A220002 -id:A220002 - cp's OEIS frontend

A220466 a((2n-1)2^p) = 4^p(n-1) + 2^(p-1)(1+2^p), p >= 0 and n >= 1.

1, 3, 2, 10, 3, 7, 4, 36, 5, 11, 6, 26, 7, 15, 8, 136, 9, 19, 10, 42, 11, 23, 12, 100, 13, 27, 14, 58, 15, 31, 16, 528, 17, 35, 18, 74, 19, 39, 20, 164, 21, 43, 22, 90, 23, 47, 24, 392, 25, 51, 26, 106, 27, 55, 28, 228, 29, 59, 30, 122, 31, 63, 32, 2080, 33, 67, 34, 138, 35

Offset: 1

Johannes W. Meijer, Dec 24 2012

The a(n) appeared in the analysis of A220002, a sequence related to the Catalan numbers.
The first Maple program makes use of a program by Peter Luschny for the calculation of the a(n) values. The second Maple program shows that this sequence has a beautiful internal structure, see the first formula, while the third Maple program makes optimal use of this internal structure for the fast calculation of a(n) values for large n.
The cross references lead to sequences that have the same internal structure as this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ralf Stephan, Some divide-and-conquer sequences with simple ordinary generating functions, The OEIS, Jan 01 2004.

Cf. A000027 (the natural numbers), A000120 (1's-counting sequence), A000265 (remove 2's from n), A001316 (Gould's sequence), A001511 (the ruler function), A003484 (Hurwitz-Radon numbers), A003602 (a fractal sequence), A006519 (highest power of 2 dividing n), A007814 (binary carry sequence), A010060 (Thue-Morse sequence), A014577 (dragon curve), A014707 (dragon curve), A025480 (nim-values), A026741, A035263 (first Feigenbaum symbolic sequence), A037227, A038712, A048460, A048896, A051176, A053381 (smooth nowhere-zero vector fields), A055975 (Gray code related), A059134, A060789, A060819, A065916, A082392, A085296, A086799, A088837, A089265, A090739, A091512, A091519, A096268, A100892, A103391, A105321 (a fractal sequence), A109168 (a continued fraction), A117973, A129760, A151930, A153733, A160467, A162728, A181988, A182241, A191488 (a companion to Gould's sequence), A193365, A220466 (this sequence).

-- Following Ralf Stephan's recurrence:
import Data.List (transpose)
a220466 n = a006519_list !! (n-1)
a220466_list = 1 : concat
   (transpose [zipWith (-) (map (* 4) a220466_list) a006519_list, [2..]])
-- Reinhard Zumkeller, Aug 31 2014

# First Maple program
a := n -> 2^padic[ordp](n, 2)*(n+1)/2 : seq(a(n), n=1..69); # Peter Luschny, Dec 24 2012
# Second Maple program
nmax:=69: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := 4^p*(n-1)  + 2^(p-1)*(1+2^p) od: od: seq(a(n), n=1..nmax);
# Third Maple program
nmax:=69: for p from 0 to ceil(simplify(log[2](nmax))) do n:=2^p: n1:=1: while n <= nmax do a(n) := 4^p*(n1-1)+2^(p-1)*(1+2^p): n:=n+2^(p+1): n1:= n1+1: od: od:  seq(a(n), n=1..nmax);

A220466 = Module[{n, p}, p = IntegerExponent[#, 2]; n = (#/2^p + 1)/2; 4^p*(n - 1) + 2^(p - 1)*(1 + 2^p)] &; Array[A220466, 50] (* JungHwan Min, Aug 22 2016 *)

a(n)=if(n%2,n\2+1,4*a(n/2)-2^valuation(n/2,2)) \\ Ralf Stephan, Dec 17 2013

a((2*n-1)*2^p) = 4^p*(n-1) + 2^(p-1)*(1+2^p), p >= 0 and n >= 1. Observe that a(2^p) = A007582(p).
a(n) = ((n+1)/2)*(A060818(n)/A060818(n-1))
a(n) = (-1/64)*(q(n+1)/q(n))/(2*n+1) with q(n) = (-1)^(n+1)*2^(4*n-5)*(2*n)!*A060818(n-1) or q(n) = (1/8)*A220002(n-1)*1/(A098597(2*n-1)/A046161(2*n))*1/(A008991(n-1)/A008992(n-1))
Recurrence: a(2n) = 4a(n) - 2^A007814(n), a(2n+1) = n+1. - Ralf Stephan, Dec 17 2013

A220412 Triangle read by rows, the coefficients of J. L. Fields generalized Bernoulli polynomials.

