A079484 -id:A079484 - cp's OEIS frontend

A000909 a(n) = (2n)!(2*n+1)! / n!^2.

1, 12, 720, 100800, 25401600, 10059033600, 5753767219200, 4487938430976000, 4577697199595520000, 5914384781877411840000, 9439358111876349296640000, 18236839872145106841108480000, 41944731705933745734549504000000

Offset: 0

N. J. A. Sloane

From Karol A. Penson, Jun 04 2009: (Start)
Integral representation of a(n) as n-th moment of a positive function W(x) = (1/4)*BesselK(1,(1/2)*sqrt(x))/Pi on the positive axis: a(n) = Integral_{x=0..oo} x^n*W(x) dx, n >= 0.
This is the solution of the Stieltjes moment problem with the moments a(n).
This solution may not be unique. (End)

E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 96.

T. D. Noe, Table of n, a(n) for n = 0..100
Index to divisibility sequences

a(n) = 4^n * A079484(n+1).

Table[(2*n)! (2*n+1)!/n!^2, {n, 0, 15}] (* T. D. Noe, Jun 20 2012 *)

a(n)=binomial(2*n,n)*(2*n+1)! \\ Charles R Greathouse IV, Jan 12 2012

A013069 Expansion of e.g.f.: exp(arcsinh(x)+log(x+1))=1+2x+3/2!x^2+3/3!x^3-3/4!x^4-15/5!*x^5...

Original entry on oeis.org

1, 2, 3, 3, -3, -15, 45, 315, -1575, -14175, 99225, 1091475, -9823275, -127702575, 1404728325, 21070924875, -273922023375, -4656674397375, 69850115960625, 1327152203251875, -22561587455281875

Offset: 0

Patrick Demichel (patrick.demichel(AT)hp.com)

sign

Cf. A046126.

a(2n) = (-1)^(n+1) * A079484(n), n>1.

With[{nn=20},CoefficientList[Series[Exp[ArcSinh[x]+Log[x+1]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 15 2024 *)

Definition clarified by Harvey P. Dale, Aug 15 2024

A046126 Denominators q[ n ] of convergents to Stern's non-simple continued fraction for Pi/2.

Original entry on oeis.org

1, 3, -3, -15, 45, 315, -1575, -14175, 99225, 1091475, -9823275, -127702575, 1404728325, 21070924875, -273922023375, -4656674397375, 69850115960625, 1327152203251875, -22561587455281875, -473793336560919375

Offset: 0

Eric W. Weisstein

Eric Weisstein's World of Mathematics, Pi.
Eric Weisstein's World of Mathematics, Pi Continued Fraction.
Index entries for sequences related to Stern's sequences

Numerators p[ n ] are (-1)^[n/2]*A001900(n). See also A013069.

Cf. A079484.

b[ n_ ] := 2-(-1)^n; a[ 1 ] := -1; a[ n_Integer?EvenQ ] := -n(n+1); a[ n_Integer?OddQ ] := -(n-2)(n-1); then use the standard algorithm to get p[ n ]/q[ n ].
a[n_] := Product[If[OddQ[k], k+2, 1-k], {k, 1, n}]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Nov 06 2012, after 1st Pari program *)

a(n)=if(n<0,0,prod(k=1,n,if(k%2,k+2,1-k)))

{a(n)=local(A); if(n<0, 0, A=matrix(2,n+1); for(k=0, n, A[2, k+1]=if(k%2, 3, 1); A[1, k+1]=if(k<2, (-1)^k, if(k%2, -(k-2)*(k-1), -k*(k+1)))); contfracpnqn(A)[2,1])} /* Michael Somos, Jul 15 2003 */

E.g.f.: exp(asinh(x))((1+x)/(1+x^2)+(2-x+x^2)/(1+x^2)^(3/2))-2. - Michael Somos, Mar 11 2004
E.g.f.: (1+3*x+2*x^3)/(1+x^2)^(3/2). - Vaclav Kotesovec, Oct 05 2013
a(n) ~ 2*(cos(Pi*n/2)+sin(Pi*n/2)) * n^(n+1) / exp(n). - Vaclav Kotesovec, Oct 05 2013

A277354 a(n) = Product_{k=1..n} (4*k^2+1).

