cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A167572 The ED3 array read by antidiagonals.

Original entry on oeis.org

1, 5, 1, 23, 11, 1, 167, 83, 17, 1, 1473, 741, 183, 23, 1, 16413, 8169, 2043, 323, 29, 1, 211479, 106107, 26529, 4409, 503, 35, 1, 3192975, 1592235, 398025, 66345, 8175, 723, 41, 1, 54010305, 27062325, 6765975, 1127655, 140865, 13677, 983, 47, 1
Offset: 1

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Author

Johannes W. Meijer, Nov 10 2009

Keywords

Comments

The coefficients in the upper right triangle of the ED3 array (m>n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED3 array (m<=n) were found with the recurrence relation, see below. We use for the array rows the letter n (>=1) and for the array columns the letter m (>=1).
For the ED1, ED2 and ED4 arrays see A167546, A167560 and A167584.

Examples

			The ED3 array begins with:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
5, 11, 17, 23, 29, 35, 41, 47, 53, 59
23, 83, 183, 323, 503, 723, 983, 1283, 1623, 2003
167, 741, 2043, 4409, 8175, 13677, 21251, 31233, 43959, 59765
1473, 8169, 26529, 66345, 140865, 266793, 464289, 756969, 1171905, 1739625
16413, 106107, 398025, 1127655, 2678325, 5623443, 10768737, 19194495, 32297805, 51834795

Links

  • Johannes W. Meijer, The four Escher-Droste arrays, jpg image, Mar 08 2013.

Crossrefs

A000012, A016969, A167573, A167574 and A167575 equal the first five rows of the array.
A167576, A167577 and A167578 equal the first three columns of the array.
A167579 equals the row sums of the ED3 array read by antidiagonals.
A167580 is a triangle related to the a(n) formulas of the rows of the ED3 array.
A167583 is a triangle related to the GF(z) formulas of the rows of the ED3 array.
Cf. A014481 (the 2^(n-1)*(n-1)!*(2*n-1) factor).
Cf. A167546 (ED1 array), A167560 (ED2 array), A167584 (ED4 array).

Formula

a(n,m) = ((2*m-1)!!/ (2*m-2*n-1)!!)*int(sinh(y*(2*n-1))/(cosh(y))^(2*m),y=0..infinity) for m>n.
The (n-1)-differences of the n-th array row lead to the recurrence relation
sum((-1)^k*binomial(n-1,k)*a(n,m-k),k=0..n-1) = 2^(n-1)*(n-1)!*(2*n-1).

A007509 Numerator of Sum_{k=0..n} (-1)^k/(2*k+1).

Original entry on oeis.org

1, 2, 13, 76, 263, 2578, 36979, 33976, 622637, 11064338, 11757173, 255865444, 1346255081, 3852854518, 116752370597, 3473755390832, 3610501179557, 3481569435902, 133330680156299, 129049485078524, 5457995496252709, 227848175409504262, 234389556075339277
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Denominators of convergents to 4/Pi. [For Brouncker's continued fraction, with numerators A025547(n+1), for n >= 0. - Wolfdieter Lang, Aug 26 2019]
See A352395 (the denominators for the present sequence) for the formula of this alternating sum, and the Abramowitz-Stegun link. - Wolfdieter Lang, Apr 06 2022

Examples

			1/1, 2/3, 13/15, 76/105, 263/315, 2578/3465, 36979/45045, 33976/45045, 622637/765765, ...

References

  • P. Beckmann, A History of Pi. Golem Press, Boulder, CO, 2nd ed., 1971, p. 131.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Denominators are given in A352395.
From Johannes W. Meijer, Nov 12 2009: (Start)
Cf. A157142 and A166107.
Appears in A167576, A167577, A167578, A024199, A167588 and A167589. (End)
Cf. A142969 for the numerators of Brouncker's continued fraction of 4/Pi - 1.

Programs

  • Magma
    [Numerator(&+[(-1)^k/(2*k+1):k in [0..n]]): n in [0..23]]; // Marius A. Burtea, Aug 26 2019
  • Maple
    A007509 := n->numer(add((-1)^k/(2*k+1),k=0..n));
  • Mathematica
    Table[Numerator[FunctionExpand[(Pi + (-1)^n(HarmonicNumber[n/2 + 1/4] - HarmonicNumber[n/2 - 1/4]))/4]], {n, 0, 20}] (* Vladimir Reshetnikov, Jan 18 2011 *)
Numerator[Table[Sum[(-1)^k/(2k+1),{k,0,n}],{n,0,30}]] (* Harvey P. Dale, Oct 22 2011 *)
Table[(-1)^k/(2k+1),{k,0,30}]//Accumulate//Numerator (* Harvey P. Dale, May 03 2019 *)

Formula

a(n) = numerator((Psi(n + 3/2) - Psi((2*n - (-1)^n)/4 + 1) - log(2) + Pi/2)/2), with the digamma function Psi(z). See the formula in A352395. - Wolfdieter Lang, Apr 06 2022
a(n) = numerator(Pi/4 + (-1)^n * (Psi((n + 5/2)/2) - Psi((n + 3/2)/2))/4). - Vaclav Kotesovec, May 16 2022

Extensions

Crossref. corrected (A025547 replaced with A352395) by Wolfdieter Lang, Apr 06 2022

A167576 The first column of the ED3 array A167572.

