A001803 -id:A001803 - cp's OEIS frontend

A162440 The pg(n) sequence that is associated with the Eta triangle A160464.

2, 16, 144, 4608, 115200, 4147200, 203212800, 26011238400, 2106910310400, 210691031040000, 25493614755840000, 3671080524840960000, 620412608698122240000, 121600871304831959040000

Offset: 2

Johannes W. Meijer, Jul 06 2009

The EG1 matrix coefficients are defined by EG1[2m-1,1] = 2*eta(2m-1) and the recurrence relation EG1[2m-1,n] = EG1[2m-1,n-1] - EG1[2m-3,n-1]/(n-1)^2 with m = .., -2, -1, 0, 1, 2, ... and n = 1, 2, 3, ... . As usual, eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function. For the EG2 matrix, the even counterpart of the EG1 matrix, see A008955.
The coefficients in the columns of the EG1 matrix, for m >= 1 and n >= 2, can be generated with GFE(z;n) = ((-1)^(n-1)*r(n)*CFN1(z,n)*GFE(z;n=1) + ETA(z,n))/pg(n) for n >= 2.
The CFN1(z,n) polynomials depend on the central factorial numbers A008955 and the ETA(z,n) are the Eta polynomials which led to the Eta triangle, see for both A160464.
The pg(n) sequence can be generated with the first Maple program and the EG1[2m-1,n] matrix coefficients can be generated with the second Maple program.
The EG1 matrix is related to the ES1 matrix, see A160464 and the formulas below.

			The first few generating functions GFE(z;n) are:
GFE(z;n=2) = ((-1)*2*(z^2 - 1)*GFE(z;n=1) + (-1))/2,
GFE(z;n=3) = ((+1)*4*(z^4 - 5*z^2 + 4)*GFE(z;n=1) + (-11 + 2*z^2))/16,
GFE(z;n=4) = ((-1)*4*(z^6-14*z^4+49*z^2-36)*GFE(z;n=1) + (-114+29*z^2-2*z^4))/144.

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.

The ETA(z, n) polynomials and the ES1 matrix lead to the Eta triangle A160464.
The CFN1(z, n), the t1(n, m) and the EG2 matrix lead to A008955.
The EG1[ -1, n] equal (1/2)*A001803(n-1)/A046161(n-1).
The r(n) sequence equals A062383(n) (n>=1).
The e(n) sequence equals A029837(n) (n>=1).
Cf. A160473 (p(n) sequence).
Cf. A162443 (BG1 matrix), A162446 (ZG1 matrix) and A162448 (LG1 matrix).

nmax := 16; seq((n-1)!^2*2^floor(ln(n-1)/ln(2)+1), n=2..nmax);
# End program 1
nmax1 := 5; coln := 4; mmax1 := nmax1: for n from 0 to nmax1 do t1(n, 0) := 1 end do: for n from 0 to nmax1 do t1(n, n) := (n!)^2 end do: for n from 1 to nmax1 do for m from 1 to n-1 do t1(n, m) := t1(n-1, m-1)*n^2 + t1(n-1, m) end do: end do: for m from 1 to mmax1 do EG1[1-2*m, 1] := evalf((2^(2*m)-1)* bernoulli(2*m)/(m)) od: EG1[1, 1] := evalf(2*ln(2)): for m from 2 to mmax1 do EG1[2*m-1, 1] := evalf(2*(1-2^(1-(2*m-1))) * Zeta(2*m-1)) od: for m from -mmax1+coln to mmax1 do EG1[2*m-1, coln]:= (-1)^(coln+1)*sum((-1)^k*t1(coln-1, k) * EG1[1-2*coln+2*m+2*k, 1], k=0..coln-1)/(coln-1)!^2 od;
# End program 2 (Edited by Johannes W. Meijer, Sep 21 2012)

