A180058 -id:A180058 - cp's OEIS frontend

A003016 Number of occurrences of n as an entry in rows <= n of Pascal's triangle (A007318).

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

Offset: 0

N. J. A. Sloane

Or, number of occurrences of n as a binomial coefficient. [Except for 1 which occurs infinitely many times. This is the only reason for the restriction "row <= n" in the definition. Any other number can only appear in rows <= n. - M. F. Hasler, Feb 16 2023]
Sequence A138496 gives record values and where they occur. - Reinhard Zumkeller, Mar 20 2008
Singmaster conjectures that this sequence is bounded. - Michael J. Hardy, Jun 09 2025

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 93, #47.
C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
H. L. Abbott, P. Erdős and D. Hanson, On the numbers of times an integer occurs as a binomial coefficient, Amer. Math. Monthly, (1974), 256-261.
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.
Kaisa Matomäki, Maksym Radziwill, Xuancheng Shao, Terence Tao, and Joni Teräväinen, Singmaster's conjecture in the interior of Pascal's triangle, arXiv:2106.03335 [math.NT], 2021.
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
Index entries for triangles and arrays related to Pascal's triangle

Cf. A003015, A059233, A138496, A180058.

a003016 n = sum $ map (fromEnum . (== n)) $
                      concat $ take (fromInteger n + 1) a007318_tabl
-- Reinhard Zumkeller, Apr 12 2012

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-François Alcover, Feb 20 2012 *)

{A003016(n)=if(n<4, [0,3,1,2][n+1], my(c=2, k=2, r=sqrtint(2*n)+1, C=r*(r-1)/2); until(, while(C= r\2 && break; C *= r-k; C \= r; r -= 1); c)} \\ M. F. Hasler, Feb 16 2023

from math import isqrt # requires python3.8 or higher
def A003016(n):
    if n < 4: return[0,3,1,2][n]
    cnt = k = 2; r = isqrt(2*n)+1; C = r*(r-1)//2
    while True:
       while C < n and k < r//2:
          C *= r-k; k += 1; C //= k
       if C == n: cnt += 2 - (r == 2*k)
       if k >= r//2: return cnt
       C *= r-k; C //= r; r -= 1 # M. F. Hasler, Feb 16 2023

More terms from Erich Friedman

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

A059233 Number of rows in which n appears in Pascal's triangle A007318.

Original entry on oeis.org

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

Offset: 2

Fabian Rothelius, Jan 20 2001

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

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

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 93, #47.
C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 96.

T. D. Noe, Table of n, a(n) for n=2..10000
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
Wikipedia, Singmaster's conjecture
Index entries for triangles and arrays related to Pascal's triangle

Cf. A003016, A003015.

a059233 n = length $ filter (n `elem`) $
                            take (fromInteger n) $ tail a007318_tabl
a059233_list = map a059233 [2..]
-- Reinhard Zumkeller, Dec 24 2012

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 *)

A059233(n)=A003016(n)\/2 \\ M. F. Hasler, Mar 01 2023

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

A376866 a(n) = smallest integer k >= 2 such that there exist n disjoint multisets of positive integers, whose corresponding multinomial coefficients equal k.

Original entry on oeis.org

2, 6, 120, 3003, 433866230594439538896000

Offset: 1

Pontus von Brömssen, Oct 07 2024

			  n |                     a(n) | disjoint multisets with multinomial coefficient a(n)
  --+--------------------------+---------------------------------------------
  1 |                        2 | {1,1}
  2 |                        6 | {2,2}, {1,5}
  3 |                      120 | {3,7}, {2,14}, {1,119}
  4 |                     3003 | {6,8}, {5,10}, {2,76}, {1,3002}
  5 | 433866230594439538896000 | {4,9,13,23}, {2,2,8,11,27}, {3,3,3,16,25}, {5,6,15,24},
    |                          | {1,433866230594439538895999}

Cf. A180058, A376369, A376376, A376828.

Edited by Max Alekseyev, May 18 2025

cp's OEIS Frontend

A003016 Number of occurrences of n as an entry in rows <= n of Pascal's triangle (A007318).

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A059233 Number of rows in which n appears in Pascal's triangle A007318.

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A376866 a(n) = smallest integer k >= 2 such that there exist n disjoint multisets of positive integers, whose corresponding multinomial coefficients equal k.

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