A050970 -id:A050970 - cp's OEIS frontend

A245296 Decimal expansion of the Landau-Kolmogorov constant C(5,1) for derivatives in the case L_infinity(infinity, infinity).

1, 0, 4, 4, 2, 5, 7, 9, 0, 9, 3, 0, 9, 7, 9, 5, 1, 4, 3, 4, 4, 5, 3, 6, 9, 6, 1, 7, 1, 5, 5, 7, 0, 2, 5, 8, 3, 0, 8, 0, 4, 2, 0, 8, 0, 4, 2, 0, 2, 5, 3, 7, 2, 0, 7, 7, 5, 7, 6, 1, 3, 4, 1, 5, 8, 0, 0, 2, 3, 2, 5, 8, 8, 8, 0, 0, 6, 2, 3, 5, 7, 8, 8, 7, 4, 4, 6, 0, 2, 0, 1, 1, 1, 9, 2, 2, 0, 2, 7, 8, 5, 4, 7, 2, 4

Offset: 1

Jean-François Alcover, Jul 17 2014

See A245198.

			1.0442579093097951434453696171557025830804208042025372077576134158002325888...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants
Eric Weisstein's MathWorld, Favard Constants

Cf. A050970, A050971, A244091, A245198.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[5,1], 10, 105] // First

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).

C(5,1) = (5*5^(4/5))/(8*2^(4/5)*3^(1/5)) = (1953125/1572864)^(1/5).

A245297 Decimal expansion of the Landau-Kolmogorov constant C(5,2) for derivatives in the case L_infinity(infinity, infinity).

Original entry on oeis.org

1, 1, 1, 6, 6, 4, 5, 9, 7, 1, 1, 0, 3, 8, 0, 9, 8, 8, 2, 6, 4, 5, 7, 1, 5, 4, 5, 1, 0, 7, 3, 1, 5, 3, 1, 7, 8, 9, 6, 6, 5, 1, 2, 0, 0, 6, 6, 9, 7, 4, 0, 4, 0, 1, 6, 4, 5, 6, 3, 4, 2, 1, 6, 0, 6, 0, 8, 1, 7, 9, 5, 2, 8, 6, 4, 8, 5, 2, 2, 2, 9, 6, 8, 4, 6, 4, 6, 0, 0, 2, 6, 2, 2, 4, 5, 4, 9, 9, 1, 2, 3

Offset: 1

Jean-François Alcover, Jul 17 2014

See A245198.

			1.1166459711038098826457154510731531789665120066974040164563421606...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants
Eric Weisstein's MathWorld, Favard Constants

Cf. A050970, A050971, A244091, A245198.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[5,2], 10, 101] // First

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).

C(5,2) = (5*5^(4/5))/(8*2^(4/5)*3^(1/5)) = (1953125/1572864)^(1/5).

A245298 Decimal expansion of the Landau-Kolmogorov constant C(5,3) for derivatives in the case L_infinity(infinity, infinity).

Original entry on oeis.org

1, 1, 1, 9, 4, 2, 3, 7, 3, 1, 7, 3, 5, 1, 0, 7, 6, 1, 1, 6, 2, 9, 7, 1, 1, 0, 8, 2, 0, 8, 1, 2, 6, 1, 0, 4, 1, 2, 4, 9, 9, 8, 5, 5, 6, 7, 0, 5, 8, 6, 0, 7, 0, 8, 6, 5, 2, 0, 9, 8, 2, 7, 9, 9, 1, 3, 1, 5, 4, 2, 2, 9, 2, 2, 9, 6, 9, 0, 4, 5, 1, 5, 2, 5, 2, 6, 2, 8, 6, 5, 9, 6, 1, 3, 0, 8, 5, 2, 2, 9, 2, 9, 5, 2

Offset: 1

Jean-François Alcover, Jul 17 2014

See A245198.

			1.11942373173510761162971108208126104124998556705860708652098279913...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants
Eric Weisstein's MathWorld, Favard Constants

Cf. A050970, A050971, A244091, A245198.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[5,3], 10, 104] // First

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).

C(5,3) = (1/2)*(15/2)^(2/5).

A245299 Decimal expansion of the Landau-Kolmogorov constant C(5,4) for derivatives in the case L_infinity(-infinity, infinity).