pg(n) = (n-1)!^2*2^floor(log(n-1)/log(2)+1) for n >= 2.
r(n) = 2^e(n) = 2^floor(log(n-1)/log(2)+1) for n >= 2.
EG1[ -1,n] = 2^(1-2*n)*(2*n-1)!/((n-1)!^2) for n >= 1.
GFE(z;n) = sum (EG1[2*m-1,n]*z^(2*m-2), m=1..infinity).
GFE(z;n) = (1-z^2/(n-1)^2)*GFE(z;n-1)-EG1[ -1,n-1]/(n-1)^2 for n = >2. with GFE(z;n=1) = 2*log(2)-Psi(z)-Psi(-z)+Psi(z/2)+Psi(-z/2) and Psi(z) is the digamma function.
EG1[2m-1,n] = (2*2^(1-2*n)*(2*n-1)!/((n-1)!^2)) * ES1[2m-1,n].

A163590 Odd part of the swinging factorial A056040.

Original entry on oeis.org

1, 1, 1, 3, 3, 15, 5, 35, 35, 315, 63, 693, 231, 3003, 429, 6435, 6435, 109395, 12155, 230945, 46189, 969969, 88179, 2028117, 676039, 16900975, 1300075, 35102025, 5014575, 145422675, 9694845, 300540195, 300540195, 9917826435, 583401555, 20419054425, 2268783825

Offset: 0

Peter Luschny, Aug 01 2009

nonn

Let n$ denote the swinging factorial. a(n) = n$ / 2^sigma(n) where sigma(n) is the exponent of 2 in the prime-factorization of n$. sigma(n) can be computed as the number of '1's in the base 2 representation of floor(n/2).

If n is even then a(n) is the numerator of the reduced ratio (n-1)!!/n!! = A001147(n-1)/A000165(n), and if n is odd then a(n) is the numerator of the reduced ratio n!!/(n-1)!! = A001147(n)/A000165(n-1). The denominators for each ratio should be compared to A060818. Here all ratios are reduced. - Anthony Hernandez, Feb 05 2020 [See the Mathematica program for a more compact form of the formula. Peter Luschny, Mar 01 2020 ]

			11$ = 2772 = 2^2*3^2*7*11. Therefore a(11) = 3^2*7*11 = 2772/4 = 693.
From _Anthony Hernandez_, Feb 04 2019: (Start)
a(7) = numerator((1*3*5*7)/(2*4*6)) = 35;
a(8) = numerator((1*3*5*7)/(2*4*6*8)) = 35;
a(9) = numerator((1*3*5*7*9)/(2*4*6*8)) = 315;
a(10) = numerator((1*3*5*7*9)/(2*4*6*8*10)) = 63. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
Peter Luschny, Swinging Factorial.

Cf. A056040, A060632, A001790, A001803.

swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
sigma := n -> 2^(add(i,i= convert(iquo(n,2),base,2))):
a := n -> swing(n)/sigma(n);

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/ f!]; a[n_] := With[{s = sf[n]}, s/2^IntegerExponent[s, 2]]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jul 26 2013 *)
r[n_] := (n - Mod[n - 1, 2])!! /(n - 1 + Mod[n - 1, 2])!! ;
Table[r[n], {n, 0, 36}] // Numerator (* Peter Luschny, Mar 01 2020 *)

A163590(n) = {
    my(a = vector(n+1)); a[1] = 1;
    for(n = 1, n,
        a[n+1] = a[n]*n^((-1)^(n+1))*2^valuation(n, 2));
a } \\ Peter Luschny, Sep 29 2019

# uses[A000120]
@CachedFunction
def swing(n):
    if n == 0: return 1
    return swing(n-1)*n if is_odd(n) else 4*swing(n-1)/n
A163590 = lambda n: swing(n)/2^A000120(n//2)
[A163590(n) for n in (0..31)]  # Peter Luschny, Nov 19 2012
# Alternatively:

@cached_function
def A163590(n):
    if n == 0: return 1
    return A163590(n - 1) * n^((-1)^(n + 1)) * 2^valuation(n, 2)
print([A163590(n) for n in (0..31)]) # Peter Luschny, Sep 29 2019

a(2*n) = A001790(n).
a(2*n+1) = A001803(n).
a(n) = a(n-1)*n^((-1)^(n+1))*2^valuation(n, 2) for n > 0. - Peter Luschny, Sep 29 2019

A001800 Coefficients of Legendre polynomials.