Original entry on oeis.org

1, 5, 85, 3145, 204425, 20646925, 2993804125, 589779412625, 151573309044625, 49261325439503125, 19753791501240753125, 9580588878101765265625, 5527999782664718558265625, 3742455852864014463945828125, 2937827844498251354197475078125

Offset: 0

Vaclav Kotesovec, Oct 10 2016

nonn

In general, for m>0, Product_{k=1..n} (m*k^2+1) is asymptotic to 2*m^(n+1/2) * n^(2*n+1) * sinh(Pi/sqrt(m)) / exp(2*n).

Cf. A079484, A101686, A101928, A274306, A277352, A277353.

Table[Product[4*k^2+1, {k, 1, n}], {n, 0, 15}]
Round@Table[2^(2 n + 1) Abs[Gamma[1 + I/2 + n]]^2 Sinh[Pi/2]/Pi, {n, 0, 15}] (* Vladimir Reshetnikov, Oct 10 2016 *)

a(n) = prod(k=1, n, (4*k^2+1)); \\ Michel Marcus, Oct 11 2016

a(n) = (-1)^(n+1) * A101928(n+2).
a(n) ~ 2^(2*n+2) * n^(2*n+1) * sinh(Pi/2) / exp(2*n).
a(n) = 2^(2*n+1) * |Gamma(1 + i/2 + n)|^2 * sinh(Pi/2)/Pi. - Vladimir Reshetnikov, Oct 10 2016

A120362 Numerators of bivariate Taylor expansion of the incomplete elliptic integral of the second kind.

Original entry on oeis.org

1, 0, -1, 0, 4, -3, 0, -16, 60, -45, 0, 64, -1008, 2520, -1575, 0, -256, 16320, -105840, 189000, -99225, 0, 1024, -261888, 4055040, -15800400, 21829500, -9823275, 0, -4096, 4193280, -149909760, 1153152000, -3178375200, 3575672100, -1404728325, 0, 16384, -67104768, 5459650560

Offset: 1

R. J. Mathar, Jun 26 2006

Table has only rows for odd h because all coefficients for even h are zero:
=====|=======================================================================
h \ s|  0     1         2          3            4            5             6
-----|-----------------------------------------------------------------------
1    |  1
3    |  0    -1
5    |  0     4        -3
7    |  0   -16        60        -45
9    |  0    64     -1008       2520        -1575
11   |  0  -256     16320    -105840       189000       -99225
13   |  0  1024   -261888    4055040    -15800400     21829500      -9823275
15   |  0 -4096   4193280 -149909760   1153152000  -3178375200    3575672100
17   |  0 16384 -67104768 5459650560 -79048569600 390486096000 -829555927200
...
From Francesco Franco, Jan 12 2016: (Start)
Conjecture:
If t(h,s) is any term of the previous table after the first column (s>0), then:
t(h,s) = -( 4*s^2*t(h-2,s) + Sum_{j=0..s-1} (t(h-2,j) + t(h,j)) ), with t(1,0) = 1, t(h,0) = 0 for h>1 and t(h,s) = 0 for odd h = 1..2*s-1.
Version without the summation:
t(h,s) = -( 4*s^2*t(h-2,s) - (4*(s-1)^2-1)*t(h-2,s-1) ).
Some example (starting from j=1 in the summation):
t(11,3) = -( 4*t(9,3)*3^2 + Sum_{j=1..2} (t(9,j) + t(11,j)) ) = -( 4*2520*9 + (64-256) + (-1008+16320) ) = -105840; second version:
t(17,5) = -( 4*5^2*t(15,5) - (4*4^2-1)*t(15,4) ) = -( 4*25*(-3178375200) - 63*1153152000 ) = 390486096000.
Also:
t(h,1) = (-1)^(h/2-1/2)*A000302(h/2-3/2) for h>1;
t(h,2) = (-1)^(h/2-3/2)*A115490(h/2-3/2) for h>3;
a(A000124(n)) = 0.
(End)

			E(m,phi) = phi - m*phi^3/3! + (4*m-3*m^2)*phi^5/5! + (-16*m+60*m^2-45*m^3)*phi^7/7! + ...
so the first row (order phi^1) is a(1,1)=1 for the coefficient of phi,
the second row (order phi^3) is a(2,0)=0 for the missing coefficient of m^0*phi^3, and a(2,1)=-1 for the coefficient of m^1*phi^3/3!.

