequence in context: A256554 A321649 A003650 * A357327 A327924 A354057
KEYWORD
easy,nice,nonn
AUTHOR
Fabian Rothelius, Jan 20 2001
STATUS
approved

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Last modified June 17 15:38 EDT 2024. Contains 373458 sequences. (Running on oeis4.)