A049606 -id:A049606 - cp's OEIS frontend

A160481 Row sums of the Beta triangle A160480.

-1, -10, -264, -13392, -1111680, -137030400, -23500108800, -5351202662400, -1562069156659200, -568747270103040000, -252681700853514240000, -134539938778433126400000, -84573370199475510312960000, -61972704966344777143418880000, -52361960516341326660973363200000

Offset: 2

Johannes W. Meijer, May 24 2009, Sep 19 2012

It is conjectured that the row sums of the Beta triangle depend on three different sequences. Two Maple algorithms are given. The first one gives the row sums according to the Beta triangle A160480 and the second one gives the row sums according to our conjecture.

Christopher P. Herzog, Kuo-Wei Huang, and Kristan Jensen, Universal Entanglement and Boundary Geometry in Conformal Field Theory, arXiv preprint arXiv:1510.00021 [hep-th], 2015.
Kuo-Wei Huang, Resummation of Multi-Stress Tensors in Higher Dimensions, arXiv:2406.07458 [hep-th], 2024. See p. 10.

A160480 is the Beta triangle.
Row sum factors A120778, A000165 and A049606.
Cf. A002474, A009445, A129890.

nmax := 14; mmax := nmax: for n from 1 to nmax do BETA(n, n) := 0 end do: m := 1: for n from m+1 to nmax do BETA(n,m) := (2*n-3)^2*BETA(n-1, m)-(2*n-4)! od: for m from 2 to mmax do for n from m+1 to nmax do BETA(n, m) := (2*n-3)^2*BETA(n-1, m) - BETA(n-1, m-1) od: od: for n from 2 to nmax do s1(n) := 0: for m from 1 to n-1 do s1(n) := s1(n) + BETA(n, m) od: od: seq(s1(n), n=2..nmax);
# End first program
nmax := nmax; A120778 := proc(n): numer(sum(binomial(2*k1, k1)/(k1+1) / 4^k1, k1=0..n)) end proc: A000165 := proc(n): 2^n*n! end proc: A049606 := proc(n): denom(2^n/n!) end proc: for n from 2 to nmax do s2(n) := (-1)*A120778(n-2)*A000165(n-2)*A049606(n-1) end do: seq(s2(n), n=2..nmax);
# End second program

BETA[2, 1] = -1; BETA[n_, 1] := BETA[n, 1] = (2*n - 3)^2*BETA[n - 1, 1] - (2*n - 4)!; BETA[n_ /; n > 2, m_ /; m > 0] /; 1 <= m <= n := BETA[n, m] = (2*n - 3)^2*BETA[n - 1, m] - BETA[n - 1, m - 1]; BETA[, ] = 0;
Table[Sum[BETA[n, m], {m, 1, n - 1}], {n, 2, 14}] (* Jean-François Alcover, Dec 13 2017 *)

Rowsums(n) = (-1)*A120778(n-2)*A000165(n-2)*A049606(n-1) for n >= 2.
Conjecture: a(n) = (2*n-3)! - 2^(2*n-3)*(n-1)!*(n-2)!, for n >= 2 (gives the first 13 terms). - Christopher P. Herzog, Nov 25 2014
Meijer's and Herzog's conjectures can also be written as: a(n) = -A129890(n-2)*A000165(n-2) = A009445(n-2) - A002474(n-2). - Peter Luschny, Dec 01 2014

a(15)-a(16) from Stefano Spezia, Jun 28 2024

A256400 Numerators of coefficients of expansion of exp( Sum_{k=0..oo} x^(2^k)/2^k ) in powers of x.

Original entry on oeis.org

1, 1, 1, 2, 2, 7, 16, 67, 88, 617, 2626, 18176, 6949, 423271, 2172172, 19151162, 58438907, 899510224, 7656246634, 1236339998, 460251204914, 6682341795214, 55431849118769, 58399157390146, 2845938531282919, 54648005951674444, 12207653488921678

Offset: 0

N. J. A. Sloane, Mar 29 2015

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..100
Koda, Tatsuhiko; Sato, Masaki; Takegahara, Yugen; 2-adic properties for the numbers of involutions in the alternating groups, J. Algebra Appl. 14 (2015), no. 4, 1550052 (21 pages). See Lemma 2.1.

For denominators see A049606. Cf. A256401/A256402.

a(18)-a(26) from Hiroaki Yamanouchi, Mar 30 2015

A212307 Numerator of n!/3^n.

