A248896 -id:A248896 - cp's OEIS frontend

A123092 Decimal expansion of Sum_{k>=1} 1/((2k-1)^2(2k+1)^2) = (Pi^2-8)/16.

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

Offset: 0

Robert G. Wilson v, Sep 27 2006

			0.116850275068084913677155687492259445957106212952549414150834336...

Erwin Kreyszig, Advanced Engineering Mathematics, 9th Edition, John Wiley and Sons, Inc., NJ, 2006, page 506.

Index entries for transcendental numbers.

Cf. A000796, A002388, A069075, A111003.

Cf. A248895, A248896.

RealDigits[Sum[1/((2k - 1)^2(2k + 1)^2), {k, Infinity}], 10, 111][[1]]

(Pi^2-8)/16 \\ Charles R Greathouse IV, Sep 30 2022

Equals (A111003-1)/2. - Hugo Pfoertner, Aug 20 2024

Equals Sum_{k>=1} 1/(4*k^2-1)^2. - Sean A. Irvine, Mar 29 2025

A248895 Decimal expansion of Sum_{i >= 1} 1/(4*i^2-1)^3.

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Mar 06 2015

			0.0373622936989363147421332343808054155321703402855879393868742479896853989...

Xavier Gourdon and Pascal Sebah, Collection of series for Pi (see paragraph 7).

Cf. A123092: Sum_{i >= 1} 1/(4*i^2-1)^2.

Cf. A248896: Sum_{i >= 1} 1/(4*i^2-1)^4.

Join[{0}, RealDigits[1/2 - 3 (Pi/8)^2, 10, 100][[1]]]

Equals 1/2 - 3*(Pi/8)^2.

A382782 Irregular triangle T(n,k) read by rows of the reduced coefficients of Pi^(2k) in the expansion of Sum_{k>=1} (1 / (4k^2-1)^n).

Original entry on oeis.org

1, -8, 1, 32, -3, -384, 30, 1, 1536, -105, -5, -30720, 1890, 105, 2, 61440, -3465, -210, -7, -10321920, 540540, 34650, 1512, 17, 4587520, -225225, -15015, -770, -17, -1486356480, 68918850, 4729725, 270270, 8415, 62, 2972712960, -130945815, -9189180, -567567, -21879, -341

Offset: 1

Sean A. Irvine, Apr 04 2025

The expansion of S(n) = Sum_{k>=1} (1 / (4*k^2-1)^n) in even powers of Pi was apparently first found by Euler and the solution for n<=4 appear in many tables of sums.

These sums have a natural denominator of 2^(2*n)*(n-1)! (or, more precisely, 2^(2*n+floor((n-1)/2))*(n-1)!), but sometimes (e.g., n=7, n=9) there are additional common factors leading to the "reduced" triangle presented here.

			S(1) = (        1                                                 ) / (2),
S(2) = (       -8 +        Pi^2                                   ) / (2^4) = A123092,
S(3) = (       32 -      3*Pi^2                                   ) / (2^5 * 2!) = A248895,
S(4) = (     -384 +     30*Pi^2 +       Pi^4                      ) / (2^7 * 3!) = A248896,
S(5) = (     1536 -    105*Pi^2 -     5*Pi^4                      ) / (2^7 * 4!),
S(6) = (   -30720 +   1890*Pi^2 +   105*Pi^4 +    2*Pi^6          ) / (2^9 * 5!),
S(7) = (    61440 -   3465*Pi^2 -   210*Pi^4 -    7*Pi^6          ) / (2^10 * 5!),
S(8) = (-10321920 + 540540*Pi^2 + 34650*Pi^4 + 1512*Pi^6 + 17*Pi^8) / (2^12 * 7!),
S(9) = (  4587520 - 225225*Pi^2 - 15015*Pi^4 -  770*Pi^6 - 17*Pi^8) / (2^18 * 5 * 7), ...

