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A003016 Number of occurrences of n as an entry in rows <= n of Pascal's triangle (A007318).
(Formerly M0227)
11
0, 3, 1, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 3, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Or, number of occurrences of n as a binomial coefficient.

Sequence A138496 gives record values and where they occur. - Reinhard Zumkeller, Mar 20 2008

REFERENCES

H. L. Abbott, P. Erdos and D. Hanson, On the numbers of times an integer occurs as a binomial coefficient, Amer. Math. Monthly, (1974), 256-261.

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

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

D. Singmaster, How often does an integer occur as a binomial coefficient?, Amer. Math. Monthly, 78 (1971), 385-386.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

R. Zumkeller, Table of n, a(n) for n = 0..10000

Daniel Kane, New Bounds on the Number of Representations of t as a Binomial Coefficient, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper A7, 2004.

Eric Weisstein's World of Mathematics, Pascal's Triangle

Index entries for triangles and arrays related to Pascal's triangle

MATHEMATICA

a[0] = 0; t = {{1}}; a[n_] := Count[ AppendTo[t, Table[ Binomial[n, k], {k, 0, n}]], n, {2}]; Table[a[n], {n, 0, 101}] (* Jean-Fran├žois Alcover, Feb 20 2012 *)

PROG

(Haskell)

a003016 n = sum $ map (fromEnum . (== n)) $

                      concat $ take (fromInteger n + 1) a007318_tabl

-- Reinhard Zumkeller, Apr 12 2012

(PARI) f(n, k)=my(g=lngamma(k+1)+log(n)); binomial(round(solve(N=k, n+k, lngamma(N+1) - lngamma(N-k+1) - g)), k)==n

a(n)=if(n<3, [0, 3, 1][n+1], 2+2*sum(k=2, (n-1)\2, f(n, k))+if(n%2, , f(n, n/2))) \\ Charles R Greathouse IV, Oct 22 2013

CROSSREFS

Cf. A003015, A059233, A138496, A180058.

Sequence in context: A185736 A144148 A085247 * A108121 A161916 A072548

Adjacent sequences:  A003013 A003014 A003015 * A003017 A003018 A003019

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Erich Friedman

Edited by N. J. A. Sloane, Nov 18 2007, at the suggestion of Max Alekseyev.

STATUS

approved

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Last modified April 24 03:45 EDT 2014. Contains 240947 sequences.