A006987 -id:A006987 - cp's OEIS frontend

A305188 Numbers that are equal to a nontrivial multinomial coefficient (i.e., equal to k!/(k1!...km!) with k1 + ... + km = k, k-2 >= k1 >= ... >= km).

6, 10, 12, 15, 20, 21, 24, 28, 30, 35, 36, 42, 45, 55, 56, 60, 66, 70, 72, 78, 84, 90, 91, 105, 110, 120, 126, 132, 136, 140, 153, 156, 165, 168, 171, 180, 182, 190, 210, 220, 231, 240, 252, 253, 272, 276, 280, 286, 300, 306, 325, 330, 336, 342, 351, 360, 364

Offset: 1

Vincent Champain, May 27 2018

nonn

This sequence answers the following question: what numbers correspond to the number of permutations of a number of items that is lower than the number of permutations itself? It means that the underlying structure has some form of redundance / symmetry.
It can be shown that:
- no prime number is part of this sequence (see A304938)
- some nonprimes are not part of the sequence (beginning with 1, 4, 8, 9, 14, 16, 18, ...)
- any number that is a factorial of another integer e is part of this sequence (k=e, k1=ki=...=ke=1).
This sequence is a generalization of A006987.
From David A. Corneth, May 28 2018: (Start)
Also the numbers that are the number of permutations of either:
- sets of balls with two distinct colors of balls where each color occurs at least twice;
- sets of balls with at least three distinct colors of balls.
(End)
From Amiram Eldar, Jul 23 2020: (Start)
The asymptotic density of this sequence is 0 (Niven, 1951).
The number of terms not exceeding x is (1 + sqrt(2)) * x^(1/2) + o(x^(1/2)) (Erdős, 1954). (End)

			a(1) = 6 because all numbers lower than 6 are either prime or a power of primes.
105 is a term of the sequence because 105 is equal to a multinomial coefficient: 105 = (4+2+1)! / (4! * 2! * 1!) and 105 is the number of ways 7 balls can be sorted where 4 are red, 2 are yellow and one is blue.
2016 is a term because 64! / (62! * 2!) = 2016. - _David A. Corneth_, May 29 2018

David A. Corneth, Table of n, a(n) for n = 1..10000
Vincent Champain, Calculation of the terms of the sequence including a k1..km list so that a(n) = k!/(k1!*...*km!) with k1+...+km = k and m > 1, 2018.
Vincent Champain, Python program for A305188
David A. Corneth, Terms with the corresponding tuples.
Paul Erdős, The Number of Multinomial Coefficients, The American Mathematical Monthly, Vol. 61, No. 1 (1954), pp. 37-39.
Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434. See theorem 2, p. 428.

Cf. A008480, A304938, A006987.

mult[w_] := Total[w]!/Times @@ (w!); L = {}; Do[ t = mult /@ Select[ IntegerPartitions@ n, #[[1]] < n-1 &]; L = Union[L, Select[t, # <= 400 &]], {n, 3, 30}]; L (* Terms < 400, Giovanni Resta, May 27 2018 *)

Python
```
# see link above
```

a(28)-a(57) from Giovanni Resta, May 27 2018

A318955 Binomial(A319500(n),a(n)) = A006917(n) with 2 <= a(n) <= A319500(n)/2.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 4, 2, 3, 2, 2, 3, 4, 2, 2, 3, 2, 2, 4, 3, 2, 5, 2, 2, 3, 2, 2, 4, 2, 3, 2, 2, 2, 3, 5, 2, 4, 2, 2, 3, 2, 2, 2, 2, 3, 2, 4, 2, 2, 5, 3, 2, 2, 2, 6, 2, 3, 2, 4, 2, 2, 2, 3, 2, 2, 2, 5, 2, 3, 4, 2, 2, 2, 3, 2, 2, 2, 6, 2, 3, 4, 2, 2, 2, 5, 2, 3, 2, 2, 2, 2, 3, 2

Offset: 1

Robert Israel, Sep 20 2018

nonn

First differs from A022912 at n=18.

			a(3) = 6 because A006987(3) = 15 = binomial(6,2).

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

Cf. A006987, A022912, A319500.

N:= 10000: # for binomial(n, k) values <= N
S:= {}:
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;
    if not member(v,S) then
      S:= S union {v};
      K[v]:= k;
    fi
od od:
A006987:= sort(convert(S,list)):
seq(K[A006987[i]],i=1..nops(A006987));

A355391 Position of first appearance of n in A181591 = binomial(bigomega(n), omega(n)).

