A001803 -id:A001803 - cp's OEIS frontend

A161198 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).

1, 1, 2, 3, 8, 4, 15, 46, 36, 8, 105, 352, 344, 128, 16, 945, 3378, 3800, 1840, 400, 32, 10395, 39048, 48556, 27840, 8080, 1152, 64, 135135, 528414, 709324, 459032, 160720, 31136, 3136, 128

Offset: 0

Johannes W. Meijer, Jun 08 2009, Jul 22 2011

The series expansion of (1-x)^((-1-2*n)/2) = sum(b(p)*x^p, p=0..infinity) for n = 0, 1, 2, .. can be described with b(p) = (F(p,n)/ (2*n-1)!!)*(binomial(2*p,p)/4^(p)) with F(x,n) = 2^n * product( x+(2*k-1)/2, k=1..n). The roots of the F(x,n) polynomials can be found at p = (1-2*k)/2 with k from 1 to n for n = 0, 1, 2, .. . The coefficients of the F(x,n) polynomials lead to the triangle given above. The triangle row sums lead to A001147.
Quite surprisingly we discovered that sum(b(p)*x^p, p=0..infinity) = (1-x)^(-1-2*n)/2, for n = -1, -2, .. . We assume that if m = n+1 then the value returned for product(f(k), k = m..n) is 1 and if m> n+1 then 1/product(f(k), k=n+1..m-1) is the value returned. Furthermore (1-2*n)!! = (-1)^(n+1)/(2*n-3)!! for n = 1, 2, 3 .. . This leads to b(p) = ((-1-2*n)!!/ G(p,n))*(binomial(2*p,p) /4^(p)) for n = -1, -2, .. . For the G(p,n) polynomials we found that G(p,n) = F(-p,-n). The roots of the G(p,n) polynomials can be found at p=(2*k-1)/2 with k from 1 to (-n) for n = -1, -2, .. . The coefficients of the G(p,n) polynomials lead to a second triangle that stands with its head on top of the first one. It is remarkable that the row sums lead once again to A001147.
These two triangles together look like an hourglass so we propose to call the F(p,n) and the G(p,n) polynomials the hourglass polynomials.
Triangle T(n,k), read by rows, given by (1, 2, 3, 4, 5, 6, 7, 8, 9, ...) DELTA (2, 0, 2, 0, 2, 0, 2, 0, 2, ...) where DELTA is the operator defined in A084938. Philippe Deléham, May 14 2015.

			From _Gary W. Adamson_, Jul 19 2011: (Start)
The first few rows of matrix M are:
  1, 2,  0,  0, 0, ...
  1, 3,  2,  0, 0, ...
  1, 4,  5,  2, 0, ...
  1, 5,  9,  7, 2, ...
  1, 6, 14, 16, 9, ... (End)
The first few G(p,n) polynomials are:
  G(p,-3) = 15 - 46*p + 36*p^2 - 8*p^3
  G(p,-2) = 3 - 8*p + 4*p^2
  G(p,-1) = 1 - 2*p
The first few F(p,n) polynomials are:
  F(p,0) = 1
  F(p,1) = 1 + 2*p
  F(p,2) = 3 + 8*p + 4*p^2
  F(p,3) = 15 + 46*p + 36*p^2 + 8*p^3
The first few rows of the upper and lower hourglass triangles are:
  [15, -46, 36, -8]
  [3, -8, 4]
  [1, -2]
  [1]
  [1, 2]
  [3, 8, 4]
  [15, 46, 36, 8]

Cf. A001790 [(1-x)^(-1/2)], A001803 [(1-x)^(-3/2)], A161199 [(1-x)^(-5/2)] and A161201 [(1-x)^(-7/2)].
Cf. A002596 [(1-x)^(1/2)], A161200 [(1-x)^(3/2)] and A161202 [(1-x)^(5/2)].
A046161 gives the denominators of the series expansions of all (1-x)^((-1-2*n)/2).
A028338 is a scaled triangle version, A039757 is a scaled signed triangle version and A109692 is a transposed scaled triangle version.
A001147 is the first left hand column and equals the row sums.
A004041 is the second left hand column divided by 2, A028339 is the third left hand column divided by 4, A028340 is the fourth left hand column divided by 8, A028341 is the fifth left hand column divided by 16.
A000012, A000290, A024196, A024197 and A024198 are the first (n-m=0), second (n-m=1), third (n-m=2), fourth (n-m=3) and fifth (n-m=4) right hand columns divided by 2^m.
A074599 * A025549 is not always equals the second left hand column.
Cf. A029635. [Gary W. Adamson, Jul 19 2011]

