A006987 -id:A006987 - cp's OEIS frontend

A095147 Odd binomial coefficients: C(n,k), 2 <= k <= n-2, sorted, duplicates removed.

15, 21, 35, 45, 55, 91, 105, 153, 165, 171, 231, 253, 325, 351, 435, 455, 465, 495, 561, 595, 703, 715, 741, 861, 903, 969, 1001, 1035, 1081, 1225, 1275, 1287, 1365, 1431, 1485, 1653, 1711, 1771, 1891, 1953, 2145, 2211, 2415, 2485, 2701, 2775, 2925, 3003

Offset: 1

Robert G. Wilson v, May 29 2004

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A007318, A006987, A047999, A095146.

Take[ Select[ Union[ Flatten[ Table[ Binomial[n, k], {n, 2, 75}, {k, 2, n - 2}]]], OddQ[ # ] &], 48]

A137905 Numbers that appear as binomial coefficients exactly twice.

Original entry on oeis.org

3, 4, 5, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87, 88

Offset: 1

David Wasserman, Feb 21 2008

Complement of A006987; a(n) = A058084(a(n)). - Reinhard Zumkeller, Mar 20 2009

			7 is a member because 7 = binomial(7, 1) = binomial(7, 6) and no other binomial coefficient equals 7. [clarified by _Jonathan Sondow_, Jan 12 2018]

Cf. A006987, A058084, A062527, A098564, A098566.

isok(n) = (sum(i=0, n, sum(j=0, i, binomial(i,j)==n)) == 2) \\ Michel Marcus, Jun 16 2013

a(n) = A185024(n+1). - Elijah Beregovsky, May 14 2019

A375572 Numbers occurring at least twice in Bernoulli's triangle A008949.

Original entry on oeis.org

1, 4, 7, 8, 11, 15, 16, 22, 26, 29, 31, 32, 37, 42, 46, 56, 57, 63, 64, 67, 79, 92, 93, 99, 106, 120, 121, 127, 128, 130, 137, 154, 163, 172, 176, 191, 211, 219, 232, 247, 254, 255, 256, 277, 299, 301, 326, 352, 378, 379, 382, 386, 407, 436, 466, 470, 497, 502

Offset: 1

Pontus von Brömssen, Aug 19 2024

nonn

Equivalently, 1 together with numbers occurring in columns k >= 2 of Bernoulli's triangle.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A006987, A008949, A375570, A375573.

isok(k) = my(nb=0); for (i=0, k, nb += #select(x->(x==k), vector(i+1, j, sum(jj=0, j-1, binomial(i, jj))))); nb >= 2; \\ Michel Marcus, Aug 22 2024

lista(nn) = my(v = vector(nn)); for (n=1, nn, my(w=vector(n+1, j, sum(jj=0, j-1, binomial(n, jj)))); for (i=1, #w, if (w[i] <= nn, v[w[i]]++));); Vec(select(x->(x>=2), v, 1)); \\ Michel Marcus, Aug 23 2024

from math import comb
from bisect import insort
def A375572_list(nmax):
    a_list = [1]
    if nmax == 1: return a_list
    nkb_list = [(2,2,4)] # List of triples (n,k,A008949(n,k)), sorted by the last element.
    while 1:
        b0 = nkb_list[0][2]
        a_list.append(b0)
        if len(a_list) == nmax: return a_list
        while 1:
            n,k,b = nkb_list[0]
            if b > b0: break
            del nkb_list[0]
            insort(nkb_list,(n+1,k,2*b-comb(n,k)),key=lambda x:x[2])
            if n == k:
                insort(nkb_list,(n+1,k+1,2**(k+1)),key=lambda x:x[2])

A319382 Binomial coefficients binomial(m,k) for 2 <= k <= m/2 in sorted order.