Original entry on oeis.org

1, 3, 30, 70, 315, 693, 12012, 25740, 109395, 230945, 1939938, 4056234, 16900975, 35102025, 1163381400, 2404321560, 9917826435, 20419054425, 167890003050, 344616322050, 1412926920405, 2893136075115, 47342226683700, 96742811049300, 395033145117975

Offset: 0

N. J. A. Sloane

nonn

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.
G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Eric Weisstein's World of Mathematics, Legendre Polynomial, eq. 28.

Cf. A001790, A001796, A001803, A008316.

Diagonal 2 of triangle A100258.

A001800:= func< n | (n+1)*(n+2)*Catalan(n+1)/2^(&+Intseq(n+2, 2)) >;
[A001800(n): n in [0..30]]; // G. C. Greubel, Apr 25 2025

wt:= proc(n) local m, r; m:=n; r:=0;
       while m>0 do r:= r+irem(m, 2, 'm') od; r
     end:
a:= n-> (n+1) *binomial(2*n+2, n+1)/2^wt(n+2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 29 2013

a[n_] := (n+1)*Binomial[2*n+2, n+1]/2^DigitCount[n+2, 2, 1]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Mar 13 2014 *)

a(n)=if(n<0,0,-polcoeff(pollegendre(n+2),n)*2^valuation((n\2*2)!,2))

def A001800(n): return (n+1)*binomial(2*n+2,n+1)//2^sum((n+2).digits(2))
print([A001800(n) for n in range(31)]) # G. C. Greubel, Apr 25 2025

a(n) = (n+1) * C(2n+2, n+1) / 2^A000120(n+2).

More terms from Michael Somos, Oct 25 2002

A002011 a(n) = 4*(2n+1)!/n!^2.

Original entry on oeis.org

4, 24, 120, 560, 2520, 11088, 48048, 205920, 875160, 3695120, 15519504, 64899744, 270415600, 1123264800, 4653525600, 19234572480, 79342611480, 326704870800, 1343120024400, 5513861152800, 22606830726480, 92580354403680, 378737813469600

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

R. C. Mullin, E. Nemeth and P. J. Schellenberg, The enumeration of almost cubic maps, pp. 281-295 in Proceedings of the Louisiana Conference on Combinatorics, Graph Theory and Computer Science. Vol. 1, edited R. C. Mullin et al., 1970.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

a(n)=4 A002457(n).
a(n) = 2 * A005430(n+1) = 4 * A002457(n).
Cf. A001803.

seq(2*n*binomial(2*n,n), n=1..23); # Zerinvary Lajos, Dec 14 2007

Table[4*(2*n + 1)!/n!^2, {n, 0, 20}] (* T. D. Noe, Aug 30 2012 *)

PARI
```
a(n)=if(n<0,0,4*(2*n+1)!/n!^2)
```

G.f.: 4*(1-4x)^(-3/2).

a(n) = 1/J(n) where J(n) = Integral_{t=0..Pi/4} (cos(t)^2 - 1/2)^(2n+1). - Benoit Cloitre, Oct 17 2006

Simpler description from Travis Kowalski (tkowalski(AT)coloradocollege.edu), Mar 20 2003

A086116 Numerator of mean deviation of a symmetrical binomial distribution on n elements.

Original entry on oeis.org

1, 1, 3, 3, 15, 15, 35, 35, 315, 315, 693, 693, 3003, 3003, 6435, 6435, 109395, 109395, 230945, 230945, 969969, 969969, 2028117, 2028117, 16900975, 16900975, 35102025, 35102025, 145422675, 145422675, 300540195, 300540195, 9917826435

