A009445 -id:A009445 - cp's OEIS frontend

A107254 a(n) = SF(2n-1)/SF(n-1)^2 where SF = A000178.

1, 1, 12, 8640, 870912000, 22122558259200000, 222531556847250309120000000, 1280394777025250130271722799104000000000, 5746332926632566442385615219551212618645504000000000000

Offset: 0

Henry Bottomley, May 14 2005

nonn

Inverse product of all matrix elements of n X n Hilbert matrix M(i,j) = 1/(i+j-1) (i,j = 1..n). - Alexander Adamchuk, Apr 12 2006

The n X n matrix with A(i,j) = 1/(i+j-1)! (i,j = 1..n) has determinant (-1)^floor(n/2)/a(n). - Mikhail Lavrov, Nov 01 2022

			a(3) = 1!*2!*3!*4!*5!/(1!*2!*1!*2!) = 34560/4 = 8640.
n = 2: HilbertMatrix[n,n]
  1/1 1/2
  1/2 1/3
so a(2) = 1 / (1 * 1/2 * 1/2 * 1/3) = 12.
The n X n Hilbert matrix begins:
  1/1 1/2 1/3 1/4  1/5  1/6  1/7  1/8 ...
  1/2 1/3 1/4 1/5  1/6  1/7  1/8  1/9 ...
  1/3 1/4 1/5 1/6  1/7  1/8  1/9 1/10 ...
  1/4 1/5 1/6 1/7  1/8  1/9 1/10 1/11 ...
  1/5 1/6 1/7 1/8  1/9 1/10 1/11 1/12 ...
  1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...

Alois P. Heinz, Table of n, a(n) for n = 0..20
Mathematics Stack Exchange, Determinant of a matrix involving factorials.
Eric Weisstein's World of Mathematics, Hilbert Matrix.

Cf. A000178, A002457, A005249, A098118.

A107254:= func< n | n eq 0 select 1 else (&*[Factorial(n+j)/Factorial(j): j in [0..n-1]]) >;
[A107254(n): n in [0..12]]; // G. C. Greubel, Apr 21 2021

a:= n-> mul((n+i)!/i!, i=0..n-1):
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 23 2012

Table[Product[(i+j-1),{i,1,n},{j,1,n}], {n,1,10}] (* Alexander Adamchuk, Apr 12 2006 *)
Table[n!*BarnesG[2n+1]/(BarnesG[n+2]*BarnesG[n+1]), {n,0,12}] (* G. C. Greubel, Apr 21 2021 *)

a = lambda n: prod(rising_factorial(k,n) for k in (1..n))
print([a(n) for n in (0..10)]) # Peter Luschny, Nov 29 2015

a(n) = n!*(n+1)!*(n+2)!*...*(2n-1)!/(0!*1!*2!*3!*...*(n-1)!) = A000178(2n-1)/A000178(n-1)^2 = A079478(n)/A000984(n) = A079478(n-1)*A009445(n-1) = A107252(n)*A000142(n) = A088020(n)/A039622(n).
a(n) = 1/Product_{j=1..n} ( Product_{i=1..n} 1/(i+j-1) ). - Alexander Adamchuk, Apr 12 2006
a(n) = 2^(n*(n-1)) * A136411(n) for n > 0 . - Robert Coquereaux, Apr 06 2013
a(n) = A136411(n) * A053763(n) for n > 0. [Following remark from Robert Coquereaux] - M. F. Hasler, Apr 06 2013
a(n) ~ A * 2^(2*n^2-1/12) * n^(n^2+1/12) / exp(3*n^2/2+1/12), where A = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Feb 10 2015
a(n) = Product_{k=1..n} rf(k,n) where rf denotes the rising factorial. - Peter Luschny, Nov 29 2015
a(n) = (n! * G(2*n+1))/(G(n+1)*G(n+2)), where G(n) is the Barnes G - function. - G. C. Greubel, Apr 21 2021

A275809 Indices of zeros in A275808.

Original entry on oeis.org

0, 5, 14, 19, 22, 54, 59, 74, 84, 89, 93, 97, 100, 111, 114, 119, 264, 269, 278, 283, 286, 366, 371, 408, 413, 422, 427, 430, 440, 463, 466, 482, 492, 497, 501, 536, 552, 557, 566, 571, 574, 579, 589, 592, 596, 601, 604, 615, 618, 623, 655, 658, 675, 685, 688, 692, 696, 701, 710, 715, 718, 1560, 1565, 1574, 1579, 1582, 1614, 1619, 1634, 1644

Offset: 0

Antti Karttunen, Aug 09 2016

Indexing begins from zero, because a(0) = 0 is a special case in this sequence.