R. J. Mathar, Chebyshev series expansion of the Elliptic Integral of the Second Kind

Cf. A010370, A079484 (diagonal).

an := proc(m,n,s) local f: f := coeftayl(EllipticE(sin(phi),m^(1/2)),phi=0,n); coeftayl(f*n!,m=0,s) ; end: nmax := 27 ; for n from 1 to nmax by 2 do for s from 0 to (n-1)/2 do printf("%d,",an(m,n,s)) ; od ; od;

a[n_, s_] := SeriesCoefficient[EllipticE[phi, m], {phi, 0, n}, {m, 0, s}]*n!; Table[a[n, s], {n, 1, 17, 2}, {s, 0, n/2}] // Flatten (* Jean-François Alcover, Jan 06 2014 *)

{T(n, k) = my(m = 2*n+1); if( k<0 || nMichael Somos, May 04 2017 */

E(m,phi) = Int_{theta=0..phi} sqrt(1-m*sin^2 theta) d theta.

E(m,phi) = Sum_{n=1,3,5,7,9,...} ( Sum_{s=0..(n-1)/2} a( (n+1)/2,s ) * m^s )*phi^n/n!.

A177698 Expansion of e.g.f.: sin(arctan(x)).

Original entry on oeis.org

0, 1, 0, -3, 0, 45, 0, -1575, 0, 99225, 0, -9823275, 0, 1404728325, 0, -273922023375, 0, 69850115960625, 0, -22561587455281875, 0, 9002073394657468125, 0, -4348001449619557104375, 0, 2500100833531245335015625, 0, -1687568062633590601135546875

Offset: 0

Michel Lagneau, May 11 2010

sign

Except periodic zeros for n even, and negative signs for n == 3 (mod 4), we find the same sequence as A079484 (determinant of M(2n-1) where M(k) is the k X k matrix with m(i,j)=j if i+j=k, m(i,j)=i otherwise).

			G.f. = x - 3*x^3 + 45*x^5 - 1575*x^7 + 99225*x^9 - 9823275*x^11 + ...
d^3y/dx^3 = 18/(1+x^2)^(5/2)*x^2 -3/(1+x^2)^(3/2) -15*x^4/(1+x^2)^(7/2).
For x = 0, we obtain a(3) = 0 - 3 + 0 = -3.

L. Comtet and M. Fiolet, Sur les dérivées successives d'une fonction implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251.

G. C. Greubel, Table of n, a(n) for n = 0..450

Cf. A079484.

m:=50; R:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!(x/Sqrt(1+x^2))); // G. C. Greubel, Sep 25 2018

a:= n-> n! * coeff(series(sin(arctan(x)), x, n+1), x, n):
seq(a(n), n=0..30);

Table[n!*SeriesCoefficient[x/Sqrt[1+x^2], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 25 2013 *)
With[{nn=30},CoefficientList[Series[Sin[ArcTan[x]],{x,0,nn}],x] Range[ 0,nn-1]!] (* Harvey P. Dale, Nov 30 2015 *)
Join[{0}, Table[DifferenceRoot[Function[{y, m}, {y[1 + m] == (n - 2 m)*y[m], y[0] == 1}]][n], {n, 1, 20}]] (* Benedict W. J. Irwin, Nov 03 2016 *)
Join[{0}, Table[Re[-((I (2 I)^n Gamma[1 + n/2] Gamma[n/2])/Pi)], {n, 1, 20}]] (* Benedict W. J. Irwin, Nov 03 2016 *)
a[ n_] := If[ EvenQ[n], 0, I^(n - 1) n!! (n - 2)!!]; (* Michael Somos, May 04 2017 *)

