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 1-3 of 3 results.

A167584 The ED4 array read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 13, 6, 1, 76, 41, 10, 1, 789, 372, 93, 14, 1, 7734, 4077, 1020, 169, 18, 1, 110937, 53106, 13269, 2212, 269, 22, 1, 1528920, 795645, 198990, 33165, 4140, 393, 26, 1, 28018665, 13536360, 3383145, 563850, 70485, 6996, 541, 30, 1
Offset: 1

Views

Author

Johannes W. Meijer, Nov 10 2009

Keywords

Comments

The coefficients in the upper right triangle of the ED4 array (m>n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED4 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 ED3 arrays see A167546, A167560 and A167572.
The Madhava-Gregory-Leibniz series representation for Pi/4 is the case m = 0 of the following more general result: for m = 0,1,2,... there holds 1/(2*m)! * Pi/4 = Sum_{k >= 0} ( (-1)^(m+k) * 1/Product_{j = -m .. m} (2*k + 1 + 2*j) ). The entries of this table are given by truncating these series to n-1 terms and then scaling by certain double factorials -- see the formula below. - Peter Bala, Nov 06 2016

Examples

			The ED4 array begins with:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1
  2, 6, 10, 14, 18, 22, 26, 30, 34, 38
  13, 41, 93, 169, 269, 393, 541, 713, 909, 1129
  76, 372, 1020, 2212, 4140, 6996, 10972, 16260, 23052, 31540
  789, 4077, 13269, 33165, 70485, 133869, 233877, 382989, 595605, 888045
  7734, 53106, 198990, 563850, 1339110, 2812194, 5389566, 9619770, 16216470, 26081490
  ...
From _Peter Bala_, Nov 06 2016: (Start)
Table extended to nonpositive values of m:
  n\m|     -4     -3    -2    -1    0
  -----------------------------------
   0 |      0      0     0     0    0
   1 |      1      1     1     1    1
   2 |    -18    -14   -10    -6   -2
   3 |    233    141    73    29    9
   4 |  -2844  -1428  -620  -228  -60
   5 |  39309  17877  7149  2325  525
  ...
Column  0: (-1)^(n+1)*(2*n - 3)!!*n. See A001193;
Column -1: (-1)^n*(2*n - 5)!!/3!!*n*(7 - 4*n^2);
Column -2: (-1)^n*(2*n - 7)!!/5!!*n(-149 + 120*n^2 - 16*n^4);
Column -3: (-1)^n*(2*n - 9)!!/7!!*n*(6483 - 6076*n^2 + 1232*n^4 - 64*n^6);
Column -4: (-1)^n*(2*n - 11)!!/9!!*n*(-477801 + 489136*n^2 - 120288*n^4 + 9984*n^6 - 256*n^8). (End)

Links

Crossrefs

A000012, A016825, A167585, A167586 and A167587 equal the first five rows of the array.
A024199, A167588 and A167589 equal the first three columns of the array.
A167590 equals the row sums of the ED4 array read by antidiagonals.
A167591 is a triangle related to the a(n) formulas of the rows of the ED4 array.
A167594 is a triangle related to the GF(z) formulas of the rows of the ED4 array.
Cf. A002866 (the 2^(n-1)*n! factor).
Cf. A167546 (ED1 array), A167560 (ED2 array), A167572 (ED3 array). Cf. A001193, A003881.

Programs

  • Maple
    T := proc (n, m) option remember;
      if n = 0 then 0
       elif n = 1 then 1
       else (4*m-2)*T(n-1,m)+(2*n+2*m-5)*(2*n-2*m-1)*T(n-2,m)
      end if;
     end proc:
#square array read by antidiagonals
seq(seq(T(n-m,m), m = 1..n-1), n = 1..10);
# Peter Bala, Nov 06 2016
  • Mathematica
    T[0, k_] := 0; T[1, k_] := 1; T[n_, k_] := T[n, k] = (4*k - 2)*T[n - 1, k] + (2*n + 2*k - 5)*(2*n - 2*k - 1)*T[n - 2, k]; Table[T[n - k, k], {n, 2, 12}, {k, 1, n - 1}] (* G. C. Greubel, Jan 20 2017 *)