nmax:=7; for n from 0 to nmax do a(n,n):=2^n: a(n,0):=doublefactorial(2*n-1) od: for n from 2 to nmax do for m from 1 to n-1 do a(n,m) := 2*a(n-1,m-1)+(2*n-1)*a(n-1,m) od: od: seq(seq(a(n,k), k=0..n), n=0..nmax);
nmax:=7: M := Matrix(1..nmax+1,1..nmax+1): A029635 := proc(n,k): binomial(n,k) + binomial(n-1,k-1) end: for i from 1 to nmax do for j from 1 to i+1 do M[i,j] := A029635(i,j-1) od: od: for n from 0 to nmax do B := M^n: for m from 0 to n do a(n,m):= B[1,m+1] od: od: seq(seq(a(n,m), m=0..n), n=0..nmax);
A161198 := proc(n,k) option remember; if k > n or k < 0 then 0 elif n = 0 and k = 0 then 1 else 2*A161198(n-1, k-1) + (2*n-1)*A161198(n-1, k) fi end:
seq(print(seq(A161198(n,k), k = 0..n)), n = 0..6);  # Peter Luschny, May 09 2013

nmax = 7; a[n_, 0] := (2*n-1)!!; a[n_, n_] := 2^n; a[n_, m_] := a[n, m] = 2*a[n-1, m-1]+(2*n-1)*a[n-1, m]; Table[a[n, m], {n, 0, nmax}, {m, 0, n}] // Flatten (* Jean-François Alcover, Feb 25 2014, after Maple *)

for(n=0,9, print(Vec(Ser( 2^n*prod( k=1,n, x+(2*k-1)/2 ),,n+1))))  \\ M. F. Hasler, Jul 23 2011

@CachedFunction
def A161198(n,k):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return 2*A161198(n-1,k-1)+(2*n-1)*A161198(n-1,k)
for n in (0..6): [A161198(n,k) for k in (0..n)]  # Peter Luschny, May 09 2013

a(n,m) := coeff(2^(n)*product((x+(2*k-1)/2),k=1..n), x, m) for n = 0, 1, .. ; m = 0, 1, .. .
a(n, m) = 2*a(n-1,m-1)+(2*n-1)*a(n-1,m) with a(n, n) = 2^n and a(n, 0) = (2*n-1)!!.
a(n,m) = the (m+1)-th term in the top row of M^n, where M is an infinite square production matrix; M[i,j] = A029635(i,j-1) = binomial(i, j-1) + binomial(i-1, j-2) with A029635 the (1.2)-Pascal triangle, see the examples and second Maple program. [Gary W. Adamson, Jul 19 2011]
T(n,k) = 2^k * A028338(n,k). - Philippe Deléham, May 14 2015

A002596 Numerators in expansion of sqrt(1+x). Absolute values give numerators in expansion of sqrt(1-x).

Original entry on oeis.org

1, 1, -1, 1, -5, 7, -21, 33, -429, 715, -2431, 4199, -29393, 52003, -185725, 334305, -9694845, 17678835, -64822395, 119409675, -883631595, 1641030105, -6116566755, 11435320455, -171529806825, 322476036831, -1215486600363, 2295919134019

Offset: 0

N. J. A. Sloane

Also, absolute values are numerators of (2n-3)!!/n! or the odd part of the (n-1)-th Catalan number.
From Dimitri Papadopoulos, Oct 28 2016: (Start)
The sum of the coefficients of the expansion of sqrt(1+x) is sqrt(2) (easy). Observation: The sum of the squares of the coefficients is 4/Pi.
Observation/conjecture: If a term of this sequence is divisible by a prime p, then that term is in a block of exactly (p^k-3)/2 consecutive terms all of which are divisible by p. Furthermore, if a(n) is the term preceding such a block then a(p*n-(p-1)/2) also precedes a block of (p^(k+1)-3)/2 terms all divisible by p.
E.g., a(4)=-5 is divisible by 5 and is in a block of (5^1 - 3)/2 = 1 consecutive terms that are all divisible by 5. Then a(5*3 - (5-1)/2) = a(13) = 52003 precedes a block of exactly (5^2 - 3)/2 = 11 terms all divisible by 5.
(End)

			sqrt(1+x) = 1 + (1/2)*x - (1/8)*x^2 + (1/16)*x^3 - (5/128)*x^4 + (7/256)*x^5 - (21/1024)*x^6 + (33/2048)*x^7 + ...
Coefficients are 1, 1/2, -1/8, 1/16, -5/128, 7/256, -21/1024, 33/2048, -429/32768, 715/65536, -2431/262144, 4199/524288, -29393/4194304, 52003/8388608, ...

B. D. Hughes, Random Walks and Random Environments, Oxford 1995, vol. 1, p. 513, Eq. (7.281).
M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 88.
Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994): 72.
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).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 6, equation 6:14:6 at page 51.

