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 1-6 of 6 results.

A307898 Expansion of 1/(1 - x * Sum_{k>=1} prime(k)*x^k).

Original entry on oeis.org

1, 0, 2, 3, 9, 19, 48, 107, 258, 594, 1405, 3277, 7693, 18004, 42203, 98834, 231592, 542497, 1271003, 2977529, 6975674, 16342011, 38285178, 89691782, 210124363, 492265243, 1153247379, 2701752062, 6329489153, 14828313076, 34738805240, 81383803849, 190660665579, 446667359857, 1046423138962
Offset: 0

Views

Author

Ilya Gutkovskiy, May 04 2019

Keywords

Comments

Antidiagonal sums of square array, in which row m equals the m-fold convolution of primes with themselves.

Crossrefs

Cf. A000040, A030017, A030018, A300662, A307899.

Programs

  • Mathematica
    nmax = 34; CoefficientList[Series[1/(1 - x Sum[Prime[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Prime[k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 34}]

Formula

Recurrence: a(n+1) = Sum_{k=1..n} prime(k)*a(n-k).

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).

A180129 Expansion of log(1/(1-Prime(x))) where Prime(x) = Sum{n>=1} A008578(n)*x^n.

Original entry on oeis.org

1, 5, 16, 49, 136, 380, 1016, 2745, 7369, 19840, 53395, 143620, 386374, 1039386, 2796001, 7521561, 20233860, 54431237, 146425762, 393900764, 1059634634, 2850529093, 7668224519, 20628334404, 55492399461, 149280418500, 401580100348, 1080292905482, 2906102072179, 7817721667285
Offset: 1

Views

Author

Vladimir Kruchinin, Aug 12 2010

Keywords

Links

Crossrefs

Cf. A008578, A300662.

Programs

  • PARI
    seq(n)={Vec(deriv(log(1/(1 - x - x*sum(k=1, n-1, prime(k)*x^k, O(x^n))))))} \\ Andrew Howroyd, Jan 04 2020

Formula

Logarithm g.f.: Sum_{n>0} a(n)*x^n/n = log(1/(1-Prime(x))).
Logarithmic derivative of A300662. - Andrew Howroyd, Jan 04 2020

Extensions

Terms a(16) and beyond from Andrew Howroyd, Jan 04 2020

A346430 E.g.f.: 1 / (1 - x - Sum_{k>=2} prime(k-1) * x^k / k!).

Original entry on oeis.org

1, 1, 4, 21, 149, 1317, 13985, 173207, 2451807, 39043963, 690844441, 13446183857, 285500221447, 6567135007015, 162678487750465, 4317650962178897, 122234460353464081, 3676789159574231397, 117102826395968235853, 3936834192059910096205, 139316727760914366716635
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 04 2021

Keywords

Crossrefs

Cf. A000040, A008578, A030017, A300632, A300662, A302194, A346791.

Programs

  • Mathematica
    nmax = 20; 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).

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).

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).
Showing 1-6 of 6 results.

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