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.

A278080 Expansion of e.g.f. (1/4!)*sin^4(x)/cos(x) (coefficients of even powers only).

Original entry on oeis.org

0, 0, 1, -5, 126, 1490, 118151, 8256885, 808428076, 100199284180, 15432169163901, 2889536106161375, 646438926423519626, 170294687860735726470, 52177485058722877649251, 18397662218707151323777465, 7396641315814156362154666776
Offset: 0

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Author

Peter Bala, Nov 10 2016

Keywords

Comments

This sequence gives the coefficients in an asymptotic expansion of a series related to the constant Pi. It can be shown that (1/4!)*Pi/4 = Sum_{k >= 1} (-1)^(k-1)/((2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)). Using Proposition 1 of Borwein et al. it can be shown that the following asymptotic expansion holds for the tails of the series: for N divisible by 4, 2*( (1/4!)*Pi/4 - Sum_{k = 1..N/2} (-1)^(k-1)/((2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)) ) ~ 1/N^5 - (-5)/N^7 + 126/N^9 - 1490/N^11 + 118151/N^13 - .... An example is given below. Cf. A024235 and A278195.

Examples

			Let N = 100000. The truncated series 2*Sum_{k = 1..N/2} (-1)^(k-1)/((2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)) = 0.065449846949787359134638(3)038183229(2)6754107(569)820314(8536)0364.... The bracketed digits show where this decimal expansion differs from that of Pi/48. The numbers 1, 5, 126, -1490 must be added to the bracketed numbers to give the correct decimal expansion to 60 digits: Pi/48 = 0.065449846949787359134638(4) 038183229(7)6754107(695)820314(7046)0364.. ..

Links

Crossrefs

Cf. A000364, A004174, A024235, A166984, A278079, A278194, A278195.

Programs

  • Maple
    A000364 := n -> abs(euler(2*n)):
seq(1/4!*(A000364(n) + (-1)^n*(9^n - 5)/4), n = 0..20);
  • Mathematica
    With[{nn=40},Take[CoefficientList[Series[1/4! Sin[x]^4/Cos[x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Feb 09 2025 *)

Formula

a(n) = [x^(2*n)/(2*n)!] ( (1/4!)*sin^4(x)/cos(x) ).
a(n) = (1/4!)*( A000364(n) + (-1)^n*(9^(n) - 5)/4 ).
a(n) = (-1)^n/(2^4*4!) * 2^(2*n)*( E(2*n,5/2) - 4*E(2*n,3/2) + 6*E(2*n,1/2) - 4*E(2*n,-1/2) + E(2*n,-3/2) ), where E(n,x) is the Euler polynomial of order n.
E.g.f. (1/4!)*sin^4(x)/cos(x) = x^4/4! - 5*x^6/6! + 126*x^8/8! + 1490*x^10/10! + ....
O.g.f. for a signed version of the sequence: Sum_{n >= 0} ( (1/2^n) * Sum_{k = 0..n} (-1)^k*binomial(n, k)/((1 - (2*k - 3)*x)*(1 - (2*k - 1)*x)*(1 - (2*k + 1)*x)*(1 - (2*k + 3)*x)*(1 - (2*k + 5)*x)) ) = 1 + 5*x^2 + 126*x^4 - 1490*x^6 + 118151*x^8 - ....

A278195 Expansion of e.g.f. (1/6!)*sin^6(x)/cos(x) (coefficients of even powers only).

Original entry on oeis.org

0, 0, 0, 1, -28, 882, -17116, 803803, 13713336, 3671012164, 506128123928, 96524822605365, 21542790273363260, 5676618945053498806, 1739246268204447115932, 613255488134158250903887, 246554708506039690689322544, 112115693433705109495581088008
Offset: 0

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Author

Peter Bala, Nov 15 2016

Keywords

Comments

This sequence gives the coefficients in an asymptotic expansion of a series related to the constant Pi. It can be shown that (1/6!)*Pi/4 = Sum_{k >= 1} (-1)^k/((2*k - 7)*(2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)*(2*k + 5)). Using Proposition 1 of Borwein et al. it can be shown that the following asymptotic expansion holds for the tails of this series: for N divisible by 4, 2*{ (1/6!)*Pi/4 - Sum_{k = 1..N/2} (-1)^k/((2*k - 7)*(2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)*(2*k + 5)) } ~ -1/N^7 + (-28)/N^9 - 882/N^11 + (-17116)/N^13 - 803803/N^15 + .... An example is given below.