T. D. Noe, Table of n, a(n) for n = 0..200
Tom Copeland, Addendum to Elliptic Lie Triad.
Eric Weisstein's World of Mathematics, Legendre Polynomial.

Denominators are A046161.
Cf. A001795.
Equals A000265(A000108(n-1)), n>0.
Absolute values are essentially A098597.
From Johannes W. Meijer, Jun 08 2009: (Start)
Cf. A161200 [(1-x)^(3/2)] and A161202 [(1-x)^(5/2)], A001803 [(1-x)^(-3/2)].
Cf. A161198 = triangle related to the series expansions of (1-x)^((-1-2*n)/2) for all values of n. (End)

[(-1)^n*Numerator((1/(1-2*n))*Binomial(2*n,n)/(4^n)): n in [0..30]]; // Vincenzo Librandi, Jan 14 2016

seq(numer(subs(k=1/2,expand(binomial(k,n)))),n=0..50); # James R. Buddenhagen, Aug 16 2014

1+InverseSeries[Series[2^p*y+y^2/2^q, {y, 0, 24}], x] (* p, q positive integers, then a(n)=numerator(y(n)). - Len Smiley, Apr 13 2000 *)
Numerator[CoefficientList[Series[Sqrt[1+x],{x,0,30}],x]] (* Harvey P. Dale, Oct 22 2011 *)
Table[Numerator[Product[(3 - 2 k)/(2 k) , {k, j}]], {j, 0, 30}] (* Dimitri Papadopoulos, Oct 22 2016 *)

x = 'x + O('x^40); apply(x->numerator(x), Vec(sqrt(1+x))) \\ Michel Marcus, Jan 14 2016

a(n+2) = C(n+1)/2^k(n+1), n >= 0; where C(n) = A000108(n), k(n) = A048881(n).
From Johannes W. Meijer, Jun 08 2009: (Start)
a(n) = (-1)^n*numerator((1/(1-2*n))*binomial(2*n,n)/(4^n)).
(1+x)^(1/2) = Sum_{n>=0} (1/(1-2*n))*binomial(2*n,n)/(4^n)*(-x)^n.
(1-x)^(1/2) = Sum_{n>=0} (1/(1-2*n))*binomial(2*n,n)/(4^n)*(x)^n. (End)
a(n) = numerator(Product_{k=1..n} (3-2*k)/(2*k)). - Dimitri Papadopoulos, Oct 22 2016

Minor correction to definition from Johannes W. Meijer, Jun 05 2009

A002454 Central factorial numbers: a(n) = 4^n * (n!)^2.

Original entry on oeis.org

1, 4, 64, 2304, 147456, 14745600, 2123366400, 416179814400, 106542032486400, 34519618525593600, 13807847410237440000, 6682998146554920960000, 3849406932415634472960000, 2602199086312968903720960000, 2040124083669367620517232640000, 1836111675302430858465509376000000

Offset: 0

N. J. A. Sloane

Denominators in the series for Bessel's J0(x) = 1 - x^2/4 + x^4/64 - x^6/2304 + ...
a(n) is the unreduced numerator in Product_{k=1..n} (4*k^2)/(4*k^2-1), therefore a(n)/A079484(n) = Pi/2 as n -> oo. - Daniel Suteu, Dec 02 2016
From Zhi-Wei Sun, Jun 26 2022: (Start)
Conjecture: Let zeta be a primitive 2n+1-th root of unity. Then the permanent of the 2n X 2n matrix [m(j,k)]_{j,k=1..2n} is a(n)/(2n+1) = ((2n)!!)^2/(2n+1), where m(j,k) is 1 or (1+zeta^(j-k))/(1-zeta^(j-k)) according as j = k or not.
The determinant of the matrix [m(j,k)]_{j,k=1..2n} was shown to be (-1)^(n-1)*((2n)!!)^2/(2n(2n+1)) by Han Wang and Zhi-Wei Sun in 2022. (End)

Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013 (20.1).
Bronstein-Semendjajew, Taschenbuch der Mathematik, 7th german ed. 1965, ch. 4.4.7
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 110.
E. L. Ince, Ordinary Differential Equations, Dover, NY, 1956; see p. 173.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
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).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 49 and 52, equations 49:6:1 and 52:6:2 at pages 483, 513.

T. D. Noe, Table of n, a(n) for n = 0..50
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 7.
Han Wang and Zhi-Wei Sun, Proof of a conjecture involving derangements and roots of unity, arXiv:2206.02589 [math.CO], 2022.
Index to divisibility sequences.
Index entries for sequences related to factorial numbers.

Cf. A000165, A001818, A079484, A197036, A334380.

J1: A002474, J2: A002506, J3: A014401.