Original entry on oeis.org

1, 4, 8, 16, 32, 24, 128, 256, 512, 48, 2048, 4096, 8192, 16384, 96, 65536, 131072, 262144, 524288, 240, 192, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 384, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 480, 768, 137438953472

Offset: 1

Gus Wiseman, Jul 04 2022

nonn

The statistic omega = A001221 counts distinct prime factors (without multiplicity).
The statistic bigomega = A001222 counts prime factors with multiplicity.
We have A181591(2^k) = k, so the sequence is fully defined. Positions meeting this maximum are A185024, complement A006987.

			The terms together with their prime indices begin:
       1: {}
       4: {1,1}
       8: {1,1,1}
      16: {1,1,1,1}
      32: {1,1,1,1,1}
      24: {1,1,1,2}
     128: {1,1,1,1,1,1,1}
     256: {1,1,1,1,1,1,1,1}
     512: {1,1,1,1,1,1,1,1,1}
      48: {1,1,1,1,2}
    2048: {1,1,1,1,1,1,1,1,1,1,1}
    4096: {1,1,1,1,1,1,1,1,1,1,1,1}
    8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
   16384: {1,1,1,1,1,1,1,1,1,1,1,1,1,1}
      96: {1,1,1,1,1,2}
   65536: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  131072: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  262144: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  524288: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
     240: {1,1,1,1,2,3}
     192: {1,1,1,1,1,1,2}

Amiram Eldar, Table of n, a(n) for n = 1..168

Positions of powers of 2 are A185024, complement A006987.
Counting multiplicity gives A355386.
The sorted version is A355392.
A000005 counts divisors.
A001221 counts prime factors without multiplicity.
A001222 count prime factors with multiplicity.
A070175 gives representatives for bigomega and omega, triangle A303555.
Cf. A022811, A056239 , A071625, A118914, A181819, A323014, A323023, A355383 (with multiplicity A339006), A355384.

s=Table[Binomial[PrimeOmega[n],PrimeNu[n]],{n,1000}];
Table[Position[s,k][[1,1]],{k,Select[Union[s],SubsetQ[s,Range[#]]&]}]

f(n) = binomial(bigomega(n), omega(n)); \\ A181591
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, Jul 10 2022

binomial(bigomega(a(n)), omega(a(n))) = n.

a(22)-a(28) from Michel Marcus, Jul 10 2022

a(29)-a(37) from Amiram Eldar, Jul 10 2022

A024746 Binomial coefficients: C(n,k), 2 <= k <= n-2, sorted.

Original entry on oeis.org

6, 10, 10, 15, 15, 20, 21, 21, 28, 28, 35, 35, 36, 36, 45, 45, 55, 55, 56, 56, 66, 66, 70, 78, 78, 84, 84, 91, 91, 105, 105, 120, 120, 120, 120, 126, 126, 136, 136, 153, 153, 165, 165, 171, 171, 190, 190, 210, 210, 210, 210, 220, 220, 231, 231, 252, 253, 253, 276, 276, 286

Offset: 1

David W. Wilson

nonn

Cf. A006987.

Sum_{n=1}^{infinity} 1/a(n) = 3/2. - Franklin T. Adams-Watters, Feb 28 2006

A024755 Binomial coefficients: C(n,k), 3 <= k <= n-3, sorted, duplicates removed.

Original entry on oeis.org

20, 35, 56, 70, 84, 120, 126, 165, 210, 220, 252, 286, 330, 364, 455, 462, 495, 560, 680, 715, 792, 816, 924, 969, 1001, 1140, 1287, 1330, 1365, 1540, 1716, 1771, 1820, 2002, 2024, 2300, 2380, 2600, 2925, 3003, 3060, 3276, 3432, 3654, 3876, 4060, 4368, 4495

Offset: 1

David W. Wilson

nonn

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

Cf. A007318 (binomial coefficients), A006987 (similar to this).

nn = 50; t = Table[Binomial[n, k], {n, 6, nn}, {k, 3, n/2}]; Union[Select[Flatten[t], # <= t[[nn - 5, 1]] &]] (* Harvey P. Dale, Mar 12 2013 *)

A051641 Palindromic binomial coefficients C(n,k) for k >= 2.

Original entry on oeis.org

3, 6, 55, 66, 171, 252, 595, 666, 969, 1001, 1771, 2002, 3003, 3003, 3003, 5005, 5995, 8008, 8778, 15051, 66066, 617716, 646646, 828828, 1269621, 1680861, 3262623, 3544453, 5073705, 5676765, 6295926, 6378736, 35133153, 61477416, 178727871

Offset: 1

Joe DeMaio (jdemaio(AT)kennesaw.edu)

			C(10,5)=252. 3003 occurs thrice because C(14,6)=C(15,5)=C(78,2)=3003.

Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, Prometheus Books, NY, 2007, page 93.