Examples

			Expansion of (1/6!)*sin^6(x)/cos(x) starts x^6/6! - 28*x^8/8! + 882*x^10/10! - 17116*x^12/12! + ....
Let N = 100000. The truncated series 2*Sum_{k = 1..N/2} (-1)^k/( (2*k - 7)*(2*k - 5)*(2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)*(2*k + 5) ) = 0.0021816615649929119711546134606107(7)5891803(617)860677(2450)20121.... The bracketed digits show where this decimal expansion differs from that of Pi/1440. The numbers -1, -28, -882 must be added to the bracketed numbers to give the correct decimal expansion: Pi/1440 = 0.0021816615649929119711546134606107(6)5891803(589)860677(1568)20121....

Links

Crossrefs

Cf. A000364, A004174, A166984, A278079, A278080, A278194.

Programs

  • Maple
    A000364 := n -> abs(euler(2*n)):
seq((1/6!)*(A000364(n) - (1/16)*((-25)^n - 7*(-9)^n + 22*(-1)^n) ), n = 0..20);
  • Mathematica
    With[{nn=40},Take[CoefficientList[Series[Sin[x]^6/Cos[x] 1/6!,{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Sep 12 2019 *)

Formula

a(n) = [x^(2*n)/(2*n)!] ( 1/6!*sin^6(x)/cos(x) ).
a(n) = (1/6!)*( A000364(n) - 1/16*((-25)^n - 7*(-9)^n + 22*(-1)^n) ).
a(n) = (-1)^(n+1)/(2^6*6!) * 2^(2*n)*( E(2*n,7/2) - 6*E(2*n,5/2) + 15*E(2*n,3/2) - 20*E(2*n,1/2) + 15*E(2*n,-1/2) - 6*E(2*n,-3/2) + E(2*n,-5/2) ), where E(n,x) is the Euler polynomial of order n.

A278079 Expansion of e.g.f. (1/3!)*sin^3(x)/cos(x) (coefficients of odd powers only).

Original entry on oeis.org

0, 1, 0, 56, 1280, 59136, 3727360, 317295616, 34977546240, 4848147562496, 825249675345920, 169237314418507776, 41153580031698534400, 11708600267324004499456, 3853197364634932928839680, 1452327126187528216207425536, 621567950620088261848869109760
Offset: 0

Views

Author

Peter Bala, Nov 10 2016

Keywords

Links

Crossrefs

Cf. A000182, A000364, A004174, A024235, A278080, A278194, A278195.

Programs

  • Maple
    seq((-1)^n*( 2/3*4^n*(4^(n+1) - 1)*bernoulli(2*n+2)/(2*n + 2) - 4^n/6 ), n = 0..20);

Formula

a(n) = [x^(2*n+1)/(2*n+1)!] ( 1/3!*sin^3(x)/cos(x) ).
a(n) = (-1)^n*( 2/3*4^n*(4^(n+1) - 1)*Bernoulli(2*n+2)/(2*n + 2) - 4^n/6 ).
a(n) = (-1)^(n+1)/(2^3*3!) * 2^(2*n+1)*( E(2*n+1,2) - 3*E(2*n+1,1) + 3*E(2*n+1,0) - E(2*n+1,-1) ), where E(n,x) is the Euler polynomial of order n.
a(n) = (-1)^(n+1)/8 * Sum_{k = 0..n} (9^(n-k) - 1)*binomial(2*n+1,2*k)*2^(2*k)* E(2*k, 1/2).
G.f. 1/3!*sin^3(x)/cos(x) = x^3/3! + 56*x^7/7! + 1280*x^9/9! + 59136*x^11/11! + ....
Showing 1-3 of 3 results.

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