[4^n*Factorial(n)^2: n in [0..15]]; // Vincenzo Librandi, Mar 15 2019

Array[4^# (#!)^2 &, 14, 0] (* Michael De Vlieger, Nov 01 2017 *)

a(n) = 4^n*(n!)^2; \\ Michel Marcus, Mar 13 2019

(-1)^n*a(n) is the coefficient of x^1 in Product_{k=0..2*n} (x+2*k-2*n). - Benoit Cloitre and Michael Somos, Nov 22 2002
E.g.f.: A(x) = arcsin(x)*sec(arcsin(x)). - Vladimir Kruchinin, Sep 12 2010
E.g.f.: arcsin(x)*sec(arcsin(x)) = arcsin(x)/sqrt(1-x^2) = x/G(0); G(k) = 2k*(x^2+1)+1-x^2*(2k+1)*(2k+2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2011
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (2*k+2)^2/(1-x/(x - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
From Ilya Gutkovskiy, Dec 02 2016: (Start)
a(n) ~ Pi*2^(2*n+1)*n^(2*n+1)/exp(2*n).
Sum_{n>=0} 1/a(n) = BesselI(0,1) = A197036. (End)
From Daniel Suteu, Dec 02 2016: (Start)
a(n) ~ 2^(2*n) * gamma(n+1/2) * gamma(n+3/2).
a(n) ~ Pi*(2*n+1)*(4*n^2-1)^n/exp(2*n). (End)
2*a(n)/(2*n+1)! = A101926(n) / A001803(n). - Daniel Suteu, Feb 03 2017
Limit_{n->oo} n*a(n)/((2n+1)!!)^2 = Pi/4. - Daniel Suteu, Nov 01 2017
Sum_{n>=0} (-1)^n/a(n) = BesselJ(0, 1) (A334380). - Amiram Eldar, Apr 09 2022
Limit_{n->oo} a(n) / (n * A001818(n)) = Pi. - Daniel Suteu, Apr 09 2022

A004731 a(0) = 1; thereafter a(n) = denominator of (n-2)!! / (n-1)!!.

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 15, 16, 35, 128, 315, 256, 693, 1024, 3003, 2048, 6435, 32768, 109395, 65536, 230945, 262144, 969969, 524288, 2028117, 4194304, 16900975, 8388608, 35102025, 33554432, 145422675, 67108864, 300540195, 2147483648, 9917826435, 4294967296, 20419054425

Offset: 0

N. J. A. Sloane

Also numerator of rational part of Haar measure on Grassmannian space G(n,1).
Also rational part of numerator of Gamma(n/2+1)/Gamma(n/2+1/2) (cf. A036039).
Let x(m) = x(m-2) + 1/x(m-1) for m >= 3, with x(1)=x(2)=1. Then the numerator of x(n+2) equals the denominator of n!!/(n+1)!! for n >= 0, where the double factorials are given by A006882. - Joseph E. Cooper III (easonrevant(AT)gmail.com), Nov 07 2010, as corrected in Cooper (2015).
Numerator of (n-1)/( (n-2)/( .../1)), with an empty fraction taken to be 1. - Flávio V. Fernandes, Jan 31 2025

			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.

T. D. Noe, Table of n, a(n) for n=0..302
Joseph E. Cooper III, A recurrence for an expression involving double factorials, arXiv:1510.00399 [math.CO], 2015.
Svante Janson, On the traveling fly problem, Graph Theory Notes of New York Vol. XXXI, 17, 1996.

Cf. A001803, A004730, A006882 (double factorials), A036069.

Cf. A036039, A046161, A001790, A001803, A101926.

import Data.Ratio ((%), numerator)
a004731 0 = 1
a004731 n = a004731_list !! n
a004731_list = map numerator ggs where
   ggs = 0 : 1 : zipWith (+) ggs (map (1 /) $ tail ggs) :: [Rational]
-- Reinhard Zumkeller, Dec 08 2011

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));
#
[1, seq(denom(doublefactorial(n-2)/doublefactorial(n-1)), n = 1..36)]; # Peter Luschny, Feb 09 2025

Table[ Denominator[ (n-2)!! / (n-1)!! ], {n, 0, 31}] (* Jean-François Alcover, Jul 16 2012 *)
Denominator[#[[1]]/#[[2]]&/@Partition[Range[-2,40]!!,2,1]] (* Harvey P. Dale, Nov 27 2014 *)
Join[{1},Table[Numerator[(n/2-1/2)!/((n/2-1)!Sqrt[Pi])], {n,1,31}]] (* Peter Luschny, Feb 08 2025 *)

f(n) = prod(i=0, (n-1)\2, n - 2*i); \\ A006882
a(n) = if (n==0, 1, denominator(f(n-2)/f(n-1))); \\ Michel Marcus, Feb 08 2025

from sympy import gcd, factorial2
def A004731(n):
    if n <= 1:
        return 1
    a, b = factorial2(n-2), factorial2(n-1)
    return b//gcd(a,b) # Chai Wah Wu, Apr 03 2021

Name corrected by Michel Marcus, Feb 08 2025

A033876 Expansion of 1/(2x) (1/(1-4*x)^(3/2)-1).