Offset: 1

Eric W. Weisstein, Jul 10 2003

From Mohamed Sabba, Apr 24 2023: (Start)
The numerators of this sequence and denominators (A086117) appear in NMR and spin physics as half of the symmetry-constrained upper bound on polarization transfer in AXn spin-1/2 systems. For example:
(a) in AX3 and AX4 spin systems, the maximum achievable transfer from Xz to Az is 3/2 = (2*3/4).
(b) in AX5 and AX6 spin systems, the maximum achievable transfer from Xz to Az is 15/8 = (2*15/16).
Note that this is different from the related adiabatic polarization transfer bounds, given by A141244.
(End)

Ole W. Sørensen, A universal bound on spin dynamics, Journal of Magnetic Resonance, 86:2 (1990) (equations 20 and 21)
Ole W. Sørensen, Polarization transfer experiments in high-resolution NMR spectroscopy, Progress in Nuclear Magnetic Resonance Spectroscopy, 21:6 (1989), (page 534)
Eric Weisstein's World of Mathematics, Random Walk 1-Dimensional
Eric Weisstein's World of Mathematics, Binomial Distribution

Cf. A001803, A086117, A141244.

Numerator[Table[If[OddQ[n], n!!/2/(n-1)!!, (n-1)!!/2/(n-2)!! ], {n, 50}]]

A173384 a(n) = 2^(2n - HammingWeight(n)) [x^n] ((x-1)^(-1) + (1-x)^(-3/2)).

Original entry on oeis.org

0, 1, 7, 19, 187, 437, 1979, 4387, 76627, 165409, 707825, 1503829, 12706671, 26713417, 111868243, 233431331, 7770342787, 16124087129, 66765132341, 137948422657, 1138049013461, 2343380261227, 9636533415373, 19787656251221

Offset: 0

Paul Curtz, Feb 17 2010

nonn

If n >= 1 it appears a(n-1) is equal to the difference between the denominator and the numerator of the ratio (2n-1)!!/(2n-2)!!. In particular a(n-1) is the difference between the denominator and the numerator of the ratio A001147(2n-2)/A000165(2n-1). See examples. - Anthony Hernandez, Feb 05 2020
It can be seen that this is true, e.g., using A001803(n) = (2n+1)!/(n!^2*2^A000120(n)) and A046161(n) = 4^n/2^A000120(n). - M. F. Hasler, Feb 07 2020
Numerators in the expansion of (1-(1-x)^(1/2))/(1-x)^(3/2). Denominators are A046161. Compare to A001790. - Thomas Curtright, Feb 09 2024

			From _Anthony Hernandez_, Feb 05 2020: (Start)
Consider n = 4. The 4th odd number is 7, and 7!!/(7-1)!! = 35/16, and a(4-1) = a(3) = 35 - 16 = 19.
Consider n = 7. The 7th odd number is 13, and 13!!/(13-1)!! = 3003/1024, and a(7-1) = a(6) = 3003 - 1024 = 1979. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Thomas Curtright and Gaurav Verma, Scattering Shadows, arXiv:2404.07745 [physics.class-ph], 2024, p. 9.

Cf. A001790, A005430, A046161.

List([0..30], n-> (NumeratorRat((2*n+1)*Binomial(2*n, n)/(4^n)) - DenominatorRat(Binomial(2*n, n)/(4^n)))); # G. C. Greubel, Dec 09 2018

[Numerator((2*n+1)*Binomial(2*n, n)/(4^n)) - Denominator(Binomial(2*n, n)/(4^n)): n in [0..30]]; // G. C. Greubel, Dec 09 2018

A046161 := proc(n) binomial(2*n,n)/4^n ; denom(%) ; end proc:
A173384 := proc(n) A001803(n)-A046161(n) ; end proc: # R. J. Mathar, Jul 06 2011

Table[Numerator[(2*n+1)*Binomial[2*n, n]/(4^n)] - Denominator[Binomial[2*n, n]/(4^n)], {n,0,30}] (* G. C. Greubel, Dec 09 2018 *)
A173384[n_] := 2^(2*n - DigitCount[n, 2, 1]) Coefficient[Series[(x - 1)^(-1) + (1 - x)^(-3/2), {x, 0, n}], x, n]
Table[A173384[n], {n, 0, 23}]  (* Peter Luschny, Feb 17 2024 *)

for(n=0,30, print1(numerator((2*n+1)*binomial(2*n, n)/(4^n)) - denominator(binomial(2*n, n)/4^n), ", ")) \\ G. C. Greubel, Dec 09 2018

[(numerator((2*n+1)*binomial(2*n, n)/(4^n)) - denominator(binomial(2*n, n)/(4^n))) for n in range(30)] # G. C. Greubel, Dec 09 2018

a(n) = A001803(n) - A046161(n). (Previous name.)