Original entry on oeis.org

1, 0, 1, 0, 1, 5, 0, 4, 21, 35, 0, 18, 101, 210, 175, 0, 48, 286, 671, 770, 385, 0, 33168, 207974, 531531, 715715, 525525, 175175, 0, 8640, 56568, 154466, 231231, 205205, 105105, 25025, 0, 1562544, 10615548, 30582796, 49534277, 49689640, 31481450, 11911900

Offset: 0

Peter Luschny, Dec 30 2012

The Fields polynomials are defined: F_{2*n}(x) = sum(k=0..n, x^k*T(n,k)/A220411(n)) and F_{2*n+1}(x) = 0. See A220002 for an application to the asymptotic expansion of the Catalan numbers.

			The coefficients T(n,k):
[0], [1]
[1], [0,  1]
[2], [0,  1,   5]
[3], [0,  4,  21,  35]
[4], [0, 18, 101, 210, 175]
[5], [0, 48, 286, 671, 770, 385]
The Fields polynomials:
F_0 (x) =  1 / 1
F_2 (x) =  x / (-6)
F_4 (x) = (5*x^2+x) / 60
F_6 (x) = (35*x^3+21*x^2+4*x) / (-504)
F_8 (x) = (175*x^4+210*x^3+101*x^2+18*x) / 2160
F_10(x) = (385*x^5+770*x^4+671*x^3+286*x^2+48*x) / (-3168)

Y. L. Luke, The Special Functions and their Approximations, Vol. 1. Academic Press, 1969, page 34.

J. L. Fields, A note on the asymptotic expansion of a ratio of gamma functions, Proc. Edinburgh Math. Soc. 15 (1) (1966), 43-45.

Cf. A220411.

FieldsPoly := proc(n,x) local recP, P; recP := proc(n,x) option remember; local k; if n = 0 then return(1) fi; -2*x*add(binomial(n-1,2*k+1)* bernoulli(2*k+2)/(2*k+2)*recP(n-2*k-2,x), k=0..(n/2-1)) end:
P := recP(n,x); (-1)^iquo(n,2)*denom(P); sort(expand(P*%)) end:
with(PolynomialTools): A220412_row := n -> CoefficientList(FieldsPoly( 2*i,x),x): seq(A220412_row(i),i=0..8);

F[0, ] = 1; F[n, x_] := F[n, x] = -2*x*Sum[Binomial[n-1, 2*k+1]*BernoulliB[2*k+2]/(2*k+2)*F[n-2*k-2, x], {k, 0, n/2-1}]; t[n_, k_] := Coefficient[(-1)^Mod[n, 2]*F[2*n, x] // Together // Numerator, x, k]; Table[t[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 09 2014 *)

@CachedFunction
def FieldsPoly(n):
    A = PolynomialRing(QQ, 'x')
    x = A.gen()
    if n == 0: return A(1)
    return -2*x*add(binomial(n-1,2*k+1)*bernoulli(2*k+2)/(2*k+2)*FieldsPoly(n-2*k-2) for k in (0..n-1))
def FieldsCoeffs(n):
    P = FieldsPoly(n)
    d = (-1)^(n//2) * denominator(P)
    return list(d * P)
A220412_row = lambda n: FieldsCoeffs(2*n)

See Y. L. Luke 2.8(3) for the generalized Bernoulli polynomials and 2.11(16) for the special case of Fields polynomials. The Maple and Sage programs give a recursive definition.

A008991 Numerators of coefficients in expansion of sqrt(sin(x)/x) (even powers only).

Original entry on oeis.org

1, -1, 1, -1, -67, -1, -64397, -113249, -3679787, -810304169, -6040635661561, -428305999661, -16827172241810597, -5620292762592913, -1550760014054450957, -4168373361283100017, -8551022502876237590534947

Offset: 0

N. J. A. Sloane

sign

G. C. Greubel, Table of n, a(n) for n = 0..255
Eric Weisstein's World of Mathematics, The Sinc Function.

Denominators are in A008992.