Original entry on oeis.org

6, 10, 15, 20, 21, 28, 35, 36, 45, 55, 56, 66, 70, 78, 84, 91, 105, 120, 120, 126, 136, 153, 165, 171, 190, 210, 210, 220, 231, 252, 253, 276, 286, 300, 325, 330, 351, 364, 378, 406, 435, 455, 462, 465, 495, 496, 528, 560, 561, 595, 630, 666, 680, 703, 715, 741, 780, 792, 816, 820, 861, 903, 924

Offset: 1

Robert Israel, Sep 18 2018

nonn

In contrast to A006987, here the duplicates are not removed. Thus 120 = binomial(10,3) = binomial(16,2) appears twice.

			The first three terms are binomial(4,2) = 6, binomial(5,2) = 10, binomial(6,2) = 15.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003015, A006987, A022911 (values of m), A022912 (values of k).

N:= 10^3: # to get terms <= N
Res:= NULL:
for n from 2 while n*(n-1)/2 <= N do
  for k from 2 to n/2 do
    v:= binomial(n,k);
    if v > N then break fi;
    Res:= Res,v
od od:
sort([Res]);

M = 10^3;
Reap[For[n = 2, n(n-1)/2 <= M, n++, For[k = 2, k <= n/2, k++, v = Binomial[n, k]; If[v > N, Break[]]; Sow[v]]]][[2, 1]] // Sort (* Jean-François Alcover, Apr 27 2019, from Maple *)

a(n) = binomial(A022911(n),A022912(n)).

A360653 Irregular table read by rows; the first row contains the value 1, and for n > 1, the n-th row lists the numbers of the form binomial(m-1, k) such that binomial(m, k) = n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 1, 3, 5, 1, 6, 1, 7, 1, 8, 1, 4, 6, 9, 1, 10, 1, 11, 1, 12, 1, 13, 1, 5, 10, 14, 1, 15, 1, 16, 1, 17, 1, 18, 1, 10, 19, 1, 6, 15, 20, 1, 21, 1, 22, 1, 23, 1, 24, 1, 25, 1, 26, 1, 7, 21, 27, 1, 28, 1, 29, 1, 30, 1, 31, 1, 32, 1, 33, 1, 15, 20, 34

Offset: 1

Rémy Sigrist, Feb 15 2023

In other words, the n-th rows lists the numbers that appear directly above n in Pascal's triangle (A007318).

The n-th row starts with 1, ends with n-1 (provided that n > 1), and contains other values iff n belongs to A006987.

			Table begins:
  n   n-th row
  --  -------------
   1  1
   2  1
   3  1, 2
   4  1, 3
   5  1, 4
   6  1, 3, 5
   7  1, 6
   8  1, 7
   9  1, 8
  10  1, 4, 6, 9
  11  1, 10
.
For n = 6:
    Pascal's triangle begins as follows:
                     1
                   1   1
                 1   2   1
               1   3   3   1
             1   4   6   4   1
           1   5  10  10   5   1
         1   6  15  20  15   6   1
    we find the value 6 in row 4 below 3 and 3, and in row 6 below 1 and 5,
    so the 6th row contains 1, 3 and 5.

Rémy Sigrist, PARI program
Index entries for triangles and arrays related to Pascal's triangle

Cf. A003016, A006987, A007318, A360654, A360655.

PARI
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See Links section.
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A375999 Narayana numbers (A001263), sorted, duplicates removed.

Original entry on oeis.org

1, 3, 6, 10, 15, 20, 21, 28, 36, 45, 50, 55, 66, 78, 91, 105, 120, 136, 153, 171, 175, 190, 196, 210, 231, 253, 276, 300, 325, 336, 351, 378, 406, 435, 465, 490, 496, 528, 540, 561, 595, 630, 666, 703, 741, 780, 820, 825, 861, 903, 946, 990, 1035, 1081, 1128

Offset: 1

Pontus von Brömssen, Sep 06 2024

nonn

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A001263, A006987, A024412, A376000, A376001.