Original entry on oeis.org

1, 4, 9, 6, 2, 7, 7, 8, 6, 9, 7, 3, 8, 8, 4, 4, 7, 3, 8, 5, 0, 8, 1, 0, 2, 1, 3, 9, 3, 2, 9, 7, 8, 2, 5, 5, 3, 3, 1, 7, 0, 0, 6, 2, 4, 7, 0, 9, 3, 2, 5, 4, 1, 0, 3, 0, 8, 7, 5, 6, 8, 6, 3, 9, 5, 0, 3, 6, 8, 0, 0, 9, 7, 2, 0, 4, 5, 0, 0, 4, 3, 3, 7, 4, 5, 7, 0, 3, 5, 8, 1, 0, 9, 0, 8, 3, 9, 6, 3, 9, 6, 9, 2, 0, 9

Offset: 1

Jean-François Alcover, Jul 17 2014

See A245198.

			1.49627786973884473850810213932978255331700624709325410308756863950368...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.3 Landau-Kolmogorov constants, p. 213.

Eric Weisstein's MathWorld, Landau-Kolmogorov Constants
Eric Weisstein's MathWorld, Favard Constants

Cf. A050970, A050971, A244091, A245198.

a[n_] := (4/Pi)*Sum[((-1)^j/(2*j+1))^(n+1), {j, 0, Infinity}]; c[n_, k_] := a[n-k]*a[n]^(-1+k/n); RealDigits[c[5,4], 10, 105] // First

C(n,k) = a(n-k)*a(n)^(-1+k/n), where a(n) = (4/Pi)*sum_{j=0..infinity}((-1)^j/(2j+1))^(n+1) or a(n) = 4*Pi^n*f(n+1), f(n) being the n-th Favard constant A050970(n)/A050971(n).

C(5,4) = (15/2)^(1/5).

A335955 a(n) = (4^n(Z(-n, 1/4) - Z(-n, 3/4)) + Z(-n, 1)(2^(n+1)-1))*A171977(n+1), where Z(n, c) is the Hurwitz zeta function.

Original entry on oeis.org

0, -1, -1, 1, 5, -1, -61, 17, 1385, -31, -50521, 691, 2702765, -5461, -199360981, 929569, 19391512145, -3202291, -2404879675441, 221930581, 370371188237525, -4722116521, -69348874393137901, 968383680827, 15514534163557086905, -14717667114151, -4087072509293123892361

Offset: 0

Peter Luschny, Jul 20 2020

sign

N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf), arXiv:math/0101168 [math.CA], 2001.
N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
Eric Weisstein's World of Mathematics, Favard Constants

Cf. A002425, A028296, A050970, A171977, A335954.

HZeta := (s, v) -> Zeta(0, s, v):
a := s -> (4^s*(HZeta(-s,1/4) - HZeta(-s,3/4)) + HZeta(-s,1)*(2^(s+1)-1))* 2^padic[ordp](2*(s+1),2): seq(a(n), n = 0..28);

a[n_] := 2^(IntegerExponent[n + 1, 2] + 1) (4^n (HurwitzZeta[-n, 1/4] - HurwitzZeta[-n, 3/4]) + HurwitzZeta[-n, 1] (2^(n + 1) - 1));
Table[FullSimplify[a[n]], {n, 0, 26}]

A002425 interleaved with A028296.
|Numerator(a(n)/n!)| = A050970(n+1) for n >= 1.
a(n) = 2*(4^n*(Z(-n, 1/4) - Z(-n, 3/4)) + Z(-n,1)*A335954(n+1)) where Z(n, c) is the Hurwitz zeta function.

cp's OEIS Frontend

A245296 Decimal expansion of the Landau-Kolmogorov constant C(5,1) for derivatives in the case L_infinity(infinity, infinity).

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A245297 Decimal expansion of the Landau-Kolmogorov constant C(5,2) for derivatives in the case L_infinity(infinity, infinity).

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A245298 Decimal expansion of the Landau-Kolmogorov constant C(5,3) for derivatives in the case L_infinity(infinity, infinity).

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A245299 Decimal expansion of the Landau-Kolmogorov constant C(5,4) for derivatives in the case L_infinity(-infinity, infinity).

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A335955 a(n) = (4^n*(Z(-n, 1/4) - Z(-n, 3/4)) + Z(-n, 1)*(2^(n+1)-1))*A171977(n+1), where Z(n, c) is the Hurwitz zeta function.

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Formula

A335955 a(n) = (4^n(Z(-n, 1/4) - Z(-n, 3/4)) + Z(-n, 1)(2^(n+1)-1))*A171977(n+1), where Z(n, c) is the Hurwitz zeta function.