Appears in A220002 and A220466.

A008991 := n -> numer(coeff(taylor(sqrt(sin(x)/x), x, 2*n+2), x, 2*n)):  seq(A008991(n), n=0..16); # Johannes W. Meijer, Feb 10 2013

Numerator[CoefficientList[Series[Sqrt[Sin[x]/x], {x, 0, 50}], x][[1 ;; -1 ;; 2]]] (* G. C. Greubel, Jul 21 2018 *)

length = 16; T = taylor(sqrt(sin(x)/x),x,0,2*length+2)
[T.coefficient(x, 2*n).numerator() for n in (0..length)]
# Peter Luschny, Dec 13 2012

sum(n>=0, A008991(n)/A008992(n) ) = A117017 - Johannes W. Meijer, Feb 10 2013

A008992 Denominators of coefficients in expansion of sqrt(sin(x)/x) (even powers only).

Original entry on oeis.org

1, 12, 1440, 24192, 29030400, 5677056, 4649508864000, 100429391462400, 39023992111104000, 100609855353520128000, 8632325589332026982400000, 6946127879462505676800000, 3060839851788186207387648000000

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..210

Numerators are in A008991.

Appears in A220002 and A220466.

A008992 := n -> denom(coeff(taylor(sqrt(sin(x)/x), x, 2*n+2), x, 2*n)): seq(A008992(n),  n=0..12); # Johannes W. Meijer, Feb 10 2013

Denominator[CoefficientList[Series[Sqrt[Sin[x]/x], {x, 0, 50}], x][[1 ;; -1 ;; 2]]] (* G. C. Greubel, Jul 21 2018 *)

length = 12; T = taylor(sqrt(sin(x)/x),x,0,2*length+2)
[T.coefficient(x,2*n).denominator() for n in (0..length)]
# Peter Luschny, Dec 13 2012

A193365 a(n) = A220371(n)/(4*A220371(n-1)).

Original entry on oeis.org

15, 126, 143, 1020, 399, 1150, 783, 8184, 1295, 3198, 1935, 9212, 2703, 6270, 3599, 65520, 4623, 10366, 5775, 25596, 7055, 15486, 8463, 73720, 9999, 21630, 11663, 50172, 13455, 28798, 15375, 524256, 17423, 36990

Offset: 1

Johannes W. Meijer, Dec 21 2012

This sequence is, via A220371, related to A220002, which is related to the Catalan numbers.

Information about the peculiar structure of the a(n) can be found in A220466.

Cf. A220466

nmax:= 34: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a(2^p*(2*n-1)) := 2^p*(2^(2*p+4)*(2*n-1)^2-1) od: od: seq(a(n), n=1..nmax);

b[n_] := b[n] = 2^(2n) Product[2i+1, {i, 1, 2n}] GCD[n!, 2^n];
a[n_] := b[n]/(4 b[n-1]);
Array[a, 34] (* Jean-François Alcover, Jun 26 2019 *)

def A193365_list(len):
    a = {}; z = 1; s = 0; p = 1
    while s < len:
        i = s; z += z
        while i < len:
            a[i] = p*((4*i+4)^2-1)
            i += z
        s += s + 1; p += p
    return [a[i] for i in range(len)]
A193365_list(30)  # Peter Luschny, Dec 22 2012

a(n) = A220371(n)/(4*A220371(n-1))

a(2^p*(2*n-1)) = 2^p*(2^(2*p+4)*(2*n-1)^2-1), p >= 0.

A220422 Numerators of coefficients of an expansion of the logarithm of the Catalan numbers.

Original entry on oeis.org

5, -1, 65, -1381, 50525, -2702761, 199360985, -19391512141, 2404879675445, -370371188237521, 69348874393137905, -15514534163557086901, 4087072509293123892365, -1252259641403629865468281, 441543893249023104553682825, -177519391579539289436664789661

Offset: 1

Peter Luschny, Dec 28 2012

sign

Let C(n) denote the Catalan numbers A000108 and S(n) = Sum_{k>=1} a(k)/(2*k*(4*n+3)^(2*k)) then log(C(n)) = (1/2)*(n*log(16)-3*log(n+3/4)-log(Pi)+S(n)).