Original entry on oeis.org

3, 15, 70, 315, 1386, 6006, 25740, 109395, 461890, 1939938, 8112468, 33801950, 140408100, 581690700, 2404321560, 9917826435, 40838108850, 167890003050, 689232644100, 2825853840810, 11572544300460, 47342226683700, 193485622098600, 790066290235950, 3223470464162676

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

a(n) is the trace of the zigzag matrix Z(n+1) (see A088961). - Paul Boddington, Nov 03 2003
The number of edges in the odd graph O_k (for k >= 2) can be computed as 0.5*(2k-1)*C(2k-2,k-1). This sequence gives the number of edges in O_k for integer values of k from k=2. - K.V.Iyer, Mar 04 2009
Apparently the number of peaks in all symmetric Dyck paths with semilength 2n+2. - David Scambler, Apr 29 2013
For n > 0, also the number of maximal and maximum cliques in the (n+2)-odd graph. - Eric W. Weisstein, Nov 30 2017

			G.f. = 3 + 15*x + 70*x^2 + 315*x^3 + 1386*x^4 + 6006*x^5 + 25740*x^6 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Maximal Clique.
Eric Weisstein's World of Mathematics, Maximum Clique.
Eric Weisstein's World of Mathematics, Odd Graph.

Cf. A000984, A001622, A001700, A001803, A002457, A037965, A088961.

a033876 n = sum $ zipWith (!!) zss [0..n] where
   zss = take (n+1) $ g (take (n+1) (1 : [0,0..])) where
       g us = (take (n+1) $ g' us) : g (0 : init us)
       g' vs = last $ take (2 * n + 3) $
                      map snd $ iterate h (0, vs ++ reverse vs)
   h (p,ws) = (1 - p, drop p $ zipWith (+) ([0] ++ ws) (ws ++ [0]))
-- Reinhard Zumkeller, Oct 25 2013

[(2*n+3)*Binomial(2*n+1, n) : n in [0..40]]; // Wesley Ivan Hurt, Nov 30 2017

[seq((n+2)*binomial(2*(n+2),n+2)/4, n=0..22)]; # Zerinvary Lajos, Jan 04 2007

Table[nn = 2 n + 1; (2 n + 1)! Coefficient[Series[Exp[x] (x^n/n!)^2/2, {x, 0, nn}], x^(2 n + 1)], {n, 30}] (* Geoffrey Critzer, Apr 19 2017 *)
Table[n Binomial[2 n, n]/4, {n, 2, 20}] (* Eric W. Weisstein, Nov 30 2017 *)
Table[(4^n Gamma[n + 3/2])/(Sqrt[Pi] Gamma[n + 1]), {n, 20}] (* Eric W. Weisstein, Nov 30 2017 *)
CoefficientList[Series[((1 - 4 x)^(-3/2) - 1)/(2 x), {x, 0, 20}], x] (* Eric W. Weisstein, Nov 30 2017 *)

x='x+O('x^66); Vec( 1/(2*x) * (1/(1-4*x)^(3/2)-1) ) \\ Joerg Arndt, May 01 2013

a(n) = (2*n+3)*binomial(2*n+1, n). - Paul Boddington, Nov 03 2003
Equals n*A000984/4, n >= 2. - Zerinvary Lajos, Jan 04 2007
For n >= 1, 1/a(n-1) = Sum_{k>=0} binomial(2*k,k)/(4^(n+k)*(n+k+1)) = int(4*t^n/sqrt(1-4*t), t=0..1/4). - Groux Roland, Jan 17 2011
G.f.: - 1/(2*x) + G(0)/(4*x), where G(k)= 1 + 1/(1 - 2*x*(2*k+3)/(2*x*(2*k+3) + (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 18 2013
a(n) = 2^(2*n+1)*binomial(n+3/2, 1/2). - Peter Luschny, May 06 2014
0 = a(n)*(16*a(n+1) - 2*a(n+2)) + a(n+1)*(-6*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Sep 17 2014
a(n-2) = n*binomial(2*n, n)/4 for n > 1. - Eric W. Weisstein, Nov 30 2017
G.f.: ((1 - 4*x)^(-3/2) - 1)/2 (by definition). - Eric W. Weisstein, Nov 30 2017
D-finite with recurrence: (n+1)*a(n) +2*(-2*n-3)*a(n-1)=0. - R. J. Mathar, Jan 28 2020
G.f.: (1F0(3/2;;4*x)-1)/(2*x). - R. J. Mathar, Jan 28 2020
From Amiram Eldar, Mar 04 2023: (Start)
Sum_{n>=0} 1/a(n) = 4*Pi/(3*sqrt(3)) - 2.
Sum_{n>=0} (-1)^n/a(n) = 2 - 8*log(phi)/sqrt(5), where phi is the golden ratio (A001622). (End)
From Mélika Tebni, Sep 04 2024: (Start)
a(n) = A037965(n+2) - A001700(n).
E.g.f.: exp(2*x)*((3+8*x)*BesselI(0, 2*x) + (1+8*x)*BesselI(1, 2*x)). (End)
a(n) = 2^n*JacobiP(n+1, 1/2, -n-1, 3). - Peter Luschny, Jan 22 2025