Formula

a(n,m) = ((2*m-3)!!/(2*(2*m-2*n-3)!!))*Integral_{y=0..oo} sinh(y*(2*n))/(cosh(y))^(2*m-1) dy for m>n.
The (n-1)-differences of the n-th array row lead to the recurrence relation
Sum_{k=0..n-1} (-1)^k*binomial(n-1,k)*a(n,m-k) = 2^(n-1)*n!
From Peter Bala, Nov 06 2016: (Start)
T(n,m) = ((2*m - 3)!!/(2*(2*m - 2*n - 3)!!)) * Sum_{k = 0..n-1} (-1)^(k+1)*binomial(2*n - k - 1, k)*2^(2*n - 2*k - 1)*1/(2*n - 2*m - 2*k + 1), for n and m >= 0.
Note the double factorial for a negative odd integer N is defined in terms of the gamma function as N!! = 2^((N+1)/2)*Gamma(N/2 + 1)/sqrt(Pi).
T(n, m) = (2*m - 3)!! * (2*n + 2*m - 3)!! * Sum_{k = 0..n-1} ( (-1)^(m + k + 1) / Product_{j = -(m-1) .. m-1} (2*k + 1 + 2*j) ).
Using this result we can extend the table to nonpositive values of m (the column index). Column 0 is a signed version of A001193. We have for m <= 0, T(n,m) = (2*n - 2*|m| - 3)!!/(2*|m| + 1)!! * Sum_{k = 0..n-1} (-1)^k*Product_{j = -|m|..|m|} (2*k + 1 + 2*j).
Recurrence: T(n, m) = (4*m - 2)*T(n-1, m) + (2*n + 2*m - 5)*(2*n - 2*m - 1)*T(n-2, m).
For a fixed value of n, the entries in row n are polynomial in the value of the column index m. The first few polynomials are [1, 4*m - 2, 12*m^2 - 8*m + 9, 32*m^3 - 16*m^2 + 120*m - 60, 80*m^4 + 952*m^2 - 768*m + 525, ...]. (End)

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

Views

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

A167589 The third column of the ED4 array A167584.

Original entry on oeis.org

1, 10, 93, 1020, 13269, 198990, 3383145, 64276920, 1349846505, 31046064210, 776157686325, 20956154152500, 607730434609725, 18839602224969750, 621707822126431425, 21759750056864358000, 805111392478121276625
Offset: 1

Views

Author

Johannes W. Meijer, Nov 10 2009

Keywords

Links

Crossrefs

Equals the third column of the ED4 array A167584.
Other columns are A024199 and A167588.
Cf. A007509 and A025547 (the sum((-1)^(k+n)/(2*k+1), k=0..n-1) factor), A001147.

Programs

  • Mathematica
    Table[(1/8)*(-1)^(n)*(2*n - 5)!!*((4*n^3 - 11*n) + (16*n^4 - 40*n^2 + 9)*(Sum[(-1)^(k + n)/(2*k + 1), {k, 0, n - 1}])), {n, 1, 50}] (* G. C. Greubel, Jun 17 2016 *)

Formula

a(n) = (1/8)*(-1)^n*(2*n-5)!!*((4*n^3-11*n)+(16*n^4-40*n^2+9)*(Sum_{k=0..n-1} (-1)^(k+n)/(2*k+1) ) ).
From Peter Bala, Nov 01 2016: (Start)
a(n) = 3*(2*n + 3)!! * Sum_{k = 0..n-1} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)*(2*k + 5)).
a(n) ~ Pi*2^(n - 5/2)*((n + 2)/e)^(n + 2).
E.g.f.: (6*arcsin(2*x) + 4*x*sqrt(1 - 4*x^2)*(5 - 8*x^2))/(32*(1 - 2*x)^(5/2)).
a(n) = 10*a(n) + (2*n - 7)*(2*n + 1)*a(n-2) with a(0) = 0, a(1) = 1.
The sequence b(n) := (2*n + 3)!! = (2*n + 4)!/((n + 2)!*2^(n+2)) = A001147(n+2) satisfies the same recurrence with b(0) = 3 and b(1) = 15. This leads to the continued fraction representation a(n) = 1/3*b(n)*( 1/(5 - 15/(10 - 7/(10 + 9/(10 + 33/(10 + ... + (2*n - 7)*(2*n + 1)/(10)))))) ) for n >= 2.
As n -> infinity, 3*a(n)/(A001147(n+2)) -> 9/4!*Pi/4 = 1/(5 - 15/(10 - 7/(10 + 9/(10 + 33/(10 + ... + (2*n - 7)*(2*n + 1)/(10 + ...)))))). (End)
Showing 1-3 of 3 results.

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