Cf. A002113, A006987, A104182.

fQ[n_] := Block[{id = IntegerDigits@n}, id == Reverse@id]; lst = {}; Do[ k = 2; While[k < n/2 + 1, b = Binomial[n, k]; If[fQ@b, AppendTo[lst, b]; Print@b]; k++ ], {n, 25000000}]; Take[ Union@ lst, 35] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Aug 20 2000

A082581 Binomial coefficients C(p, k), 2<=k<=p-2, sorted, with duplicates removed, p being prime.

Original entry on oeis.org

10, 21, 35, 55, 78, 136, 165, 171, 253, 286, 330, 406, 462, 465, 666, 680, 715, 820, 903, 969, 1081, 1287, 1378, 1711, 1716, 1771, 1830, 2211, 2380, 2485, 2628, 3081, 3403, 3654, 3876, 3916, 4495, 4656, 5050, 5253, 5671, 5886, 6188, 6328, 7770, 8001, 8515

Offset: 1

Lekraj Beedassy, May 07 2003

nonn

Cf. A006987, A082583.

with(combinat): S := {}: for n from 3 to 80 do p := ithprime(n): for k from 2 to p-2 do S := S union {binomial(p, k)}: od: od: S := sort(convert(S, list)): for j from 1 to 100 do printf(`%d,`,S[j]) od: # James Sellers, May 19 2003

More terms from James Sellers, May 19 2003

A095146 Even binomial coefficients: C(n,k), 2 <= k <= n-2, sorted, duplicates removed.

Original entry on oeis.org

6, 10, 20, 28, 36, 56, 66, 70, 78, 84, 120, 126, 136, 190, 210, 220, 252, 276, 286, 300, 330, 364, 378, 406, 462, 496, 528, 560, 630, 666, 680, 780, 792, 816, 820, 924, 946, 990, 1128, 1140, 1176, 1326, 1330, 1378, 1540, 1596, 1716, 1770, 1820, 1830, 2002

Offset: 1

Robert G. Wilson v, May 29 2004

Cf. A007318, A006987, A047999, A095147.

Take[ Select[ Union[ Flatten[ Table[ Binomial[n, k], {n, 2, 61}, {k, 2, n - 2}]]], EvenQ[ # ] &], 51]

A210576 Positive integers that cannot be expressed as sum of one or more nontrivial binomial coefficients.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 17, 19, 23, 29

Offset: 1

Douglas Latimer, Mar 22 2012

The nontrivial binomial coefficients are C(n,k), 2 <= k <= n-2 (A006987).
I conjectured that the sequence is finite, consisting of the terms listed.
This conjecture is now proved. - Douglas Latimer, Apr 10 2013
Note that this sequence allows the same binomial coefficient to be used multiple times. - T. D. Noe, Apr 12 2013
These are the only values of the angular momentum for which a wavefunction with such an angular momentum and the symmetry of a dodecahedron is impossible. [Baez] - Andrey Zabolotskiy, Mar 28 2018

			The smallest terms in the sequence are 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14 because 6, 10 and 15 cannot be terms, as these are the lowest nontrivial binomial coefficients; 12 and 16 cannot be terms, as these are the lowest sums of two nontrivial binomial coefficients; and sums of three or more nontrivial binomial coefficients cannot exclude any of the listed terms.

John Baez, Quantum Mechanics and the Dodecahedron, Dec 31 2017.
Douglas Latimer, Computation of Terms <= 30.
Douglas Latimer, Terms Listed Are the Entire Sequence.

A210578 contains many of the integers that cannot be elements of this sequence.
Cf. A006987 and A007318.
Positions of zeros in A008651. Cf. A005796.

cp's OEIS Frontend

A305188 Numbers that are equal to a nontrivial multinomial coefficient (i.e., equal to k!/(k1!*...*km!) with k1 + ... + km = k, k-2 >= k1 >= ... >= km).

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A318955 Binomial(A319500(n),a(n)) = A006917(n) with 2 <= a(n) <= A319500(n)/2.

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A355391 Position of first appearance of n in A181591 = binomial(bigomega(n), omega(n)).

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A024746 Binomial coefficients: C(n,k), 2 <= k <= n-2, sorted.

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A024755 Binomial coefficients: C(n,k), 3 <= k <= n-3, sorted, duplicates removed.

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A051641 Palindromic binomial coefficients C(n,k) for k >= 2.

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A082581 Binomial coefficients C(p, k), 2<=k<=p-2, sorted, with duplicates removed, p being prime.

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Maple

Extensions

A095146 Even binomial coefficients: C(n,k), 2 <= k <= n-2, sorted, duplicates removed.

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Mathematica

A210576 Positive integers that cannot be expressed as sum of one or more nontrivial binomial coefficients.

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A305188 Numbers that are equal to a nontrivial multinomial coefficient (i.e., equal to k!/(k1!...km!) with k1 + ... + km = k, k-2 >= k1 >= ... >= km).