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Number of rows in which n appears in Pascal's triangle A007318.
9

%I #22 Mar 24 2023 12:49:43

%S 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,

%T 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,

%U 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

%N Number of rows in which n appears in Pascal's triangle A007318.

%C 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

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 93, #47.

%D C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 96.

%H T. D. Noe, <a href="/A059233/b059233.txt">Table of n, a(n) for n=2..10000</a>

%H D. Singmaster, <a href="http://www.jstor.org/stable/2316907">How often does an integer occur as a binomial coefficient?</a>, Amer. Math. Monthly, 78 (1971), 385-386.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PascalsTriangle.html">Pascal's Triangle</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Singmaster&#39;s_conjecture">Singmaster's conjecture</a>

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%F 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

%F a(n) = ceiling(A003016(n)/2). - _M. F. Hasler_, Mar 01 2023

%e 6 appears in both row 4 and row 6 in Pascal's triangle, therefore a(6) = 2.

%t 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 *)

%o (Haskell)

%o a059233 n = length $ filter (n `elem`) $

%o take (fromInteger n) $ tail a007318_tabl

%o a059233_list = map a059233 [2..]

%o -- _Reinhard Zumkeller_, Dec 24 2012

%o (PARI) A059233(n)=A003016(n)\/2 \\ _M. F. Hasler_, Mar 01 2023

%Y Cf. A003016, A003015.

%K easy,nice,nonn

%O 2,5

%A _Fabian Rothelius_, Jan 20 2001