A220412 -id:A220412 - cp's OEIS frontend

A220002 Numerators of the coefficients of an asymptotic expansion in even powers of the Catalan numbers.

1, 5, 21, 715, -162877, 19840275, -7176079695, 1829885835675, -5009184735027165, 2216222559226679575, -2463196751104762933637, 1679951011110471133453965, -5519118103058048675551057049, 5373485053345792589762994345215, -12239617587594386225052760043303511

Offset: 0

Peter Luschny, Dec 27 2012

sign

Let N = 4*n+3 and A = sum_{k>=0} a(k)/(A123854(k)*N^(2*k)) then
C(n) ~ 8*4^n*A/(N*sqrt(N*Pi)), C(n) = (4^n/sqrt(Pi))*(Gamma(n+1/2)/ Gamma(n+2)) the Catalan numbers A000108.
The asymptotic expansion of the Catalan numbers considered here is based on the Taylor expansion of square root of the sine cardinal. This asymptotic series involves only even powers of N, making it more efficient than the asymptotic series based on Stirling's approximation to the central binomial which involves all powers (see for example: D. E. Knuth, 7.2.1.6 formula (16)). The series is discussed by Kessler and Schiff but is included as a special case in the asymptotic expansion given by J. L. Fields for quotients Gamma(x+a)/Gamma(x+b) and discussed by Y. L. Luke (p. 34-35), apparently overlooked by Kessler and Schiff.

			With N = 4*n+3 the first few terms of A are A = 1 + 5/(4*N^2) + 21/(32*N^4) + 715/(128*N^6) - 162877/(2048*N^8) + 19840275/(8192*N^10). With this A C(n) = round(8*4^n*A/(N*sqrt(N*Pi))) for n = 0..39 (if computed with sufficient numerical precision).

Donald E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 4: Generating All Trees—History of Combinatorial Generation, 2006.
Y. L. Luke, The Special Functions and their Approximations, Vol. 1. Academic Press, 1969.

Peter Luschny, Table of n, a(n) for n = 0..99
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.

The logarithmic version is A220422. Appears in A193365 and A220466.

Cf. A220412.

A220002 := proc(n) local s; s := n -> `if`(n > 0, s(iquo(n,2))+n, 0);
(-1)^n*mul(4*i+2, i = 1..2*n)*2^s(iquo(n,2))*coeff(taylor(sqrt(sin(x)/x), x,2*n+2), x, 2*n) end: seq(A220002(n), n = 0..14);
# Second program illustrating J. L. Fields expansion of gamma quotients.
A220002 := proc(n) local recF, binSum, swing;
binSum := n -> add(i,i=convert(n,base,2));
swing := n -> n!/iquo(n, 2)!^2;
recF := proc(n, x) option remember; `if`(n=0, 1, -2*x*add(binomial(n-1,2*k+1)*bernoulli(2*k+2)/(2*k+2)*recF(n-2*k-2,x),k=0..n/2-1)) end: recF(2*n,-1/4)*2^(3*n-binSum(n))*swing(4*n+1) end:

max = 14; CoefficientList[ Series[ Sqrt[ Sinc[x]], {x, 0, 2*max+1}], x^2][[1 ;; max+1]]*Table[ (-1)^n*Product[ (2*k+1), {k, 1, 2*n}], {n, 0, max}] // Numerator (* Jean-François Alcover, Jun 26 2013 *)

length = 15; T = taylor(sqrt(sin(x)/x),x,0,2*length+2)
def A005187(n): return A005187(n//2) + n if n > 0 else 0
def A220002(n):
    P = mul(4*i+2 for i in (1..2*n)) << A005187(n//2)
    return (-1)^n*P*T.coefficient(x, 2*n)
[A220002(n) for n in range(length)]

# Second program illustrating the connection with the Euler numbers.
def A220002_list(n):
    S = lambda n: sum((4-euler_number(2*k))/(4*k*x^(2*k)) for k in (1..n))
    T = taylor(exp(S(2*n+1)),x,infinity,2*n-1).coefficients()
    return [t[0].numerator() for t in T][::-1]
A220002_list(15)

Let [x^n]T(f(x)) denote the coefficient of x^n in the Taylor expansion of f(x) then r(n) = (-1)^n*prod_{i=1..2n}(2i+1)*[x^(2*n)]T(sqrt(sin(x)/x)) is the rational coefficient of the asymptotic expansion (in N=4*n+3) and a(n) = numerator(r(n)) = r(n)*2^(3*n-bs(n)), where bs(n) is the binary sum of n (A000120).
Also a(n) = numerator([x^(2*n)]T(exp(S))) where S = sum_{k>=1}((4-E(2*k))/ (4*k)*x^(2*k)) and E(n) the Euler numbers A122045.
Also a(n) = sf(4*n+1)*2^(3*n-bs(n))*F_{2*n}(-1/4) where sf(n) is the swinging factorial A056040, bs(n) the binary sum of n and F_{n}(x) J. L. Fields' generalized Bernoulli polynomials A220412.
In terms of sequences this means
r(n) = (-1)^n*A103639(n)*A008991(n)/A008992(n),
a(n) = (-1)^n*A220371(n)*A008991(n)/A008992(n).
Note that a(n) = r(n)*A123854(n) and A123854(n) = 2^A004134(n) = 8^n/2^A000120(n).
Formula from Johannes W. Meijer:
a(n) = d(n+1)*A098597(2*n+1)*(A008991(n)/A008992(n)) with d(1) = 1 and
d(n+1) = -4*(2*n+1)*A161151(n)*d(n),
d(n+1) = (-1)^n*2^(-1)*(2*(n+1))!*A060818(n)*A048896(n).