Terms A009445(n)-1, [0, 5, 119, 5039, 362879, 39916799, ...] form a subsequence, and also the terms of A010050(n)-2, [0, 22, 718, 40318, 3628798, ...] form a subsequence.

Antti Karttunen, Table of n, a(n) for n = 0..1384
Index entries for sequences related to factorial base representation

Cf. A275736, A275808.
Cf. A275810 (first differences).
Subsequence of A275813 and of A275805 (after the initial 0).
Cf. also A009445, A010050.

A354211 a(n) is the numerator of Sum_{k=0..n} 1 / (2*k+1)!.

Original entry on oeis.org

1, 7, 47, 5923, 426457, 15636757, 7318002277, 1536780478171, 603180793741, 142957467201379447, 60042136224579367741, 10127106976545720025649, 18228792557782296046168201, 12796612375563171824410077103, 3463616416319098507140327535879, 1380498543075754976417359117871773

Offset: 0

Ilya Gutkovskiy, May 24 2022

			1, 7/6, 47/40, 5923/5040, 426457/362880, 15636757/13305600, 7318002277/6227020800, ...

Cf. A009445, A053557, A061354, A073742, A103816, A120265, A143382, A289381, A354331 (denominators), A354332, A354334.

Table[Sum[1/(2 k + 1)!, {k, 0, n}], {n, 0, 15}] // Numerator
nmax = 15; CoefficientList[Series[Sinh[Sqrt[x]]/(Sqrt[x] (1 - x)), {x, 0, nmax}], x] // Numerator

a(n) = numerator(sum(k=0, n, 1/(2*k+1)!)); \\ Michel Marcus, May 24 2022

from fractions import Fraction
from math import factorial
def A354211(n): return sum(Fraction(1,factorial(2*k+1)) for k in range(n+1)).numerator # Chai Wah Wu, May 24 2022

Numerators of coefficients in expansion of sinh(sqrt(x)) / (sqrt(x) * (1 - x)).

A354331 a(n) is the denominator of Sum_{k=0..n} 1 / (2*k+1)!.

Original entry on oeis.org

1, 6, 40, 5040, 362880, 13305600, 6227020800, 1307674368000, 513257472000, 121645100408832000, 51090942171709440000, 8617338912961658880000, 15511210043330985984000000, 10888869450418352160768000000, 2947253997913233984847872000000, 1174691236311131831103651840000000

Offset: 0

Ilya Gutkovskiy, May 24 2022

			1, 7/6, 47/40, 5923/5040, 426457/362880, 15636757/13305600, 7318002277/6227020800, ...

Cf. A009445, A053556, A061355, A073742, A143383, A289488, A354211 (numerators), A354333, A354335.

Table[Sum[1/(2 k + 1)!, {k, 0, n}], {n, 0, 15}] // Denominator
nmax = 15; CoefficientList[Series[Sinh[Sqrt[x]]/(Sqrt[x] (1 - x)), {x, 0, nmax}], x] // Denominator

a(n) = denominator(sum(k=0, n, 1/(2*k+1)!)); \\ Michel Marcus, May 24 2022

from fractions import Fraction
from math import factorial
def A354331(n): return sum(Fraction(1,factorial(2*k+1)) for k in range(n+1)).denominator # Chai Wah Wu, May 24 2022

Denominators of coefficients in expansion of sinh(sqrt(x)) / (sqrt(x) * (1 - x)).

A354332 a(n) is the numerator of Sum_{k=0..n} (-1)^k / (2*k+1)!.

Original entry on oeis.org

1, 5, 101, 4241, 305353, 33588829, 209594293, 1100370038249, 23023126954133, 102360822438075317, 42991545423991633141, 4350744396907953273869, 13052233190723859821607001, 9162667699888149594768114701, 7440086172309177470951709137213, 364172638960396581472899447242531

Offset: 0

Ilya Gutkovskiy, May 24 2022

			1, 5/6, 101/120, 4241/5040, 305353/362880, 33588829/39916800, 209594293/249080832, ...

Cf. A009445, A049469, A053557, A061354, A103816, A120265, A143382, A354211, A354298, A354333 (denominators), A354334.