from bisect import insort
from itertools import islice
def A375999_generator():
    yield 1
    nkN_list = [(3, 2, 3)] # List of triples (n, k, A001263(n, k)), sorted by the last element.
    while 1:
        N0 = nkN_list[0][2]
        yield N0
        while 1:
            n, k, N = nkN_list[0]
            if N > N0: break
            del nkN_list[0]
            insort(nkN_list, (n+1, k, n*(n+1)*N//((n-k+1)*(n-k+2))), key=lambda x:x[2])
            if n == 2*k-1:
                insort(nkN_list, (n+2, k+1, 4*n*(n+2)*N//(k+1)**2), key=lambda x:x[2])
def A375999_list(nmax):
    return list(islice(A375999_generator(),nmax))

A376000 Numbers that can be written as a Narayana number (A001263) in at least 2 ways.

Original entry on oeis.org

1, 6, 10, 15, 21, 28, 36, 45, 50, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 196, 210, 231, 253, 276, 300, 325, 336, 351, 378, 406, 435, 465, 490, 496, 528, 540, 561, 595, 630, 666, 703, 741, 780, 820, 825, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1210

Offset: 1

Pontus von Brömssen, Sep 06 2024

nonn

All Narayana numbers A001263(n,k) with n != 2*k-1, are terms since A001263(n,k) = A001263(n,n+1-k). In particular, all positive triangular numbers except 3 are terms. Are there any other terms, i.e., is there a number A001263(2*k-1,k), k >= 2, that can be written as a Narayana number in another way? Any such number would also be a term of A376001.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A000217, A001263, A006987, A098564, A374796, A375572, A375999, A376001.

from bisect import insort
from itertools import islice
def A376000_generator():
    yield 1
    nkN_list = [(3, 2, 3)] # List of triples (n, k, A001263(n, k)), sorted by the last element.
    while 1:
        N0 = nkN_list[0][2]
        c = 0
        while 1:
            n, k, N = nkN_list[0]
            if N > N0:
                if c >= 2: yield N0
                break
            central = n==2*k-1
            c += 2-central
            del nkN_list[0]
            insort(nkN_list, (n+1, k, n*(n+1)*N//((n-k+1)*(n-k+2))), key=lambda x:x[2])
            if central:
                insort(nkN_list, (n+2, k+1, 4*n*(n+2)*N//(k+1)**2), key=lambda x:x[2])
def A376000_list(nmax):
    return list(islice(A376000_generator(),nmax))

A062527 Smallest number (>1) which appears at least n times in Pascal's triangle.

Original entry on oeis.org

2, 3, 6, 10, 120, 120, 3003, 3003

Offset: 1

Henry Bottomley, Jul 10 2001

Singmaster's conjecture is that this sequence is finite.

			a(8)=3003 since 3003 =C(3003,1) =C(3003,3002) =C(78,2) =C(78,76) =C(15,5) =C(15,10) =C(14,6) = C(14,8).

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.
FTL Magazine, One Thousand and One Coincidences
D. Singmaster, How often does an integer occur as a binomial coefficient?, Amer. Math. Monthly, 78 (1971), 385-386.
David Singmaster, Repeated binomial coefficients and Fibonacci numbers, Fibonacci Quarterly 13 (1975) 295-298.

Cf. A003015, A003016, A006987, A007318, A059233.

(* Computation lasts a few minutes *) max = 4000; Clear[cnt]; cnt[] = 0; Do[b = Binomial[n, k]; If[b <= max, cnt[b] += 1], {n, 0, max}, {k, 1, n - 1}]; sel = Select[Table[{b, cnt[b]}, {b, 1, max }], #[[2]] >= 1&]; a[n] := Select[sel, #[[2]] >= n&][[1, 1]]; Array[a, 8] (* Jean-François Alcover, Oct 05 2015 *)

A210577 Natural numbers equal to the sum of two nontrivial binomial coefficients, sorted, duplicates removed.