Let r(n) = (-2)^n*Sum_{j=0..n-1} binomial(n,j)*Bernoulli(j+n+1, 1/2)/(j+n+1) then a(n) = numerator(r(n)). - Peter Luschny, Jun 20 2017

New name using an expansion of Thomas Curtright by Peter Luschny, Feb 17 2024

A036069 Denominator of rational part of Haar measure on Grassmannian space G(n,1).

Original entry on oeis.org

1, 2, 1, 4, 3, 16, 5, 32, 35, 256, 63, 512, 231, 2048, 429, 4096, 6435, 65536, 12155, 131072, 46189, 524288, 88179, 1048576, 676039, 8388608, 1300075, 16777216, 5014575, 67108864, 9694845, 134217728, 300540195

Offset: 0

N. J. A. Sloane

Also rational part of denominator of Gamma(n/2+1)/Gamma(n/2+1/2) (cf. A004731).

			1, 1, 1/2*Pi, 2, 3/4*Pi, 8/3, 15/16*Pi, 16/5, 35/32*Pi, 128/35, 315/256*Pi, ...
The sequence Gamma(n/2+1)/Gamma(n/2+1/2), n >= 0, begins 1/Pi^(1/2), (1/2)*Pi^(1/2), 2/Pi^(1/2), (3/4)*Pi^(1/2), (8/3)/Pi^(1/2), (15/16)*Pi^(1/2), (16/5)/Pi^(1/2), ...

D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge, p. 67.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A004731.
Bisections are A001790 and A101926.
Cf. A004731, A046161, A001790, A001803, A101926.

if n mod 2 = 0 then k := n/2; 2*k*Pi*binomial(2*k-1,k)/4^k else k := (n-1)/2; 4^k/binomial(2*k,k); fi;
f:=n->simplify(GAMMA(n/2+1)/GAMMA(n/2+1/2));

Table[ Denominator[ Gamma[n/2+1]/Gamma[n/2+1/2]*Sqrt[Pi]^(1 - 2 Mod[n, 2])], {n, 0, 32}] (* Jean-François Alcover, Jul 16 2012 *)

A277233 Numerators of the partial sums of the squares of the expansion coefficients of 1/sqrt(1-x). Also the numerators of the Landau constants.

Original entry on oeis.org

1, 5, 89, 381, 25609, 106405, 1755841, 7207405, 1886504905, 7693763645, 125233642041, 508710104205, 33014475398641, 133748096600189, 2165115508033649, 8754452051708621, 9054883309760265929, 36559890613417481741, 590105629859261338481, 2379942639329101454549

Offset: 0

Wolfdieter Lang, Nov 12 2016

This is the instance m=1/2 of the partial sums r(m,n) = Sum_{k=0..n} (risefac(m,k)/ k!)^2, where risefac(x,k) = Product_{j=0..k-1} (x+j), and risefac(x,0) = 1.
The limit n -> oo does not exist. It would be hypergeometric([1/2,1/2],[1],z -> 1), which diverges.
The partial sums of the cubes converge for |m| < 2/3. See Morley's series under A277232 (for m=1/2).
a(n)/A056982(n) are the Landau constants. These constants are defined as G(n) = Sum_{j=0..n} g(j)^2, where g(n) = (2*n)!/(2^n*n!)^2 = A001790(n)/A046161(n). - Peter Luschny, Sep 27 2019

			The rationals r(n) begin: 1, 5/4, 89/64, 381/256, 25609/16384, 106405/65536, 1755841/1048576, 7207405/4194304, 1886504905/1073741824, 7693763645/4294967296, ...

