A220411 -id:A220411 - cp's OEIS frontend

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

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.

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

A276997 Denominators of coefficients of polynomials arising from applying the complete Bell polynomials to k!B_k(x)/(k*(k-1)) with B_k(x) the Bernoulli polynomials.

Original entry on oeis.org

1, 1, 1, 6, 1, 1, 1, 2, 2, 1, 60, 1, 1, 1, 1, 1, 6, 2, 3, 1, 1, 504, 4, 4, 1, 1, 1, 1, 1, 24, 8, 12, 2, 2, 2, 1, 2160, 18, 9, 3, 2, 1, 3, 1, 1, 1, 60, 4, 6, 1, 5, 1, 1, 1, 1, 3168, 48, 16, 6, 3, 2, 2, 1, 2, 1, 1, 1, 288, 32, 144, 12, 12, 4, 2, 1, 6, 2, 1

Offset: 0

Peter Luschny, Oct 01 2016

For formulas and references see A276996.

Compare T(n,0) with A220411.

			Triangle starts:
     1;
     1,  1;
     6,  1,  1;
     1,  2,  2,  1;
    60,  1,  1,  1, 1;
     1,  6,  2,  3, 1, 1;
   504,  4,  4,  1, 1, 1, 1;
     1, 24,  8, 12, 2, 2, 2, 1;
  2160, 18,  9,  3, 2, 1, 3, 1, 1;

Cf. A276996 (numerators), A220411.

A276997_row := proc(n) local p;
p := (n,x) -> CompleteBellB(n,0,seq((k-2)!*bernoulli(k,x),k=2..n)):
seq(denom(coeff(p(n,x),x,k)), k=0..n) end:
seq(A276997_row(n), n=0..11);

CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
p[n_, x_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, x], {k, 2, n}]]];
row[0] = {1}; row[1] = {1, 1}; row[n_] := CoefficientList[p[n, x], x] // Denominator;
Table[row[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Sep 09 2018 *)

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

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A006934 A series for Pi.

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A276997 Denominators of coefficients of polynomials arising from applying the complete Bell polynomials to k!B_k(x)/(k*(k-1)) with B_k(x) the Bernoulli polynomials.

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Maple

Mathematica