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 1-7 of 7 results.

A300662 Expansion of 1/(1 - x - Sum_{k>=2} prime(k-1)*x^k).

Original entry on oeis.org

1, 1, 3, 8, 22, 59, 160, 429, 1155, 3105, 8354, 22474, 60457, 162636, 437509, 1176941, 3166097, 8517138, 22912002, 61635707, 165806564, 446037175, 1199887133, 3227823181, 8683185454, 23358686444, 62837334885, 169039070970, 454732963567, 1223279724439, 3290751724917
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 10 2018

Keywords

Comments

Invert transform of A008578.

Links

Crossrefs

Cf. A000040, A008578, A023626, A030011, A030012, A030013, A030014, A030015, A030016, A030017, A030018, A292744, A300632.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
     `if`(j=1, 1, ithprime(j-1))*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 10 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[1/(1 - x - Sum[Prime[k - 1] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
p[1] = 1; p[n_] := p[n] = Prime[n - 1]; a[n_] := a[n] = Sum[p[k] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 30}]

Formula

G.f.: 1/(1 - Sum_{k>=1} A008578(k)*x^k).

A343622 E.g.f.: log(1 + x + Sum_{k>=2} prime(k-1) * x^k / k!).

Original entry on oeis.org

1, 1, -1, -1, 6, -1, -79, 214, 1378, -11321, -14855, 611932, -1739312, -34374895, 311453831, 1548864398, -42005057494, 66254532775, 5287751144127, -45726542532086, -568193240268798, 12768316133375343, 16933257518347115, -3008868695961855284, 21477836260078982762
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 04 2021

Keywords

Crossrefs

Cf. A000040, A007446, A007447, A008578, A300632, A336185, A343623.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Log[1 + x + Sum[Prime[k - 1] x^k/k!, {k, 2, nmax}]], {x, 0, nmax}], x] Range[0, nmax]! // Rest

Formula

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

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!).

A343623 E.g.f.: -log(1 - x - Sum_{k>=2} prime(k-1) * x^k / k!).

Original entry on oeis.org

1, 3, 11, 59, 416, 3683, 39093, 484220, 6854176, 109150227, 1931303809, 37589753206, 798135918850, 18358887315769, 454779141016707, 12070296596154136, 341715021307029876, 10278722402921420619, 327369178071821161755, 11005696560250745851048, 389469699942038630639524
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 04 2021

Keywords

Crossrefs

Cf. A000040, A007446, A007447, A008578, A300632, A336185, A343622.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[-Log[1 - x - Sum[Prime[k - 1] x^k/k!, {k, 2, nmax}]], {x, 0, nmax}], x] Range[0, nmax]! // Rest

Formula

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

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).

A353079 Exponential transform of odd primes.

Original entry on oeis.org

1, 3, 14, 79, 521, 3876, 31935, 287225, 2791122, 29066589, 322292257, 3784650052, 46857941291, 609360372095, 8296220760974, 117914344818807, 1745211622467633, 26838798853062516, 428009369349905497, 7065576909286562195, 120545067517808693300, 2122393931891338237325, 38512344746420591905771
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 22 2022

Keywords

Crossrefs

Cf. A007446, A065091, A300632.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      ithprime(j+1)*binomial(n-1, j-1), j=1..n))
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 27 2022
  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[Sum[Prime[k + 1] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Prime[k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

Formula

E.g.f.: exp( Sum_{k>=1} prime(k+1) * x^k / k! ).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * prime(k+1) * a(n-k).
Showing 1-7 of 7 results.

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