A101926 a(n) = 2^A101925(n).

Original entry on oeis.org

2, 4, 16, 32, 256, 512, 2048, 4096, 65536, 131072, 524288, 1048576, 8388608, 16777216, 67108864, 134217728, 4294967296, 8589934592, 34359738368, 68719476736, 549755813888, 1099511627776, 4398046511104, 8796093022208

Offset: 0

Ralf Stephan, Dec 28 2004

nonn

a(n) is the numerator of 2^(2*n+1)*(n!)^2/(2*n+1)/(2*n)!. The corresponding denominator is A001803. - Daniel Suteu, Feb 03 2017
a(n) is the numerator of Integral_{x=-oo..oo} sech(x)^(2*n+2) dx. The corresponding denominator is A001803(n). - Mohammed Yaseen, Jul 25 2023
a(n) is the denominator of (1/Pi) * Integral_{x=0..Pi/2} sin(x)^(2*n) dx. The corresponding numerator is A001790(n). - Mohammed Yaseen, Sep 19 2023
a(n) = numerator(Pi*binomial(n, -1/2)). - Peter Luschny, Dec 05 2024

Index to divisibility sequences

Bisection of A036069 and of A086117.

Cf. A001803, A001790, A101925.

denom((binomial(2n,n)*4^-n)/2); # Stephen Crowley, Mar 05 2007

Table[Numerator[Beta[1, n + 1, 1/2]], {n, 0, 22}] (* Gerry Martens, Nov 13 2016 *)

More terms from Joshua Zucker, May 15 2006

A161199 Numerators in expansion of (1-x)^(-5/2).

Original entry on oeis.org

1, 5, 35, 105, 1155, 3003, 15015, 36465, 692835, 1616615, 7436429, 16900975, 152108775, 339319575, 1502700975, 3305942145, 115707975075, 251835004575, 1091285019825, 2354878200675, 20251952525805, 43397041126725, 185423721177825, 395033145117975

Offset: 0

Johannes W. Meijer, Jun 08 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A161198 (triangle for (1-x)^((-1-2*n)/2) for all values of n).
Cf. A046161 (denominators for (1-x)^(-5/2)).
Numerators of [x^n]( (1-x)^(p/2) ): A161202 (p=5), A161200 (p=3), A002596 (p=1), A001790 (p=-1), A001803 (p=-3), this sequence (p=-5), A161201 (p=-7).

A161199:= func< n | Numerator( Binomial(n+3,3)*Catalan(n+2)/2^(2*n+1) ) >;
[A161199(n): n in [0..30]]; // G. C. Greubel, Sep 24 2024

Numerator[CoefficientList[Series[(1-x)^(-5/2),{x,0,30}],x]] (* or *) Numerator[Table[(4n^2+8n+3)/3 Binomial[2n,n]/4^n,{n,0,30}]] (* Harvey P. Dale, Oct 15 2011 *)

def A161199(n): return numerator((-1)^n*binomial(-5/2,n))
[A161199(n) for n in range(31)] # G. C. Greubel, Sep 24 2024

a(n) = numerator(((3 + 8*n + 4*n^2)/3)*binomial(2*n,n)/(4^n)).

a(n) = denominator((3/2)*Integral_{x=0..1} x^n*sqrt(1-x) dx), where the integral is sqrt(Pi)*n!/Gamma(n+5/2) = n!/( (n+3/2)*(n+1/2)*(n-1/2)*...*(1/2)). - Groux Roland, Feb 23 2011

A161202 Numerators in expansion of (1-x)^(5/2).