Original entry on oeis.org

1, 1, 2, 2, 8, 40, 80, 560, 4480, 4480, 44800, 492800, 1971200, 25625600, 358758400, 1793792000, 28700672000, 487911424000, 975822848000, 18540634112000, 370812682240000, 2595688775680000, 57105153064960000, 1313418520494080000, 10507348163952640000

Offset: 0

Jean-François Alcover and Paul Curtz, Oct 30 2013

Also the 3rd column of A152656 (or of A216919).

Cf. A001316, A049606, A125824 (denominators), A152656, A216919.

Cf. A000142, A060828.

Table[Numerator[n!/3^n], {n, 0, 32}]
(* or *) CoefficientList[Series[Exp[3x], {x, 0, 32}], x] // Denominator

a(n) = numerator(n!/3^n); \\ Michel Marcus, Oct 30 2013

a(n) = Product_{i=1..n} A038502(i). - Tom Edgar, Mar 22 2014

a(n) = A000142(n)/A060828(n). - Ridouane Oudra, Sep 23 2024

A306184 a(n) = (2n+1)!! mod (2n)!! where k!! = A006882(k).

Original entry on oeis.org

1, 7, 9, 177, 2715, 42975, 91665, 3493665, 97345395, 2601636975, 70985324025, 57891366225, 9411029102475, 476966861546175, 20499289200014625, 847876038362978625, 35160445175104123875, 1487419121780448231375, 945654757149212735625, 357657177058846280240625

Offset: 1

Alex Ratushnyak, Jan 27 2019

nonn

a(n) is divisible by A049606(n). - Robert Israel, Jan 28 2019

			a(3) = A006882(7) mod A006882(6) = (7*5*3) mod (6*4*2) = 105 mod 48 = 9.

Robert Israel, Table of n, a(n) for n = 1..403

Cf. A006882, A049606, A122649, A129890, A232617, A232618, A306185.

f:= n -> doublefactorial(2*n+1) mod doublefactorial(2*n):
map(f, [$1..40]); # Robert Israel, Jan 28 2019

Mod[#[[2]],#[[1]]]&/@Partition[Range[2,42]!!,2] (* Harvey P. Dale, May 29 2025 *)

o=e=1
for n in range(2, 99, 2):
  o*=n+1
  e*=n
  print(o%e, end=', ')

a(n) = A006882(2*n+1) mod A006882(2*n).

A336940 Number of odd divisors of n!.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 12, 12, 20, 30, 60, 72, 144, 216, 336, 336, 672, 864, 1728, 2160, 3200, 4800, 9600, 10560, 14784, 22176, 28224, 35280, 70560, 86400, 172800, 172800, 245760, 368640, 497664, 559872, 1119744, 1679616, 2363904, 2626560, 5253120, 6451200, 12902400, 16128000

Offset: 0

Gus Wiseman, Aug 23 2020

nonn

			The a(1) = 1 through a(8) = 12 divisors:
  1  1  1  1  1   1   1    1
        3  3  3   3   3    3
              5   5   5    5
              15  9   7    7
                  15  9    9
                  45  15   15
                      21   21
                      35   35
                      45   45
                      63   63
                      105  105
                      315  315

Seiichi Manyama, Table of n, a(n) for n = 0..10000

A049606 gives the maximum among these divisors, with quotient A060818.
A337257 is the even version.
A000265 gives the maximum odd divisor of n.
A001227 counts odd divisors.
A183063 counts even divisors.
Cf. A000005, A001013, A001055, A006939, A113474, A124010, A253249.
Factorial numbers: A000142, A022559, A027423 (divisors), A048656, A071626, A076716 (factorizations), A325272, A325273, A325617, A336414, A336498.

Table[Length[Select[Divisors[n!],OddQ]],{n,0,15}]

a(n) = sumdiv(n!, d, d%2); \\ Michel Marcus, Aug 24 2020

a(n) = numdiv(prod(k=1, n, k >> valuation(k, 2))); \\ Michel Marcus, Aug 27 2020

a(n) = A001227(n!).
a(n) = A000005(A049606(n)).
a(n) + A337257(n) = A027423(n) = A000005(n!).
From Seiichi Manyama, Aug 27 2020: (Start)
If p is odd prime, a(p) = 2 * a(p-1).
a(n) = A027423(n) / A113474(n) for n > 0. (End)

a(36)-a(44) from Seiichi Manyama, Aug 26 2020

A337257 Number of even divisors of n!.

