A324467 -id:A324467 - cp's OEIS frontend

A324152 a(0)=1; for n>0, a(n) = (3/((n+1)(n+2)(n+3))) * multinomial(4*n;n,n,n,n).

1, 3, 126, 9240, 900900, 104756652, 13742520792, 1968826448160, 301700280152700, 48756255150603000, 8226155369009738160, 1438285479229504301760, 259131100507849025033760, 47897087290614993606462000, 9050997011303368719799740000

Offset: 0

Michael De Vlieger and N. J. A. Sloane, Mar 01 2019

nonn

It is conjectured that a(n) is always an integer.

If all terms except the first are doubled, we get A324478, which IS known to be integral.

N. J. A. Sloane, Table of n, a(n) for n = 0..360
Luis Fredes, Avelio Sepulveda, Tree-decorated planar maps, arXiv:1901.04981 [math.CO], 2019. See Remark 4.6.

Cf. A000108, A324151, A324465 (exponent of 2), A324467, A324478.

[1] cat [n le 1 select 3 else Self(n-1)*4*(4*n-3)*(4*n-2)*(4*n-1)/((n)^2*(n+3)): n in [1..20]]; // Vincenzo Librandi, Mar 11 2019

c[m_, n_] := m Product[1/(n + i), {i, m}] (Multinomial @@ ConstantArray[n, m + 1]); {1}~Join~Array[c[3, #] &, 14] (* Michael De Vlieger, Mar 01 2019 *)
Flatten[{1, Table[3*(4*n)! / ((n!)^3 * (n+3)!), {n, 1, 15}]}] (* Vaclav Kotesovec, Jul 21 2019 *)

a(n+1) = a(n)*4*(4*n+1)*(4*n+2)*(4*n+3)/((n+1)^2*(n+4)) for n>0.
From Vaclav Kotesovec, Jul 21 2019: (Start)
For n>0, a(n) = 3*(4*n)! / ((n!)^3 * (n+3)!).
a(n) ~ 3 * 2^(8*n - 1/2) / (Pi^(3/2) * n^(9/2)). (End)

A324465 Exponent of highest power of 2 that divides A324152(n).

Original entry on oeis.org

0, 0, 1, 3, 2, 2, 3, 5, 2, 3, 4, 6, 5, 4, 5, 7, 2, 3, 4, 6, 5, 5, 6, 8, 5, 6, 7, 9, 8, 6, 7, 9, 2, 3, 4, 6, 5, 5, 6, 8, 5, 6, 7, 9, 8, 7, 8, 10, 5, 6, 7, 9, 8, 8, 9, 11, 8, 9, 10, 12, 11, 8, 9, 11, 2, 3, 4, 6, 5, 5, 6, 8, 5, 6, 7, 9, 8, 7, 8, 10, 5, 6, 7, 9

Offset: 0

N. J. A. Sloane, Mar 01 2019

nonn

First occurrence of k=0,1,2,...: 0, 2, 4, 3, 10, 7, 11, 15, 23, 27, 47, 55, 59, 111, 119, 123, 239, 247, 251, 495, 503, 507, 1007, 1015, 1019, 2031, 2039, 2043, 4079, 4087, 4091, 8175, 8183, 8187, 16367, 16375, 16379, 32751, 32759, 32763, 65519, 65527, 65531, 131055, 131063, 131067, ..., . Robert G. Wilson v, Mar 01 2019

Robert G. Wilson v, Table of n, a(n) for n = 0..10000 (first 501 terms from N. J. A. Sloane)

Cf. A000120 (binary weight), A007814, A324152, A324467.

f[n_] := IntegerExponent[(3/((n + 1)(n + 2)(n + 3)))*Multinomial[n, n, n, n], 2]; f[0] = 0; Array[f, 84, 0] (* Robert G. Wilson v, Mar 01 2019 *)

a(n) = 3*hammingweight(n) - valuation((n+1)*(n+2)*(n+3), 2); \\ Michel Marcus, Jul 10 2022

a(n) = 3*wt(n) - (2-adic valuation of (n+1)*(n+2)*(n+3))
= 3*A000120(n) - (A007814(n+1)+A007814(n+2)+A007814(n+3)).
E.g. if n = 14 = 1110_2, with weight 3, we get a(14) = 3*3 - 2-adic valuation of 15*16*17 = 9 - 4 = 5.