{a(n) = if( n%2==0, 0, n<0, 1 / self()(-n), n! * binomial(n-1,n\2) * 2^(1-n) * (-1)^(n\2))}; /* Michael Somos, May 04 2017 */

x='x+O('x^50); concat([0], Vec(x/sqrt(1+x^2))) \\ G. C. Greubel, Sep 25 2018

E.g.f.: sin(arctan(x)) = x/sqrt(1+x^2).
E.g.f.: x/(G(0)+x) where G(k)= 1 - 2*x/(1 + 1/G(k+1) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 08 2012
a(n) ~ 2*sin(Pi*n/2)*n^n/exp(n). - Vaclav Kotesovec, Sep 25 2013
From Benedict W. J. Irwin, Nov 03 2016: (Start)
a(n) = y(n,n), n>0, where y(m+1,n) = (n-2*m)*y(m,n), with y(0,n)=1, for all n.
a(n) = Real part of -i*(2*i)^n*Gamma(1 + n/2)*Gamma(n/2)/Pi. (End)
From Michael Somos, May 04 2017: (Start)
a(n) = -n * (n-2) * a(n-2) for all n in Z.
a(n) = 1 / a(-n) for all odd n in Z.
a(n) = n! * binomial(n-1,(n-1)/2) * 2^(1-n) * (-1)^((n-1)/2) if n is odd > 0.
a(2*n + 1) = (-1)^n * A079484(n). (End)

A296788 Expansion of e.g.f. exp(x*arcsinh(x)) (even powers only).

Original entry on oeis.org

1, 2, 8, 54, 104, 18810, -1648428, 247726374, -49445941200, 12841169289714, -4206667789245780, 1697448414191239710, -827415782970517712376, 479396168140498731959850, -325673237888367403728512700, 256401822876859593450127851030, -231597610351491427264049084814240

Offset: 0

Ilya Gutkovskiy, Dec 20 2017

sign

			exp(x*arcsinh(x)) = 1 + 2*x^2/2! + 8*x^4/4! + 54*x^6/6! + 104*x^8/8! + ...

Cf. A001818, A006228, A009214, A009233, A079484, A259647, A296787, A296789.

nmax = 16; Table[(CoefficientList[Series[Exp[x ArcSinh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 16; Table[(CoefficientList[Series[(x + Sqrt[1 + x^2])^x, {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] exp(x*arcsinh(x)).

a(n) ~ -(-1)^n * 2^(2*n) * n^(2*n-1) / exp(2*n + Pi/2). - Vaclav Kotesovec, Dec 21 2017

A306364 Triangular array of the number of binary, rooted, leaf-labeled tree topologies with n leaves and k cherries, n >= 2, 1 <= k <= floor(n/2).

Original entry on oeis.org

1, 3, 12, 3, 60, 45, 360, 540, 45, 2520, 6300, 1575, 20160, 75600, 37800, 1575, 181440, 952560, 793800, 99225, 1814400, 12700800, 15876000, 3969000, 99225, 19958400, 179625600, 314344800, 130977000, 9823275

Offset: 2

Noah A Rosenberg, Feb 10 2019

A cherry is an internal node with exactly two descendant leaves. Each binary, rooted, leaf-labeled tree topology with n leaves has at least 1 cherry and at most floor(n/2) cherries.

			For n=4 leaves A, B, C, and D, a(4,1)=12 and a(4,2)=3. The 12 labeled topologies with 1 cherry are (((A,B),C),D), (((A,B),D),C), (((A,C),B),D), (((A,C),D),B), (((A,D),B),C), (((A,D),C),B), (((B,C),A),D), (((B,C),D),A), (((B,D),A),C), (((B,D),C),A), (((C,D),A),B), (((C,D),B),A). The 3 labeled topologies with 2 cherries are ((A,B),(C,D)), ((A,C),(B,D)), ((A,D),(B,C)).
Triangular array begins:
        1;
        3;
       12,        3;
       60,       45;
      360,      540,       45;
     2520,     6300,     1575;
    20160,    75600,    37800,    1575;
   181440,   952560,   793800,   99225;
  1814400, 12700800, 15876000, 3969000, 99225;
  ...

Robert S. Maier, Triangular recurrences, generalized Eulerian numbers, and related number triangles, Adv. Appl. Math. 146 (2023), 102485.
Noah A. Rosenberg, Enumeration of lonely pairs of gene trees and species trees by means of antipodal cherries, Adv. Appl. Math. 102 (2019), 1-17.
Taoyang Wu and Kwok Pui Choi, On joint subtree distributions under two evolutionary models, Theor. Pop. Biol. 108 (2016), 13-23.