A220411 The denominators of J. L. Fields generalized Bernoulli polynomials.

Original entry on oeis.org

1, -6, 60, -504, 2160, -3168, 786240, -51840, 1762560, -82736640, 1437004800, -858470400, 101896704000, -1881169920, 8659353600, -855305256960, 14071151001600, -150493593600, 8252165711462400, -70431001804800, 24434139856896000, -1076294062964736000

Offset: 0

Peter Luschny, Dec 30 2012

sign

See A220412 for definitions and references.

F[0, ] = 1; F[n, x_] := F[n, x] = -2x Sum[Binomial[n-1, 2k+1] BernoulliB[2k+2]/(2k+2) F[n-2k-2, x], {k, 0, n/2-1}] // Expand;
a[n_] := (-1)^n Denominator[Together[F[2n, x]]];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jun 27 2019 *)

A239792 Numerator of b_{2n}(1/4), where b_{n}(x) are Nörlund's generalized Bernoulli polynomials.

Original entry on oeis.org

1, -1, 3, -61, 1261, -4977, 999645, -16820653, 288427601, -1975649524361, 250373334235999, -741069328361243, 2017175162278526957, -16484758150014378103, 1866091048556360006871, -747145289541069391049541, 558035966935526487401599645, -94004035636878314426017611

Offset: 0

Peter Luschny, Mar 26 2014

Y. L. Luke, The Special Functions and their Approximations, Vol. 1. Academic Press, 1969, page 34.
N. E. Nörlund, Vorlesungen über Differenzenrechnung, Berlin, 1924.

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. A220412, A239793 (denominators).

b := proc(n) option remember; if n < 1 then 1 else
-add(binomial(n-1, k-1)*bernoulli(k)*b(n-k)/k, k= 2..n)/2 fi end:
A239792 := n -> numer(b(2*n));
seq(A239792(n), n=0..17);

b[n_] := b[n] = If[n < 1, 1, -Sum[Binomial[n - 1, k - 1] BernoulliB[k] b[n - k]/k, {k, 2, n}]/2];
a[n_] := b[2n] // Numerator;
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jun 28 2019, from Maple *)

Let b(n) = -Sum_{k=2..n} (C(n-1, k-1)*Bernoulli(k)*b(n-k)/k)/2 for n>0 and otherwise 1. Then a(n) = numerator(b(2*n)).

A239793 Denominator of b_{2n}(1/4), where b_{n}(x) are Nörlund's generalized Bernoulli polynomials.

Original entry on oeis.org

1, 24, 320, 10752, 184320, 360448, 23855104, 94371840, 285212672, 267764367360, 3720515420160, 987842478080, 201004469452800, 103903848824832, 637716744110080, 11997870882291712, 368450744514248704, 2251799813685248, 164633587978155851776, 9367487224930631680

Offset: 0

Peter Luschny, Mar 26 2014

See A239792 for references.

Cf. A220412, A239792 (numerators).

b := proc(n) option remember; if n < 1 then 1 else
-add(binomial(n-1, k-1)*bernoulli(k)*b(n-k)/k, k= 2..n)/2 fi end:
A239793 := n -> denom(b(2*n));
seq(A239793(n), n=0..19);

b[n_] := b[n] = If[n < 1, 1, -Sum[Binomial[n - 1, k - 1] BernoulliB[k] b[n - k]/k, {k, 2, n}]/2];
a[n_] := b[2 n] // Denominator;
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jun 28 2019 *)

Let b(n) = -sum_{2<=k<=n}(C(n-1, k-1)*Bernoulli(k)*b(n-k)/k)/2 for n>0 and otherwise 1. Then a(n) = denominator(b(2*n)).

A006934 A series for Pi.