E. P. Adams, Smithsonian Mathematical Formulae and Tables of Elliptic Functions, 1922 (eq. 6.911).

I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products (6th ed.), 2000, (eq. 0.235).
Sean A. Irvine, Computing A382782, 2025.
Sean A. Irvine, Java program (github)
L. B. W. Jolley, Summation of Series, Dover, (1961) (eq. 373).

Cf. A123092 (n=2), A248895 (n=3), A248896 (n=4).

Cf. A382783, A382784.

A382784 Irregular triangle T(n,k) read by rows of the coefficients of Pi^(2k) in the expansion of Sum_{k>=1} (1 / (4k^2-1)^n) with denominator 2^(2n)*(n-1)!.

Original entry on oeis.org

2, -8, 1, 64, -6, -768, 60, 2, 12288, -840, -40, -245760, 15120, 840, 16, 5898240, -332640, -20160, -672, -165150720, 8648640, 554400, 24192, 272, 5284823040, -259459200, -17297280, -887040, -19584, -190253629440, 8821612800, 605404800, 34594560, 1077120, 7936, 7610145177600, -335221286400, -23524300800, -1452971520, -56010240, -872960

Offset: 1

Sean A. Irvine, Apr 04 2025

sign

See A382782 for a version of this triangle where common factors have been removed.

			Triangle begins:
S(1) =  (2) / (2^2 * 0!),
S(2) = -(8 - Pi^2) / (2^4 * 1!) = A123092,
S(3) =  (64 - 6*Pi^2) / (2^6 * 2!) = A248895,
S(4) = -(768 - 60*Pi^2 - 2*Pi^4)/ (2^8 * 3!) = A248896,
S(5) =  (12288 - 840*Pi^2 - 40*Pi^4) / (2^10 * 4!),
S(6) = -(245760 - 15120*Pi^2 - 840*Pi^4 - 16*Pi^6) / (2^12 * 5!),
S(7) =  (5898240 - 332640*Pi^2 - 20160*Pi^4 - 672*Pi^6) / (2^14 * 6!),
S(8) = -(165150720 - 8648640*Pi^2 - 554400*Pi^4 - 24192*Pi^6 - 272*Pi^8) / (2^16 * 7!),
S(9) =  (5284823040 - 259459200*Pi^2 - 17297280*Pi^4 - 887040*Pi^6 - 19584*Pi^8) / (2^18 * 8!), ...

Cf. A123092 (n=2), A248895 (n=3), A248896 (n=4).

Cf. A382782.

cp's OEIS Frontend

A123092 Decimal expansion of Sum_{k>=1} 1/((2k-1)^2(2k+1)^2) = (Pi^2-8)/16.

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A248895 Decimal expansion of Sum_{i >= 1} 1/(4*i^2-1)^3.

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A382782 Irregular triangle T(n,k) read by rows of the reduced coefficients of Pi^(2k) in the expansion of Sum_{k>=1} (1 / (4k^2-1)^n).

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A382784 Irregular triangle T(n,k) read by rows of the coefficients of Pi^(2k) in the expansion of Sum_{k>=1} (1 / (4k^2-1)^n) with denominator 2^(2n)*(n-1)!.

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cp's OEIS Frontend

A123092 Decimal expansion of Sum_{k>=1} 1/((2k-1)^2(2k+1)^2) = (Pi^2-8)/16.

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A248895 Decimal expansion of Sum_{i >= 1} 1/(4*i^2-1)^3.

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A382782 Irregular triangle T(n,k) read by rows of the reduced coefficients of Pi^(2*k) in the expansion of Sum_{k>=1} (1 / (4*k^2-1)^n).

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A382784 Irregular triangle T(n,k) read by rows of the coefficients of Pi^(2k) in the expansion of Sum_{k>=1} (1 / (4k^2-1)^n) with denominator 2^(2n)*(n-1)!.

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A382782 Irregular triangle T(n,k) read by rows of the reduced coefficients of Pi^(2k) in the expansion of Sum_{k>=1} (1 / (4k^2-1)^n).