Row sums equal A001147(n-1).
Column k=1 gives A001710.
T(2n,n) gives A079484(n-1).

Table[n! (n - 2)!/(2^(2 k - 1) (n - 2 k)! k! (k - 1)!), {n, 2, 15}, {k, 1, Floor[n/2]}]

T(n,k) = n! (n-2)! / (2^(2k-1) (n-2k)! k! (k-1)! ).

A013155 Expansion of e.g.f. exp(arctanh(x)+log(x+1)).

Original entry on oeis.org

1, 2, 3, 6, 21, 90, 495, 3150, 23625, 198450, 1885275, 19646550, 225935325, 2809456650, 37927664775, 547844046750, 8491582724625, 139700231921250, 2444754058621875, 45123174910563750, 879901910755993125, 18004146789314936250, 387089155970271129375, 8696002899239114208750

Offset: 0

Patrick Demichel (patrick.demichel(AT)hp.com)

nonn

			G.f.= 1+2*x+3/2!*x^2+6/3!*x^3+21/4!*x^4+90/5!*x^5...

Andrew Howroyd, Table of n, a(n) for n = 0..200

a(2n+1) = 2 * A079484(n+1).

With[{nn=20},CoefficientList[Series[Exp[ArcTanh[x]+Log[x+1]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 05 2021 *)

my(x='x+O('x^25)); Vec(serlaplace(exp(atanh(x)+log(x+1)))) \\ Christian Krause, Jan 05 2024

a(n) = 2*a(n-1) + ((2-n)^2-1)*a(n-2). - Christian Krause, Jan 05 2024

Definition clarified by Harvey P. Dale, Oct 05 2021

Terms a(21) and beyond from Andrew Howroyd, Jan 05 2024

A211163 Numerator of (-1/Pi^n) * integral_{0..1} (log(1-1/x)^n) dx.

Original entry on oeis.org

2, 0, 8, 0, 32, 0, 128, 0, 2560, 0, 1415168, 0, 57344, 0, 118521856, 0, 5749735424, 0, 91546451968, 0, 1792043646976, 0, 1982765704675328, 0, 286994513002496, 0, 3187598700536922112, 0, 4625594563496048066560

Offset: 2

Jean-François Alcover, Jan 30 2013

Conjecture: sequence of denominators is A141459.

			2*Pi^2/3, 0, 8*Pi^4/15, 0, 32*Pi^6/21, 0, 128*Pi^8/15, 0, 2560*Pi^10/33, ...

Cf. A079484 (Gerry Martens's Pari program uses this integral).

a[n_] := (-1/Pi^n)*Numerator[Integrate[Log[1 - 1/x]^n, {x, 0, 1}]]; Table[Print[an = a[n]]; an, {n, 2, 30}]

cp's OEIS Frontend

A000909 a(n) = (2*n)!*(2*n+1)! / n!^2.

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A013069 Expansion of e.g.f.: exp(arcsinh(x)+log(x+1))=1+2*x+3/2!*x^2+3/3!*x^3-3/4!*x^4-15/5!*x^5...

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A046126 Denominators q[ n ] of convergents to Stern's non-simple continued fraction for Pi/2.

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PARI

PARI

Formula

A277354 a(n) = Product_{k=1..n} (4*k^2+1).

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PARI

Formula

A120362 Numerators of bivariate Taylor expansion of the incomplete elliptic integral of the second kind.

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Maple

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A177698 Expansion of e.g.f.: sin(arctan(x)).

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Magma

Maple

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PARI

PARI

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A296788 Expansion of e.g.f. exp(x*arcsinh(x)) (even powers only).

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A306364 Triangular array of the number of binary, rooted, leaf-labeled tree topologies with n leaves and k cherries, n >= 2, 1 <= k <= floor(n/2).

A000909 a(n) = (2n)!(2*n+1)! / n!^2.

A013069 Expansion of e.g.f.: exp(arcsinh(x)+log(x+1))=1+2x+3/2!x^2+3/3!x^3-3/4!x^4-15/5!*x^5...