Original entry on oeis.org

1, -5, 15, -5, -5, -3, -5, -5, -45, -55, -143, -195, -1105, -1615, -4845, -7429, -185725, -294975, -950475, -1550775, -10235115, -17058525, -57378675, -97294275, -1329688425, -2287064091, -7916760315, -13781027215

Offset: 0

Johannes W. Meijer, Jun 08 2009

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

Cf. A046161 (denominators).
Cf. A161198 (triangle of coefficients of (1-x)^((-1-2*n)/2)).
Numerators of [x^n]( (1-x)^(p/2) ): this sequence (p=5), A161200 (p=3), A002596 (p=1), A001790 (p=-1), A001803 (p=-3), A161199 (p=-5), A161201 (p=-7).

A161202:= func< n | -Numerator(15*(n+1)*Catalan(n)/(4^n*(2*n-1)*(2*n-3)*(2*n-5))) >;
[A161202(n): n in [0..30]]; // G. C. Greubel, Sep 24 2024

Numerator[CoefficientList[Series[(1-x)^(5/2),{x,0,30}],x]] (* Harvey P. Dale, Aug 22 2011 *)
Table[(-1)^n*Numerator[Binomial[5/2, n]], {n,0,30}] (* G. C. Greubel, Sep 24 2024 *)

def A161202(n): return (-1)^n*numerator(binomial(5/2,n))
[A161202(n) for n in range(31)] # G. C. Greubel, Sep 24 2024

a(n) = numerator( (15/(15-46*n+36*n^2-8*n^3))*binomial(2*n,n)/(4^n) ).

a(n) = (-1)^n*numerator( binomial(5/2, n) ). - G. C. Greubel, Sep 24 2024

A162443 Numerators of the BG1[ -5,n] coefficients of the BG1 matrix.

Original entry on oeis.org

5, 66, 680, 2576, 33408, 14080, 545792, 481280, 29523968, 73465856, 27525120, 856162304, 1153433600, 18798870528, 86603988992, 2080374784, 2385854332928, 3216930504704, 71829033058304, 7593502179328, 281749854617600

Offset: 1

Johannes W. Meijer, Jul 06 2009

The BG1 matrix coefficients are defined by BG1[2m-1,1] = 2*beta(2m) and the recurrence relation BG1[2m-1,n] = BG1[2m-1,n-1] - BG1[2m-3,n-1]/(2*n-3)^2 with m = .. , -2, -1, 0, 1, 2, .. and n = 1, 2, 3, .. . As usual beta(m) = sum((-1)^k/(1+2*k)^m, k=0..infinity). For the BG2 matrix, the even counterpart of the BG1 matrix, see A008956.
We discovered that the n-th term of the row coefficients can be generated with BG1[1-2*m,n] = RBS1(1-2*m,n)* 4^(n-1)*((n-1)!)^2/ (2*n-2)! for m >= 1. For the BS1(1-2*m,n) polynomials see A160485.
The coefficients in the columns of the BG1 matrix, for m >= 1 and n >= 2, can be generated with GFB(z;n) = ((-1)^(n+1)*CFN2(z;n)*GFB(z;n=1) + BETA(z;n))/((2*n-3)!!)^2 for n >= 2. For the CFN2(z;n) and the Beta polynomials see A160480.
The BG1[ -5,n] sequence can be generated with the first Maple program and the BG1[2*m-1,n] matrix coefficients can be generated with the second Maple program.
The BG1 matrix is related to the BS1 matrix, see A160480 and the formulas below.

			The first few formulas for the BG1[1-2*m,n] matrix coefficients are:
BG1[ -1,n] = (1)*4^(n-1)*(n-1)!^2/(2*n-2)!
BG1[ -3,n] = (1-2*n)*4^(n-1)*(n-1)!^2/(2*n-2)!
BG1[ -5,n] = (1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!
The first few generating functions GFB(z;n) are:
GFB(z;2) = ((-1)*(z^2-1)*GFB(z;1) + (-1))/1
GFB(z;3) = ((+1)*(z^4-10*z^2+9)*GFB(z;1) + (-11 + z^2))/9
GFB(z;4) = ((-1)*( z^6- 35*z^4+259*z^2-225)*GFB(z;1) + (-299 + 36*z^2 - z^4))/225

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.
J. M. Amigo, Relations among Sums of Reciprocal Powers Part II, International Journal of Mathematics and Mathematical Sciences , Volume 2008 (2008), pp. 1-20.