Original entry on oeis.org

0, 0, 1, 2, 6, 12, 24, 48, 84, 140, 240, 480, 720, 1440, 2376, 3696, 5040, 10080, 13824, 27648, 38880, 57600, 91200, 182400, 232320, 325248, 510048, 649152, 882000, 1764000, 2246400, 4492800, 5356800, 7618560, 11796480, 15925248

Offset: 0

Gus Wiseman, Aug 23 2020

nonn

			The a(2) = 1 through a(5) = 12 divisors:
  2  2  2   2
     6  4   4
        6   6
        8   8
        12  10
        24  12
            20
            24
            30
            40
            60
            120

Seiichi Manyama, Table of n, a(n) for n = 0..10000

A336940 is the odd version.
A000265 gives the maximum odd divisor of n.
A001227 counts odd divisors.
A183063 counts even divisors.
Cf. A000005, A001013, A001055, A006939, A049606, A124010, A253249.
Factorial numbers: A000142, A022559, A027423 (divisors), A048656, A071626, A076716 (factorizations), A325272, A325273, A325617, A336414, A336498.

Table[Length[Select[Divisors[n!],EvenQ]],{n,0,15}]

a(n) = sumdiv(n!, d, !(d%2)); \\ Michel Marcus, Aug 24 2020

a(n) = A183063(n!).

A336940(n) + a(n) = A027423(n) = A000005(n!).

A129195 a(n) = denominator(n!/4^n).

Original entry on oeis.org

1, 4, 8, 32, 32, 128, 256, 1024, 512, 2048, 4096, 16384, 16384, 65536, 131072, 524288, 131072, 524288, 1048576, 4194304, 4194304, 16777216, 33554432, 134217728, 67108864, 268435456, 536870912, 2147483648, 2147483648, 8589934592, 17179869184, 68719476736

Offset: 0

Paul Barry, Apr 02 2007

Cf. A049606, A092391.

Table[Denominator[n!/4^n],{n,0,30}] (* Harvey P. Dale, Mar 05 2013 *)

a(n) = denominator((1/(2*Pi))*int(exp(i*4*t)(-((Pi-t)/i)^n),t,0,2*Pi)), i=sqrt(-1).

a(n) = 2^A092391(n).

A129915 Irregular triangle read by rows: T(n, k) = f(n, A113474(n-1) - k), where f(n, k) = (n-1)!/2^k if (n-1)!/2^k is an integer, otherwise f(n, k) = 0.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 3, 6, 12, 24, 15, 30, 60, 120, 45, 90, 180, 360, 720, 315, 630, 1260, 2520, 5040, 315, 630, 1260, 2520, 5040, 10080, 20160, 40320, 2835, 5670, 11340, 22680, 45360, 90720, 181440, 362880, 14175, 28350, 56700, 113400, 226800, 453600

Offset: 1

Roger L. Bagula, Jun 05 2007

			Irregular triangle begins as:
    1;
    1;
    1,   2;
    3,   6;
    3,   6,   12,   24;
   15,  30,   60,  120;
   45,  90,  180,  360,  720;
  315, 630, 1260, 2520, 5040;
  315, 630, 1260, 2520, 5040, 10080, 20160, 40320;

G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened

Cf. A049606, A067655, A113474.

A113474:= func< n | n+1 - Multiplicity(Intseq(n, 2), 1) >;
f:= func< n,k | IsIntegral(Factorial(n-1)/2^k) select Factorial(n-1)/2^k else 0 >;
A129915:= func< n,k | f(n, A113474(n-1) - k) >;
[A129915(n,k): k in [1..A113474(n-1)], n in [1..12]]; // G. C. Greubel, Sep 28 2024

A113474[n_]:= n+1 - DigitCount[n, 2, 1];
f[n_, k_]:= If[IntegerQ[(n-1)!/2^k], (n-1)!/2^k, 0];
A129915[n_, k_]:= f[n, A113474[n-1]-k];
Table[A129915[n,k], {n,15}, {k,A113474[n-1]}]//Flatten (* modified by G. C. Greubel, Sep 28 2024 *)

def A113474(n): return n+1 - sum((n+0).digits(2))
def f(n,k): return factorial(n-1)/2^k if (factorial(n-1)/2^k).is_integer() else 0
def A129915(n,k): return f(n, A113474(n-1) - k)
flatten([[A129915(n,k) for k in range(1, A113474(n-1)+1)] for n in range(1,16)]) # G. C. Greubel, Sep 28 2024

T(n, k) = f(n, A113474(n-1) - k), where f(n, k) = (n-1)!/2^k if (n-1)!/2^k is an integer, otherwise f(n, k) = 0, for n >= 1, 1 <= k <= A113474(n-1).