A324466 Exponent of highest power of 2 that divides multinomial(3*n;n,n,n).

Original entry on oeis.org

0, 1, 1, 4, 1, 2, 4, 6, 1, 2, 2, 7, 4, 5, 6, 8, 1, 2, 2, 5, 2, 3, 7, 9, 4, 5, 5, 9, 6, 7, 8, 10, 1, 2, 2, 5, 2, 3, 5, 7, 2, 3, 3, 10, 7, 8, 9, 11, 4, 5, 5, 8, 5, 6, 9, 11, 6, 7, 7, 11, 8, 9, 10, 12, 1, 2, 2, 5, 2, 3, 5, 7, 2, 3, 3, 8, 5, 6, 7, 9, 2, 3, 3, 6, 3, 4, 10, 12, 7, 8, 8

Offset: 0

N. J. A. Sloane, Mar 02 2019

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000120 (analog for binomial coefficients), A006480, A007814, A324467.

[Valuation(Factorial(3*n)/Factorial(n)^3, 2): n in [0..100]]; // Vincenzo Librandi, Mar 10 2019

[seq(padic[ordp](combinat:-multinomial(3*n,n,n,n),2),n=0..128)];
# alternative:
f:= proc(n) local r,t;
  t:= 0; r:= 3*n;
  while r > 1 do t:= t + floor(r) - 3*floor(r/3); r:= r/2; od;
  t
end proc:
map(f, [$0..200]); # Robert Israel, Mar 03 2019

Table[IntegerExponent[(3 n)!/n!^3, 2], {n, 0, 100}] (* Vincenzo Librandi Mar 10 2019 *)

a(n) = valuation((3*n)!/n!^3, 2); \\ Michel Marcus, Mar 04 2019

from math import factorial
def A324466(n): return (~(m:=factorial(3*n)//factorial(n)**3)& m-1).bit_length() # Chai Wah Wu, Jul 07 2022

a(2*n) = a(n). - Robert Israel, Mar 04 2019
From Amiram Eldar, Feb 21 2021: (Start)
a(n) = A007814(A006480(n)).
a(n) = 3*A000120(n) - A000120(3*n). (End)

A324469 Exponent of highest power of 3 that divides multinomial(4*n;n,n,n,n).

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 2, 5, 6, 1, 2, 3, 2, 3, 6, 4, 6, 7, 2, 3, 4, 5, 6, 8, 6, 8, 9, 1, 2, 3, 2, 3, 5, 3, 6, 7, 2, 3, 4, 3, 4, 8, 6, 8, 9, 4, 5, 6, 6, 7, 9, 7, 9, 10, 2, 3, 4, 3, 4, 6, 4, 9, 10, 5, 6, 7, 6, 7, 10, 8, 10, 11, 6, 7, 8, 8, 9, 11, 9, 11, 12, 1, 2

Offset: 0

N. J. A. Sloane, Mar 03 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Analogs for binomial and trinomials: A000989, A053735. See also A324467.

Cf. A007949 (3-adic valuation of n), A008977.

[seq(padic[ordp](combinat[multinomial](4*n, n$4), 3), n=0..128)];

s[n_] := Plus @@ IntegerDigits[n, 3]; a[n_] := 2*s[n] - s[4*n]/2; Array[a, 100, 0] (* Amiram Eldar, Feb 21 2021 *)

a(n) = 2*A000989(n) + A000989(2*n). - Charlie Neder, Mar 09 2019
From Amiram Eldar, Feb 21 2021: (Start)
a(n) = A007949(A008977(n)).
a(n) = 2*A053735(n) - A053735(4*n)/2. (End)

cp's OEIS Frontend

A324152 a(0)=1; for n>0, a(n) = (3/((n+1)*(n+2)*(n+3))) * multinomial(4*n;n,n,n,n).

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A324465 Exponent of highest power of 2 that divides A324152(n).

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A324466 Exponent of highest power of 2 that divides multinomial(3*n;n,n,n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

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Formula

A324469 Exponent of highest power of 3 that divides multinomial(4*n;n,n,n,n).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A324152 a(0)=1; for n>0, a(n) = (3/((n+1)(n+2)(n+3))) * multinomial(4*n;n,n,n,n).