Table[Sum[(-1)^k/(2 k + 1)!, {k, 0, n}], {n, 0, 15}] // Numerator
nmax = 15; CoefficientList[Series[Sin[Sqrt[x]]/(Sqrt[x] (1 - x)), {x, 0, nmax}], x] // Numerator

a(n) = numerator(sum(k=0, n, (-1)^k/(2*k+1)!)); \\ Michel Marcus, May 24 2022

from fractions import Fraction
from math import factorial
def A354332(n): return sum(Fraction(-1 if k % 2 else 1,factorial(2*k+1)) for k in range(n+1)).numerator # Chai Wah Wu, May 24 2022

Numerators of coefficients in expansion of sin(sqrt(x)) / (sqrt(x) * (1 - x)).

A354333 a(n) is the denominator of Sum_{k=0..n} (-1)^k / (2*k+1)!.

Original entry on oeis.org

1, 6, 120, 5040, 362880, 39916800, 249080832, 1307674368000, 27360571392000, 121645100408832000, 51090942171709440000, 5170403347776995328000, 15511210043330985984000000, 10888869450418352160768000000, 8841761993739701954543616000000, 432780981798838043038187520000000

Offset: 0

Ilya Gutkovskiy, May 24 2022

			1, 5/6, 101/120, 4241/5040, 305353/362880, 33588829/39916800, 209594293/249080832, ...

Cf. A009445, A049469, A053556, A061355, A143383, A354299, A354331, A354332 (numerators), A354335.

Table[Sum[(-1)^k/(2 k + 1)!, {k, 0, n}], {n, 0, 15}] // Denominator
nmax = 15; CoefficientList[Series[Sin[Sqrt[x]]/(Sqrt[x] (1 - x)), {x, 0, nmax}], x] // Denominator

a(n) = denominator(sum(k=0, n, (-1)^k/(2*k+1)!)); \\ Michel Marcus, May 24 2022

from fractions import Fraction
from math import factorial
def A354333(n): return sum(Fraction(-1 if k % 2 else 1,factorial(2*k+1)) for k in range(n+1)).denominator # Chai Wah Wu, May 24 2022

Denominators of coefficients in expansion of sin(sqrt(x)) / (sqrt(x) * (1 - x)).

A191662 a(n) = n! / A000034(n-1).

Original entry on oeis.org

1, 1, 6, 12, 120, 360, 5040, 20160, 362880, 1814400, 39916800, 239500800, 6227020800, 43589145600, 1307674368000, 10461394944000, 355687428096000, 3201186852864000, 121645100408832000, 1216451004088320000, 51090942171709440000, 562000363888803840000

Offset: 1

Paul Curtz, Jun 10 2011

The a(n) are the denominators in the formulas of the k-dimensional square pyramidal numbers:
A005408 = (2*n+1)/1                       = 1, 3,  5,   7,   9, ... (k=1)
A000290 = (n^2)/1                         = 1, 4,  9,  16,  25, ... (k=2)
A000330 = n*(n+1)*(2*n+1)/6               = 1, 5, 14,  30,  55, ... (k=3)
A002415 = (n^2)*(n^2-1)/12                = 1, 6, 20,  50, 105, ... (k=4)
A005585 = n*(n+1)*(n+2)*(n+3)*(2*n+3)/120 = 1, 7, 27,  77, 182, ... (k=5)
A040977 = (n^2)*(n^2-1)*(n^2-4)/360       = 1, 8, 35, 112, 294, ... (k=6)
A050486 (k=7), A053347 (k=8), A054333 (k=9), A054334 (k=10), A057788 (k=11).
The first superdiagonal of this array appears in A029651. - Paul Curtz, Jul 04 2011
The general formula for the k-dimensional square pyramidal numbers is (2*n+k)*binomial(n+k-1,k-1)/k, k >= 1, n >= 0, see A097207. - Johannes W. Meijer, Jun 22 2011

Cf. A000142, A009445, A002674, A064680.
Cf. A000290, A000330, A002415, A005408, A005585, A029651, A040977, A050486, A053347, A054333, A054334, A057788.
Cf. A052612.

A191662:= proc(n): n!/A000034(n-1) end: A000034 := proc(n) op((n mod 2)+1, [1, 2]) ; end proc: seq(A191662(n),n=1..17); # Johannes W. Meijer, Jun 22 2011

Array[If[EvenQ[#],#!/2,#!]&,20] (* Harvey P. Dale, Mar 14 2014 *)

a(2*n-1) = (2*n-1)!, a(2*n) = (2*n)!/2.
a(n+1) = A064680(n+1) * a(n).
From Amiram Eldar, Jul 06 2022: (Start)
Sum_{n>=1} 1/a(n) = sinh(1) + 2*cosh(1) - 2.
Sum_{n>=1} (-1)^(n+1)/a(n) = sinh(1) - 2*cosh(1) + 2. (End)
D-finite with recurrence: a(n) - (n-1)*n*a(n-2) = 0 for n >= 3 with a(1)=a(2)=1. - Georg Fischer, Nov 25 2022
a(n) = A052612(n)/2 for n >= 1. - Alois P. Heinz, Sep 05 2023

More terms from Harvey P. Dale, Mar 14 2014

A085990 Number of topological types of polygons with 2n different sides.