Edited by G. C. Greubel, Sep 28 2024

A135354 a(0)=1, a(n) = largest divisor of n! that is coprime to a(n-1).

Original entry on oeis.org

1, 1, 2, 3, 8, 15, 16, 315, 128, 2835, 256, 155925, 1024, 6081075, 2048, 638512875, 32768, 10854718875, 65536, 1856156927625, 262144, 194896477400625, 524288, 49308808782358125, 4194304, 3698160658676859375, 8388608, 1298054391195577640625, 33554432, 263505041412702261046875, 67108864

Offset: 0

Leroy Quet, Dec 07 2007

Robert Israel, Table of n, a(n) for n = 0..504

Cf. A060818, A049606.

f:= proc(n,a)
  local P,R,i;
  P:= select(t -> isprime(t) and igcd(t,a)=1, [2,seq(i,i=3..n,2)]);
  R:= map(proc(p) local k; add(floor(n/p^k), k=1 ..ilog[p](n)) end proc, P);
  mul(P[i]^R[i],i=1..nops(P));
end proc:
R:= 1: r:= 1: for i from 1 to 50 do r:= f(i,r); R:= R,r od:
R; # Robert Israel, Jul 21 2024

a = {1}; For[n = 1, n < 25, n++, AppendTo[a, Select[Divisors[n! ], GCD[a[[ -1]], # ] == 1 &][[ -1]]]]; a (* Stefan Steinerberger, Dec 10 2007 *)
ldnf[{n_,a_}]:={n+1,Max[Select[Divisors[(n+1)!],CoprimeQ[#,a]&]]}; Transpose[ NestList[ldnf,{0,1},30]][[2]] (* Harvey P. Dale, Jan 21 2016 *)

a(2n) = the largest power of 2 that divides (2n)!. a(2n+1) = the largest odd divisor of (2n+1)! = (2n+1)!/a(2n).

More terms from Stefan Steinerberger, Dec 10 2007

More terms from Robert Israel, Jul 21 2024

A140105 Trailing zeros removed from n! in binary.

Original entry on oeis.org

1, 1, 1, 11, 11, 1111, 101101, 100111011, 100111011, 101100010011, 11011101011111, 100110000100010101, 1110010001100111111, 10111001100101000110011, 10100010011000011101100101, 100110000011101110111011101011, 100110000011101110111011101011

Offset: 0

Ben Branman, Jun 03 2008

nonn

			a(3)=11 because 3! in binary is 110, which is 11 when the zero is removed.

Robert Israel, Table of n, a(n) for n = 0..192

Cf. A007088, A049606.

seq(convert(n!/2^padic:-ordp(n!,2),binary), n=0..30); # Robert Israel, May 01 2019

a[n_]:=FromDigits[Drop[IntegerDigits[n!,2],-IntegerExponent[n!,2]]];Array[a,17,0] (* James C. McMahon, Jul 04 2025 *)

a(n) = A007088(A049606(n)). - Robert Israel, May 01 2019

a(13)-a(16) from James C. McMahon, Jul 04 2025

cp's OEIS Frontend

A160481 Row sums of the Beta triangle A160480.

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A256400 Numerators of coefficients of expansion of exp( Sum_{k=0..oo} x^(2^k)/2^k ) in powers of x.

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A212307 Numerator of n!/3^n.

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A306184 a(n) = (2n+1)!! mod (2n)!! where k!! = A006882(k).

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A336940 Number of odd divisors of n!.

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A337257 Number of even divisors of n!.

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A129195 a(n) = denominator(n!/4^n).

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A129915 Irregular triangle read by rows: T(n, k) = f(n, A113474(n-1) - k), where f(n, k) = (n-1)!/2^k if (n-1)!/2^k is an integer, otherwise f(n, k) = 0.

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