Seiichi Manyama, Table of n, a(n) for n = 0..831
Edmund Landau, Abschätzung der Koeffzientensumme einer Potenzreihe, Arch. Math. Phys. 21 (1913), 42-50. [Accessible in the USA through the Hathi Trust Digital Library.]
Edmund Landau, Abschätzung der Koeffzientensumme einer Potenzreihe (Zweite Abhandlung), Arch. Math. Phys. 21 (1913), 250-255. [Accessible in the USA through the Hathi Trust Digital Library.]
Cristinel Mortici, Sharp bounds of the Landau constants, Math. Comp. 80 (2011), pp. 1011-1018.
G. N. Watson, The constants of Landau and Lebesgue, Quart. J. Math. Oxford Ser. 1:2 (1930), pp. 310-318.

Denominators are A056982.

Cf. A001803/A046161, A277232/A241756, A001790/A046161, A005187.

a := n -> numer(add(binomial(-1/2, j)^2, j=0..n));
seq(a(n), n=0..19); # Peter Luschny, Sep 26 2019
# Alternatively:
G := proc(x) hypergeom([1/2,1/2], [1], x)/(1-x) end: ser := series(G(x), x, 20):
[seq(coeff(ser,x,n), n=0..19)]: numer(%); # Peter Luschny, Sep 28 2019

Accumulate[CoefficientList[Series[1/Sqrt[1-x],{x,0,20}],x]^2]//Numerator (* Harvey P. Dale, Feb 10 2019 *)
G[x_] := (2 EllipticK[x])/(Pi (1 - x));
CoefficientList[Series[G[x], {x, 0, 19}], x] // Numerator (* Peter Luschny, Sep 28 2019 *)

def A277233(n):
    return sum((A001790(k)*(2^(A005187(n) - A005187(k))))^2 for k in (0..n))
print([A277233(n) for n in (0..19)]) # Peter Luschny, Sep 30 2019

a(n) = numerator(r(n)), with the fractional
r(n) = Sum_{k=0..n} (risefac(1/2,k)/k!)^2;
r(n) = Sum_{k=0..n} (binomial(-1/2,k))^2;
r(n) = Sum_{k=0..n} ((2*k-1)!!/(2*k)!!)^2.
The rising factorial has been defined in a comment above. The double factorials are given in A001147 and A000165 with (-1)!! := 1.
r(n) ~ (log(n+3/4) + EulerGamma + 4*log(2))/Pi. - Peter Luschny, Sep 27 2019
Rational generating function: (2*K(x))/(Pi*(1-x)) where K is the complete elliptic integral of the first kind. - Peter Luschny, Sep 28 2019
a(n) = Sum_{k=0..n}(A001790(k)*(2^(A005187(n) - A005187(k))))^2. - Peter Luschny, Sep 30 2019

A093581 Numerators of odd moments in the distribution of chord lengths for picked at random on the circumference of a unit circle.

Original entry on oeis.org

4, 32, 512, 4096, 131072, 1048576, 16777216, 134217728, 8589934592, 68719476736, 1099511627776, 8796093022208, 281474976710656, 2251799813685248, 36028797018963968, 288230376151711744

Offset: 1

Eric W. Weisstein, Apr 01 2004

Presumably this is the same as A102557? - Andrew S. Plewe, Apr 18 2007

A102557(n) equals a(n) for n <= 55000. - G. C. Greubel, Oct 20 2024

			1, 4/Pi, 2, 32/(3*Pi), 6, 512/(15*Pi), 20, 4096/(35*Pi), ...

G. C. Greubel, Table of n, a(n) for n = 1..828
Eric Weisstein's World of Mathematics, Circle Line Picking

Cf. A000984, A001803, A061549, A102557.

Denominators are A001803*Pi.