Original entry on oeis.org

1, 1, 21, 671, 180323, 20898423, 7426362705, 1874409467055, 5099063967524835, 2246777786836681835, 2490122296790918386363, 1694873049836486741425113, 5559749161756484280905626951, 5406810236613380495234085140851, 12304442295910538475633061651918089

Offset: 0

Simon Plouffe and N. J. A. Sloane

nonn

Formula (21) in Luke (see ref.): Let y = 4*n+1. Then for n -> oo
Pi ~ 4*(n!)^4*2^(4*n)/(y*((2*n)!)^2)*(sum_{k>=0}((-1)^k*y^(-2*k)* A006934(k)/A123854(k)))^2. (Luke does not reference the sequences in this form.) - Peter Luschny, Mar 23 2014
This might be related to the numerators of eq. (18) in N. Elezovic' "Asymptotic Expansions of Central Binomial...", J. Int. Seq. 17 (2014) # 14.2.1. - R. J. Mathar, Mar 23 2014
Several references give an erroneous value of 1874409465055 instead of a(7) in the formula for pi. - M. F. Hasler, Mar 23 2014

Y. L. Luke, The Special Functions and their Approximation, Vol. 1, Academic Press, NY, 1969, see p. 36.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. L. Fields, A note on the asymptotic expansion of a ratio of gamma functions, Proc. Edinburgh Math. Soc. (15), 43-45, 1966.
A. Gil, J. Segura, N. M. Temme, Fast and accurate computation of the Weber parabolic cylinder function W(a,x), IMA J. Num. Anal. 31 (2011), 1194-1216, eq (3.8).
A. Lupas, Re: Pi Calculation ?, on mathforum.org, Feb 15 2003.
C. Mortici, On some accurate estimates of pi, Bull. Math. Anal. Appl. 2(4) (2010) 137-139. (Formula (1.5), same typo as in Luke)
Index entries for sequences related to the number Pi

Cf. A088802, A123854, A220412.

A006934_list := proc(n) local k, f, bp;
bp := proc(n,x) option remember; local k; if n = 0 then 1 else -x*add(binomial(n-1,2*k+1)*bernoulli(2*k+2)/(k+1)*bp(n-2*k-2,x), k=0..n/2-1) fi end:
f := n -> 2^(3*n-add(i, i=convert(n,base,2)));
add(bp(2*k,1/4)*binomial(4*k,2*k)*x^(2*k), k=0..n-1);
seq((-1)^k*f(k)*coeff(%,x,2*k), k=0..n-1) end:
A006934_list(15);  # Peter Luschny, Mar 23 2014
# Second solution, without using Nörlund's generalized Bernoulli polynomials, based on Euler numbers:
A006934_list := proc(n) local a,c,j;
c := n -> 4^n/2^add(i, i=convert(n,base,2));
a := [seq((-4)^j*euler(2*j)/(4*j), j=1..n)];
expand(exp(add(a[j]*x^(-j), j=1..n))); taylor(%, x=infinity, n+2);
subs(x=1/x, convert(%,polynom)): seq(c(iquo(j,2))*coeff(%,x,j), j=0..n) end:
A006934_list(14); # Peter Luschny, Apr 08 2014

A006934List[n_] := Module[{c, a, s, sx}, c[k_] := 4^k/2^Total[ IntegerDigits[k, 2]]; a = Table[(-4)^j EulerE[2j]/(4j), {j, 1, n}]; s[x_] = Series[Exp[Sum[a[[j]] x^(-j), {j, 1, n}]], {x, Infinity, n+2}] // Normal; sx = s[1/x]; Table[c[Quotient[j, 2]] Coefficient[sx, x, j], {j, 0, n}]];
A006934List[14] (* Jean-François Alcover, Jun 02 2019, from second Maple program *)

@CachedFunction
def p(n):
    if n < 2: return 1
    return -add(binomial(n-1,k-1)*bernoulli(k)*p(n-k)/k for k in range(2,n+1,2))/2
def A006934(n): return (-1)^n*p(2*n)*binomial(4*n,2*n)*2^(3*n-sum(n.digits(2)))
[A006934(n) for n in (0..14)]  # Peter Luschny, Mar 24 2014

Let p(n,x) = sum(k=0..n, x^k*A220412(n,k))/A220411(n) then a(n) = (-1)^n*p(n,1/4)*A123854(n)*A001448(n). - Peter Luschny, Mar 23 2014

Pi = lim_{n->oo} 2^{4n+2}/((4n+1)*C(2n,n)^2)*(sum_{k=0..oo} (-1)^k*a(k)/(A123854(k)*(4n+1)^{2k}))^2. - M. F. Hasler, Mar 23 2014

a(7) corrected, a(8)-a(14) from Peter Luschny, Mar 23 2014

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A220002 Numerators of the coefficients of an asymptotic expansion in even powers of the Catalan numbers.

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A220411 The denominators of J. L. Fields generalized Bernoulli polynomials.

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A239792 Numerator of b_{2n}(1/4), where b_{n}(x) are Nörlund's generalized Bernoulli polynomials.

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A239793 Denominator of b_{2n}(1/4), where b_{n}(x) are Nörlund's generalized Bernoulli polynomials.

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A006934 A series for Pi.

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A239795 a(n) = A239793(n)/2^(3*n).

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