A162444 are the denominators of the BG1[ -5, n] matrix coefficients.
The BG1[ -3, n] equal (-1)*A002595(n-1)/A055786(n-1) for n >= 1.
The BG1[ -1, n] equal A046161(n-1)/A001790(n-1) for n >= 1.
The cs(n) equal A046161(n-2)/A001803(n-2) for n >= 2.
The BETA(z, n) polynomials and the BS1 matrix lead to the Beta triangle A160480.
The CFN2(z, n), the t2(n, m) and the BG2 matrix lead to A008956.
Cf. A000364, A001818 and A160485.
Cf. A162443 (BG1 matrix), A162446 (ZG1 matrix) and A162448 (LG1 matrix).

a := proc(n): numer((1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!) end proc: seq(a(n), n=1..21);
# End program 1
nmax1 := 5; coln := 3; Digits := 20: mmax1 := nmax1: for n from 0 to nmax1 do t2(n, 0) := 1 od: for n from 0 to nmax1 do t2(n, n) := doublefactorial(2*n-1)^2 od: for n from 1 to nmax1 do for m from 1 to n-1 do t2(n, m) := (2*n-1)^2* t2(n-1, m-1) + t2(n-1, m) od: od: for m from 1 to mmax1 do BG1[1-2*m, 1] := euler(2*m-2) od: for m from 1 to mmax1 do BG1[2*m-1, 1] := Re(evalf(2*sum((-1)^k1/(1+2*k1)^(2*m), k1=0..infinity))) od: for m from -mmax1 +coln to mmax1 do BG1[2*m-1, coln] := (-1)^(coln+1)*sum((-1)^k1*t2(coln-1, k1)*BG1[2*m-(2*coln-1)+2*k1, 1], k1=0..coln-1)/doublefactorial(2*coln-3)^2 od;
# End program 2
# Maple programs edited by Johannes W. Meijer, Sep 25 2012

a(n) = numer(BG1[ -5,n]) and A162444(n) = denom(BG1[ -5,n]) with BG1[ -5,n] = (1-8*n+12*n^2)*4^(n-1)*(n-1)!^2/(2*n-2)!.
The generating functions GFB(z;n) of the coefficients in the matrix columns are defined by
GFB(z;n) = sum(BG1[2*m-1,n]*z^(2*m-2), m=1..infinity).
GFB(z;n) = (1-z^2/(2*n-3)^2)*GFB(n-1) - 4^(n-2)*(n-2)!^2/((2*n-4)!*(2*n-3)^2) for n => 2 with GFB(z;n=1) = 1/(z*cos(Pi*z/2))*int(sin(z*t)/sin(t),t=0..Pi/2).
The column sums cs(n) = sum(BG1[2*m-1,n]*z^(2*m-2), m=1..infinity) = 4^(n-1)/((2*n-2)*binomial(2*n-2,n-1)) for n >= 2.
BG1[2*m-1,n] = (n-1)!^2*4^(n-1)*BS1[2*m-1,n]/(2*n-2)!

A161201 Numerators in expansion of (1-x)^(-7/2).

Original entry on oeis.org

1, 7, 63, 231, 3003, 9009, 51051, 138567, 2909907, 7436429, 37182145, 91265265, 882230895, 2103781365, 9917826435, 23141595015, 856239015555, 1964313035685, 8948537162565, 20251952525805, 182267572732245

Offset: 0

Johannes W. Meijer, Jun 08 2009

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

Cf. A046161 (denominators).
Cf. A161198 (triangle of coefficients of (1-x)^((-1-2*n)/2)).
Numerators of [x^n]( (1-x)^(p/2) ): A161202 (p=5), A161200 (p=3), A002596 (p=1), A001790 (p=-1), A001803 (p=-3), A161199 (p=-5), this sequence (p=-7).

A161201:= func< n | Numerator((n+1)*(2*n+1)*(2*n+3)*(2*n+5)*Catalan(n)/(15*4^n)) >;
[A161201(n): n in [0..30]]; // G. C. Greubel, Sep 24 2024

CoefficientList[Series[(1-x)^(-7/2),{x,0,20}],x]//Numerator (* Harvey P. Dale, Jan 14 2020 *)
Table[(-1)^n*Numerator[Binomial[-7/2, n]], {n, 0, 30}] (* G. C. Greubel, Sep 24 2024 *)

def A161201(n): return (-1)^n*numerator(binomial(-7/2,n))
[A161201(n) for n in range(31)] # G. C. Greubel, Sep 24 2024

a(n) = numerator(((15+46*n+36*n^2+8*n^3)/15)*binomial(2*n,n)/(4^n)).

a(n) = (-1)^n*numerator( binomial(-7/2, n) ). - G. C. Greubel, Sep 24 2024

cp's OEIS Frontend

A161198 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).

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A002596 Numerators in expansion of sqrt(1+x). Absolute values give numerators in expansion of sqrt(1-x).

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A002454 Central factorial numbers: a(n) = 4^n * (n!)^2.

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A004731 a(0) = 1; thereafter a(n) = denominator of (n-2)!! / (n-1)!!.

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A033876 Expansion of 1/(2*x) * (1/(1-4*x)^(3/2)-1).

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A101926 a(n) = 2^A101925(n).

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A033876 Expansion of 1/(2x) (1/(1-4*x)^(3/2)-1).