A093581:= func< n | Power(2, 4*n-2-(&+Intseq(2*(n-1), 2))) >;
[A093581(n): n in [1..30]]; // G. C. Greubel, Oct 20 2024

Table[Power[2, 4*n-2 - DigitCount[n-1,2,1]], {n, 30}] (* G. C. Greubel, Oct 20 2024 *)

def A093581(n): return pow(2, 4*n-2 - sum((2*n-2).digits(2)))
[A093581(n) for n in range(1,31)] # G. C. Greubel, Oct 20 2024

a(n) = 4*A061549(n-1).

A144815 Numerators of triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the coefficient of x^(2k+1) in polynomial t_n(x), used to define continuous and n times differentiable sigmoidal transfer functions.

Original entry on oeis.org

1, 3, -1, 15, -5, 3, 35, -35, 21, -5, 315, -105, 189, -45, 35, 693, -1155, 693, -495, 385, -63, 3003, -3003, 9009, -2145, 5005, -819, 231, 6435, -15015, 27027, -32175, 25025, -12285, 3465, -429, 109395, -36465, 153153, -109395, 425425, -69615, 58905, -7293, 6435

Offset: 0

Alois P. Heinz, Sep 21 2008

All even coefficients of t_n have to be 0, because t_n is defined to be point-symmetric with respect to the origin, with vanishing n-th derivative for x=1.

A sigmoidal transfer function sigma_n: R->[ -1,1] can be defined as sigma_n(x) = 1 if x>1, sigma_n(x) = t_n(x) if x in [ -1,1] and sigma_n(x) = -1 if x<-1.

			1, 3/2, -1/2, 15/8, -5/4, 3/8, 35/16, -35/16, 21/16, -5/16, 315/128, -105/32, 189/64, -45/32, 35/128, 693/256, -1155/256, 693/128, -495/128, 385/256, -63/256 ... = A144815/A144816
As triangle:
    1;
    3/2,     -1/2;
   15/8,     -5/4,    3/8;
   35/16,   -35/16,  21/16,  -5/16;
  315/128, -105/32, 189/64, -45/32, 35/128;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Alois P. Heinz, Animation of sigma_n(x) and their derivatives for n=0..15

Denominators of T(n,k): A144816.
Column k=0 gives A001803.
Diagonal gives (-1)^n A001790(n).
Cf. A144702, A144703.

t:= proc(n) option remember; local f,i,x; f:= unapply(simplify(sum('cat(a||(2*i+1)) *x^(2*i+1)', 'i'=0..n) ), x); unapply(subs(solve({f(1)=1, seq((D@@i)(f)(1)=0, i=1..n)}, {seq(cat(a||(2*i+1)), i=0..n)}), sum('cat(a||(2*i+1)) *x^(2*i+1)', 'i'=0..n) ), x); end: T:= (n,k)-> coeff(t(n)(x), x, 2*k+1): seq(seq(numer(T(n,k)), k=0..n), n=0..10);

row[n_] := Module[{f, a, eq}, f = Function[x, Sum[a[2*k+1]*x^(2*k+1), {k, 0, n}]]; eq = Table[Derivative[k][f][1] == If[k == 0, 1, 0], {k, 0, n}]; Table[a[2*k+1], {k, 0, n}] /. Solve[eq] // First]; Table[row[n] // Numerator, {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 03 2014 *)
Flatten[Table[Numerator[CoefficientList[Hypergeometric2F1[1/2,1-n,3/2,x^2]*(2*n)!/(n!*(n-1)!*2^(2*n-1)),x^2]],{n,1,9}]] (* Eugeniy Sokol, Aug 20 2019 *)

See program.

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A162440 The pg(n) sequence that is associated with the Eta triangle A160464.

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A163590 Odd part of the swinging factorial A056040.

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A001800 Coefficients of Legendre polynomials.

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A002011 a(n) = 4*(2n+1)!/n!^2.

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A086116 Numerator of mean deviation of a symmetrical binomial distribution on n elements.

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A173384 a(n) = 2^(2*n - HammingWeight(n)) * [x^n] ((x-1)^(-1) + (1-x)^(-3/2)).

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A173384 a(n) = 2^(2n - HammingWeight(n)) [x^n] ((x-1)^(-1) + (1-x)^(-3/2)).