			Let N = 4*n+3 then log(C(n)) = (n*log(16)-3*log(n+3/4)-log(Pi))/2+a(1)/(4*N^2)+a(2)/(8*N^4)+a(3)/(12*N^6)+a(4)/(16*N^8)+O(1/N^10).

Y. L. Luke, The Special Functions and their Approximations, Vol. 1. Academic Press, 1969.

J. L. Fields, A note on the asymptotic expansion of a ratio of gamma functions, Proc. Edinburgh Math. Soc. 15 (1) (1966), 43-45.
D. Kessler and J. Schiff, The asymptotics of factorials, binomial coefficients and Catalan numbers. April 2006.

Cf. A000108, A122045.

The exponential version is A220002.

Maple
```
A220422 := n -> 4 - euler(2*n):
```

def A220422Generator() :
    A = {-1:0, 0:1};
    k = 0; e = 1; i = 0; m = 0
    while True:
        An = 0; A[k + e] = 0; e = -e
        for j in (0..i) :
            An += A[k]; A[k] = An; k += e
        if e < 0 :
            yield 4 - A[-m]*(-1)^m
            m += 1
        i += 1
A220422 = A220422Generator()
[next(A220422) for n in (1..16)]

a(n) = -4^(2*n+1)*B_{2*n+1}(-1/4)/(2*n+1), B_{n}(x) the Bernoulli polynomials.

a(n) = 4 - E(2*n), E the Euler numbers A122045.

A239739 a(n) = n4^(2n+1).

Original entry on oeis.org

0, 64, 2048, 49152, 1048576, 20971520, 402653184, 7516192768, 137438953472, 2473901162496, 43980465111040, 774056185954304, 13510798882111488, 234187180623265792, 4035225266123964416, 69175290276410818560, 1180591620717411303424, 20070057552195992158208

Offset: 0

Peter Luschny, Mar 26 2014

Appears in asymptotic expansions of the logarithm of the central binomial and the Catalan numbers. (See Kessler and Schiff, page 2.)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Kessler and J. Schiff, The asymptotics of factorials, binomial coefficients and Catalan numbers. April 2006.
Index entries for linear recurrences with constant coefficients, signature (32,-256).

Cf. A219931, A220002, A220422, A267796, A013709.

[n*4^(2*n+1): n in [0..25]]; // Vincenzo Librandi, Apr 25 2014

CoefficientList[Series[64 x /(1 - 16 x)^2, {x, 0, 20}], x] (* Vincenzo Librandi, Apr 25 2014 *)
LinearRecurrence[{32,-256},{0,64},20] (* Harvey P. Dale, May 06 2021 *)

G.f.: 64*x / (1 - 16*x)^2. [Bruno Berselli, Mar 26 2014]
(n-1)*a(n) - 16*n*a(n-1) = 0. [Bruno Berselli, Mar 26 2014]
a(n) = n*A013709(n). - Michel Marcus, Jan 30 2016

cp's OEIS Frontend

A220466 a((2*n-1)*2^p) = 4^p*(n-1) + 2^(p-1)*(1+2^p), p >= 0 and n >= 1.

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A220412 Triangle read by rows, the coefficients of J. L. Fields generalized Bernoulli polynomials.

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Sage

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A008991 Numerators of coefficients in expansion of sqrt(sin(x)/x) (even powers only).

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Maple

Mathematica

Sage

Formula

A008992 Denominators of coefficients in expansion of sqrt(sin(x)/x) (even powers only).

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Maple

Mathematica

Sage

A193365 a(n) = A220371(n)/(4*A220371(n-1)).

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Sage

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A220422 Numerators of coefficients of an expansion of the logarithm of the Catalan numbers.

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Maple

Sage

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A239739 a(n) = n*4^(2*n+1).

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A220466 a((2n-1)2^p) = 4^p(n-1) + 2^(p-1)(1+2^p), p >= 0 and n >= 1.

A239739 a(n) = n4^(2n+1).