Original entry on oeis.org

0, 3, 60, 2520, 181440, 19958400, 3113510400, 653837184000, 177843714048000, 60822550204416000, 25545471085854720000, 12926008369442488320000, 7755605021665492992000000, 5444434725209176080384000000, 4420880996869850977271808000000

Offset: 1

Sergey L. Dolmatov, Almir Dzhumaev (aalma(AT)mail.ru), Aug 18 2003

a(n) equals (-1)^n times the coefficient of sqrt(1-x^2)*(arcsin x)^2 in int (arcsin x)^(2n-1) dx. - John M. Campbell, Jul 20 2011

For n >= 4, also the number of distinct adjacency matrices of the n-Moebius ladder. - Eric W. Weisstein, Mar 31 2017

			For example: if n=1 then no polygon exists with 2 different sides. If n=2 then the polygon has 4 different sides A, B, C, D. In this case 3 different types of such 4-angle exist: (A, B, C, D), (A, B, D, C), (A, C, B, D).

Eric Weisstein's World of Mathematics, Adjacency Matrix
Eric Weisstein's World of Mathematics, Moebius Ladder

Cf. A009445.

A085990:=n->`if`(n=1,0,(2*n-1)!/2): seq(A085990(n), n=1..20); # Wesley Ivan Hurt, Mar 31 2015

nn = 32; a = Log[1/(1 - x^2)^(1/4)] - x^2/4; Prepend[Select[Range[0, nn]! CoefficientList[Series[a, {x, 0, nn}], x], # > 0 &], 0] (* Geoffrey Critzer, Dec 10 2011 *)
Join[{0}, (2 Range[2, 20] - 1)!/2] (* Wesley Ivan Hurt, Mar 31 2015 *)

a(n)=(2*n-1)!\2 \\ Charles R Greathouse IV, Dec 10 2011

a(n) = (n-1)*(2*n-1)*(2*n-3)!

a(n) = (2n-1)!/2 = A009445(n)/2, for n>1. - Wesley Ivan Hurt, Mar 31 2015

A136579 Triangle read by rows: A128174 * A136572.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 1, 0, 6, 1, 0, 2, 0, 24, 0, 1, 0, 6, 0, 120, 1, 0, 2, 0, 24, 0, 720, 0, 1, 0, 6, 0, 120, 0, 5040, 1, 0, 2, 0, 24, 0, 720, 0, 40320

Offset: 0

Gary W. Adamson, Jan 09 2008

Row sums = A136580: 1, 1, 3, 7, 27, 127, ...

			First few rows of the triangle:
  1;
  0, 1;
  1, 0, 2;
  0, 1, 0, 6;
  1, 0, 2, 0, 24;
  0, 1, 0, 6,  0, 120;
  1, 0, 2, 0, 24,   0, 720;
  ...

Cf. A136580, A128174, A136572, A000142.

A128174 * A136572 Triangle, even rows = even n! interspersed with zeros. Odd n rows, = odd n! interspersed with zeros.

T(2*i,2*k) = (2*k)! = A010050(k). T(2*i+1,2*k+1) = (2*k+1)! = A009445(k). - R. J. Mathar, Jun 04 2021

A160481 Row sums of the Beta triangle A160480.

Original entry on oeis.org

-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

cp's OEIS Frontend

A107254 a(n) = SF(2n-1)/SF(n-1)^2 where SF = A000178.

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A275809 Indices of zeros in A275808.

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A354211 a(n) is the numerator of Sum_{k=0..n} 1 / (2*k+1)!.

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A354331 a(n) is the denominator of Sum_{k=0..n} 1 / (2*k+1)!.

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A354332 a(n) is the numerator of Sum_{k=0..n} (-1)^k / (2*k+1)!.

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A354333 a(n) is the denominator of Sum_{k=0..n} (-1)^k / (2*k+1)!.

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A191662 a(n) = n! / A000034(n-1).

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A085990 Number of topological types of polygons with 2n different sides.

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