Original entry on oeis.org

12, 16, 20, 21, 25, 26, 27, 30, 31, 34, 35, 36, 38, 40, 41, 42, 43, 45, 46, 48, 49, 50, 51, 55, 56, 57, 60, 61, 62, 63, 64, 65, 66, 70, 71, 72, 73, 75, 76, 77, 80, 81, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 97, 98, 99, 100, 101, 102, 104, 105, 106, 110

Offset: 1

Douglas Latimer, Mar 22 2012

Nontrivial binomial coefficients are C(n,k) with 2 <= k <= n-2.

			a(1) = 12 since 6 is the lowest nontrivial binomial coefficient and 6+6 = 12.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Two-term sums of members of A006987.

Cf. A007318.

lim = 110; bc = {}; n = 4; While[c = Select[Binomial[n, Range[2, Floor[n/2]]], # <= lim &]; Length[c] > 0, bc = Join[bc, c]; n++]; bc = Sort[bc]; Select[Union[Flatten[Outer[Plus, bc, bc]]], # <= lim &] (* T. D. Noe, Mar 22 2012 *)

list(lim)=my(v=List(), t, u=v); for(n=4, sqrtint(2*lim)+1, for(k=2, n\2, t=binomial(n, k); if(t>lim, break, listput(v, t)))); v=vecsort(Vec(v), , 8); for(i=1,#v,for(j=1,i,if(v[i]+v[j]>lim,break,listput(u,v[i]+v[j]))));vecsort(Vec(u),,8) \\ Charles R Greathouse IV, Apr 03 2012

A210578 Natural numbers that can be expressed as the sum of one or two nontrivial binomial coefficients, sorted, duplicates removed.

Original entry on oeis.org

6, 10, 12, 15, 16, 20, 21, 25, 26, 27, 28, 30, 31, 34, 35, 36, 38, 40, 41, 42, 43, 45, 46, 48, 49, 50, 51, 55, 56, 57, 60, 61, 62, 63, 64, 65, 66, 70, 71, 72, 73, 75, 76, 77, 78, 80, 81, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 97, 98, 99, 100, 101, 102

Offset: 1

Douglas Latimer, Mar 22 2012

Recall that the nontrivial binomial coefficients are C(n,k), 2 <= k <= n-2.

			a(1) = 6, since 6 is the lowest nontrivial binomial coefficient.
a(2) = 10, since 10 is the lowest nontrivial binomial coefficient except 6.
a(3) = 12 , since 6 is the lowest nontrivial binomial coefficient and 6+6 = 12 .

Robert Israel, Table of n, a(n) for n = 1..10000

This is A006987 combined with two-term sums of members of A006987. Cf. A210577 and A007318 .

N:= 200: # to get all terms <= N
BC:= {}:
for k from 2 do
   if binomial(k+2,k) > N then break fi;
   for n from k+2 do
     v:= binomial(n,k);
     if v > N then break fi;
     BC:= BC union {v};
   od;
od:
A:= BC:
for i from 1 to nops(BC) do
  A:= A union select(`<=`,map(`+`,BC[1..i],BC[i]),N)
od:
sort(convert(A,list)): # Robert Israel, Apr 27 2017

cp's OEIS Frontend

A095147 Odd binomial coefficients: C(n,k), 2 <= k <= n-2, sorted, duplicates removed.

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A137905 Numbers that appear as binomial coefficients exactly twice.

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A375572 Numbers occurring at least twice in Bernoulli's triangle A008949.

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PARI

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A319382 Binomial coefficients binomial(m,k) for 2 <= k <= m/2 in sorted order.

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Maple

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A360653 Irregular table read by rows; the first row contains the value 1, and for n > 1, the n-th row lists the numbers of the form binomial(m-1, k) such that binomial(m, k) = n.

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PARI

A375999 Narayana numbers (A001263), sorted, duplicates removed.

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Python

A376000 Numbers that can be written as a Narayana number (A001263) in at least 2 ways.

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Python

A062527 Smallest number (>1) which appears at least n times in Pascal's triangle.

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Mathematica

A210577 Natural numbers equal to the sum of two nontrivial binomial coefficients, sorted, duplicates removed.