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.

Previous Showing 11-20 of 20 results.

A346791 E.g.f.: 1 / (1 + x + Sum_{k>=2} prime(k-1) * x^k / k!).

Original entry on oeis.org

1, -1, 0, 3, -5, -17, 103, 57, -2707, 6785, 84135, -659369, -2129683, 55537445, -103722105, -4630217025, 37357780827, 334163569535, -7214177094045, -2126819153101, 1233139349668817, -8794491537166765, -184459444459530193, 3483053621920936363, 15570880115951580635
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 04 2021

Keywords

Crossrefs

Cf. A000040, A008578, A030018, A300632, A300662, A302194, A346430.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[1/(1 + x + Sum[Prime[k - 1] x^k/k!, {k, 2, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n,k) * A008578(k) * a(n-k).

A225132 Convolutory inverse of the Thue Morse sequence.

Original entry on oeis.org

1, -1, 1, -2, 3, -4, 5, -8, 12, -16, 22, -32, 45, -62, 87, -124, 175, -244, 343, -484, 679, -952, 1339, -1884, 2646, -3716, 5224, -7342, 10313, -14490, 20365, -28618, 40210, -56502, 79400, -111570, 156769, -220290, 309553, -434974, 611210, -858864, 1206862
Offset: 1

Views

Author

Clark Kimberling, Apr 29 2013

Keywords

Comments

Coefficients in 1/(1+g(x)), where g is the generating function of the Thue-Morse sequence, A010060. Conjecture: a(n+1)/a(n) -> -1.405177106052... .

Examples

			(1,1,0,1,0,0,1,1,...)**(1,-1,1,-2,3,-4,5,-8,...) = (1,0,0,0,0,...), where ** denotes convolution.

Links

Crossrefs

Cf. A030018, A010060.

Programs

  • Mathematica
    r[n_] := If[n == 0, 0, If[Mod[n, 2] == 0, r[n/2], 1 - r[(n - 1)/2]]]; Table[r[n], {n, 1, 30}]; (* A010060 *) k[n_] := k[n] = 0; k[1] = 1; s[n_] := s[n] = (k[n] - Sum[r[k]*s[n - k + 1], {k, 2, n}])/r[1]; Table[s[n], {n, 1, 60}]

A247818 Decimal expansion of 1/(theta*P'(theta)), a constant appearing in the asymptotic evaluation of the coefficients q_n in 1/(1+P(x)), where P(x) is the generating function of the primes and theta the unique zero of P(x) in [-3/4, 0].

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Sep 24 2014

Keywords

Examples

			-0.622306574570085664621341181270009605130784301479...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.5. Kalmár’s Composition Constant, p. 294 and p. 551.

Crossrefs

Cf. A030018, A072508, A088751.

Programs

  • Mathematica
    digits = 100; P[x_] := 1 + Sum[Prime[n]*x^n, {n, 1, 1000}]; PPrime[x_] := Sum[n*Prime[n]*x^(n-1), {n, 1, 1000}]; theta = x /. FindRoot[P[x] == 0, {x, -3/4}, WorkingPrecision -> digits+5]; RealDigits[1/(theta*PPrime[theta]), 10, digits] // First

Formula

q_n ~ (1/(theta*P'(theta))) * (1/theta^n).

A300661 Expansion of e.g.f. exp(-Sum_{k>=1} prime(k)*x^k/k!).

Original entry on oeis.org

1, -2, 1, 5, 4, -53, -177, 282, 5759, 20355, -83420, -1420133, -6245485, 29035652, 648899541, 4034393367, -10488623858, -464971765297, -4310935438663, -3489419105786, 446500913437911, 6423072226704027, 30987397708208720, -462727554963927783, -11862200720684515159
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 10 2018

Keywords

Examples

			E.g.f.: A(x) = 1 - 2*x/1! + x^2/2! + 5*x^3/3! + 4*x^4/4! - 53*x^5/5! - 177*x^6/6! + 282*x^7/7! + ...

Links

Crossrefs

Cf. A000040, A007446, A007447, A030017, A030018, A300632.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, -add(a(n-j)*
      ithprime(j)*binomial(n-1, j-1), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 10 2018
  • Mathematica
    nmax = 24; CoefficientList[Series[Exp[-Sum[Prime[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[-Prime[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}]

Formula

E.g.f.: exp(-Sum_{k>=1} A000040(k)*x^k/k!).

A303073 L.g.f.: log(1 + Sum_{k>=1} prime(k)*x^k) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

2, 2, 5, 2, 12, -13, 16, -30, 41, -18, 46, -73, 132, -278, 315, -318, 580, -805, 1218, -1998, 2665, -3958, 5936, -7761, 11612, -17678, 25313, -38134, 54754, -76833, 114392, -166334, 240685, -356454, 515996, -748441, 1095572, -1581482, 2303163, -3375550, 4903684, -7149365, 10417010, -15111622
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 18 2018

Keywords

Examples

			L.g.f.: L(x) = 2*x + 2*x^2/2 + 5*x^3/3 + 2*x^4/4 + 12*x^5/5 - 13*x^6/6 + 16*x^7/7 - 30*x^8/8 + 41*x^9/9 - 18*x^10/10 + ...
exp(L(x)) = 1 + 2*x + 3*x^2 + 5*x^3 + 7*x^4 + 11*x^5 + 13*x^6 + 17*x^7 + 19*x^8 + 23*x^9 + 29*x^10 + ... + A000040(n)*x^n + ...

Crossrefs

Cf. A000040, A007446, A007447, A008578, A030017, A030018, A302194.

Programs

  • Mathematica
    nmax = 44; Rest[CoefficientList[Series[Log[1 + Sum[Prime[k] x^k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]

A346792 G.f.: 1 / (1 + x + Sum_{k>=2} prime(k-1) * x^k).