Original entry on oeis.org

1, 5, 23, 167, 1473, 16413, 211479, 3192975, 54010305, 1030249845, 21566327895, 497334999735, 12405876372225, 335591130336525, 9716331072597975, 301633179343890975, 9941514351641143425, 348336799875365041125
Offset: 1

Views

Author

Johannes W. Meijer, Nov 10 2009

Keywords

Comments

Basically a(n) measures the difference between the Euler factorial n! and the Luschny factorial L(n) at half-integer values. For the Luschny factorial see the link. The formula given in the Maple section is a variant of a formula given by Cyril Damamme in A135457. - Peter Luschny, Jul 18 2015

Examples

			G.f. = x + 5*x^2 + 23*x^3 + 167*x^4 + 1473*x^5 + 16413*x^6 + ...

Links

Crossrefs

Equals the first column of the ED3 array A167572.
Equals the first right hand column of A167583.
Other columns are A167577 and A167578.
Cf. A097801 (the 2*(-1)^n*(2*n-5)!! factor).
Cf. A007509 and A025547 (the sum((-1)^(k+n)/(2*k+1), k=0..n-1) factor).
Cf. A024199 and A135457.

Programs

  • Maple
    L := x -> (1+x*(Psi(1-x/2)-Psi(1/2-x/2)))/(-x)!:
a := x -> (L(x-1/2)-(x-1/2)!)*2^(x-1)*sqrt(Pi):
seq(simplify(a(n)),n=1..18); # Peter Luschny, Jul 18 2015
a := proc(n) option remember: if n=1 then 1 else (2*n-1)*a(n-1)+2*(-1)^n*doublefactorial(2*n-5) fi: end: seq(a(n),n=1..18); # Johannes W. Meijer, Jul 20 2015
  • Mathematica
    a[ n_] := If[ n < 1, 0, (2 n - 3)!! ((-1)^n - I (4 n - 2) Sum[ I^k / k, {k, 1, 2 n - 1, 2}])]; (* Michael Somos, Jul 20 2015 *)
a[ n_] := If[ n < 1, 0, (2 n - 3)!! ((-1)^n + (4 n - 2) Sum[ KroneckerSymbol[ -4, k]/ k, {k, 2 n - 1}])]; (* Michael Somos, Jan 31 2019 *)
  • PARI
    {a(n) = if( n<1, 0, prod(k=1, n-1, 2*k - 1) * ((-1)^n - (4*n - 2) * sum(k=1, n, (-1)^k / (2*k - 1))))}; /* Michael Somos, Jul 20 2015 */

Formula

a(n) = (-1)^n*(2*n-3)!!*(1 + (4*n-2)*Sum_{k=0..n-1} (-1)^(k+n)/(2*k+1)).
a(n) = (2*n-1)*a(n-1) + 2*(-1)^n*(2*n-5)!! with a(1) = 1.
a(n) = 4*a(n-1) + (4*n^2 - 16*n + 15)*a(n-2) with a(1) = 1 and a(2) = 5 [Superseeker].
0 = a(n)*a(n+1)*(-440*a(n+2) - 220*a(n+3) + 55*a(n+4)) + a(n)*a(n+2)*(536*a(n+2) - 118*a(n+3) - 4*a(n+4)) + a(n)*a(n+3)*(-4*a(n+3) + a(n+4)) + a(n+1)^2*(-220*a(n+2) - 32*a(n+3) + 8*a(n+4)) + a(n+1)*a(n+2)*(+71*a(n+2) + 4*a(n+3) - 2*a(n+4)) + a(n+2)^2*(-4*a(n+2) + a(n+3)) if n>0. - Michael Somos, Jul 19 2015
a(n) = (-1 + (n-1/2)*LerchPhi(-1,1,n+1/2) + (-n+1/2)*LerchPhi(-1,1,-n+1/2))/(1-2*n)!!. - Johannes W. Meijer, Jul 20 2015
a(n) = A024199(n) + A135457(n). - Cyril Damamme, Jul 22 2015
a(n) = ((-1)^n/(2*n - 1) + Pi/2 - (-1)^n LerchPhi(-1, 1, n + 1/2)) (2*n - 1)!!. - Michael Somos, Jan 31 2019

A167578 The third column of the ED3 array A167572.

Original entry on oeis.org

1, 17, 183, 2043, 26529, 398025, 6765975, 128556675, 2699661825, 62092533825, 1552309291575, 41912411683275, 1215458905032225, 37679245697871225, 1243414695550433175, 43519523831289457875, 1610222144582102522625
Offset: 1

Views

Author

Johannes W. Meijer, Nov 10 2009

Keywords

Links

Crossrefs

Equals the third column of the ED3 array A167572.
Other columns are A167576 and A167577.
Cf. A007509 and A025547 (the sum((-1)^(k+n)/(2*k+1), k=0..n-1) factor).

Programs

  • Mathematica
    Table[(1/4)*(-1)^(n)*(2*n - 7)!!*((8*n^4 - 20*n^3 - 22*n^2 + 55*n + 12) + (32*n^5 - 80*n^4 - 80*n^3 + 200*n^2 + 18*n - 45)*(Sum[(-1)^(k + n)/(2*k + 1), {k, 0, n - 1}])), {n, 1, 50}] (* G. C. Greubel, Jun 16 2016 *)

Formula

a(n) = (1/4)*(-1)^(n)*(2*n-7)!!*((8*n^4-20*n^3-22*n^2+55*n+12)+(32*n^5-80*n^4-80*n^3+200*n^2+18*n-45)*(sum((-1)^(k+n)/(2*k+1),k=0..n-1))).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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