Original entry on oeis.org

1, -1, -1, 0, 0, 1, 0, 3, 1, -1, -4, -10, 3, 0, 9, 19, 9, 2, -44, -27, -40, -3, 95, 75, 156, -36, -181, -274, -349, 81, 205, 982, 832, 35, -596, -2587, -1803, -1259, 2118, 5876, 5365, 4922, -6811, -12175, -17181, -12932, 14144, 28575, 53548, 27663, -19181
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 04 2021

Keywords

Crossrefs

Cf. A000040, A008578, A030017, A030018, A300662, A346791.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/(1 + x + Sum[Prime[k - 1] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A008578(k) * a(n-k).

A074142 a(n) is the coefficient of x^n in x/(1 + Sum_{k>=1} (1/2)*(prime(k+1) - 1)*x^k).

Original entry on oeis.org

1, -1, -1, 0, 0, 2, 1, 0, -2, -5, 2, 3, 5, 6, -10, -12, -9, 11, 32, 11, -5, -55, -61, 29, 84, 129, 9, -188, -232, -136, 322, 567, 255, -354, -1185, -840, 585, 2038, 2318, -594, -3909, -4761, -929, 7387, 10441, 3930, -11137, -23097, -12215, 16547, 44716, 36786, -23108
Offset: 1

Views

Author

Zak Seidov, Sep 16 2002

Keywords

Comments

The series reciprocal to the series with coefficients in A005097 has (integer) coefficients with irregular signs and values. In contrast the series reciprocal to the series with coefficients = primes themselves has coefficients (A030018) with alternating signs and regular growth. The radius of convergence (defined from consecutive coefficients ratio) of that series is 0.686777834460.

Crossrefs

Cf. A005097, A030018.

Formula

G.f.: x / (1 + Sum_{i>=1} A005097(i)*x^i).

A333372 a(n) = prime(n)# - Sum_{k=1..n-1} prime(k)# * a(n-k), where prime()# = A002110.

Original entry on oeis.org

1, 2, 2, 14, 110, 1526, 20858, 388718, 7614806, 183489830, 5561422394, 174355099682, 6611001621542, 274364327631194, 11859712864132730, 562348710109678226, 30121397642062848278, 1792060484721674304638, 109581661931207939415266, 7388364108380826136619810
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2020

Keywords

Links

Crossrefs

Cf. A000698, A002110, A003319, A030018, A167894, A307364.

Programs

  • Mathematica
    primorial[n_] := Product[Prime[k], {k, 1, n}]; a[n_] := a[n] = primorial[n] - Sum[primorial[k] a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 0, 19}]

Formula

G.f.: 2 - 1 / (Sum_{k>=0} prime(k)# * x^k).

A353156 a(0) = 1; a(n) = -Sum_{k=1..n} prime(k+1) * a(n-k).

Original entry on oeis.org

1, -3, 4, -4, 2, 6, -22, 46, -74, 86, -40, -120, 450, -958, 1506, -1694, 744, 2500, -9184, 19422, -30450, 34032, -14178, -52286, 188038, -394724, 615102, -681110, 268666, 1089974, -3847390, 8021030, -12426638, 13632728, -5063588, -22711916, 78708912, -162966020, 251005706
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 27 2022

Keywords

Crossrefs

Cf. A006005, A030017, A030018, A065091, A292744, A300662, A307898, A307899, A346792.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = -Sum[Prime[k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 38}]
nmax = 38; CoefficientList[Series[1/(1 + Sum[Prime[k + 1] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1 / (1 + Sum_{k>=1} prime(k+1) * x^k).

A368862 Numerators of an infinite series that converges to the negative inverse of Backhouse's constant (A088751).

Original entry on oeis.org

-1, -3, 1, 1, -1, 5, -19, -9, 41, -103, 17, 289, -169, 331, -689, -4991, 3999, 7833, -6509, 21827, -22165, -87637, 119441, -190981, -152513, 1482023, -425985, -1045091, 1071237, -14108791, 5845271, 39852203, -35832801, 54451699, 44061359, -435442725, 261309855, -22217917
Offset: 1

Views

Author

Raul Prisacariu, Jan 08 2024

Keywords

Comments

Whittaker's root series formula is applied to 1 + Sum_{k>=1} prime(k) x^k. The following infinite series that converges to the negative inverse of Backhouse's constant (-x) is obtained:
x = -1/(1*2) - 3/(2*1) + 1/(1*1) + 1/(1*2) - 1/(2*3) + 5/(3*7) - 19/(7*10) - 9/(10*13) + 41/(13*21) - 103/(21*26) + 17/(26*33) + 289/(33*53) ...
The denominators of the infinite series are obtained by multiplying the absolute values of 2 consecutive terms from the sequence A030018.

Examples

			a(1) = -1;
a(2) = -3;
a(3) = -det ToeplitzMatrix((3,2),(3,5)) = 1;
a(4) = -det ToeplitzMatrix((3,2,1),(3,5,7)) = 1;
a(5) = -det ToeplitzMatrix((3,2,1,0),(3,5,7,11)) = -1;
a(6) = -det ToeplitzMatrix((3,2,1,0,0),(3,5,7,11,13)) = 5;
a(7) = -det ToeplitzMatrix((3,2,1,0,0,0),(3,5,7,11,13,17)) = -19.

Links

Crossrefs

Cf. A088751, A030018, A072508.

Formula

a(1) = -1.
For n > 1, a(n) = -det ToeplitzMatrix((c(2),c(1),c(0),0,0,...,0),(c(2),c(3),c(4),...,c(n))), where c(0)=1 and c(n) is the n-th prime number.

Extensions

a(21)-a(38) from Stefano Spezia, Jan 09 2024
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