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 11-17 of 17 results.

A294458 E.g.f.: Product_{n>=1} (1 - x^(2*n-1))^(1/(2*n-1)).

Original entry on oeis.org

1, -1, 0, -2, 8, -24, 64, -160, 8448, -86912, 509696, -1449216, 44615680, -366395392, 3315376128, -190488356864, 4591008579584, -33244620718080, 86342088982528, -2543409132470272, 136456182420996096, -5644134983026343936, 103753337226615848960
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2017

Keywords

Links

Crossrefs

Cf. A081362, A206303, A285069, A294459.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, d%2)*x^k/k))))

Formula

E.g.f.: exp(-Sum_{n>=1} A001227(n) * x^n / n).
a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} A001227(k)*a(n-k)/(n-k)! for n > 0.

A295833 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^((-1)^k/k).

Original entry on oeis.org

1, -1, 3, -11, 47, -279, 2089, -16057, 137409, -1417553, 15656651, -187422531, 2501688463, -34832785831, 529520417217, -8723102543009, 146573712239489, -2670058109819937, 52017332039568019, -1041334898093864443, 22335551258991482991, -502509800119879530551, 11641825391540821682393
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2017

Keywords

Examples

			E.g.f.: Sum_{n>=0} a(n)*x^n/n! = ((1 + x^2)^(1/2)*(1 + x^4)^(1/4)*(1 + x^6)^(1/6)* ...)/((1 + x)*(1 + x^3)^(1/3)*(1 + x^5)^(1/5)* ...) = 1 - x + 3*x^2/2! - 11*x^3/3! + 47*x^4/4! - 279*x^5/5! + 2089*x^6/6! - 16057*x^7/7! + ...

Crossrefs

Cf. A028342, A168243, A206303, A284474, A294356, A295792, A295834.

Programs

  • Maple
    a:=series(mul((1+x^k)^((-1)^k/k),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[Product[(1 + x^k)^((-1)^k/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A295834 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^((-1)^(k+1)/k).

Original entry on oeis.org

1, 1, -1, -1, 11, 19, -311, -1919, 20201, 154169, -1363249, -14236289, 140759299, 1213688059, -33239720359, -257577468511, 11707385639249, 119005356808561, -3416942071608929, -43117983466829441, 893917358612502011, 13133282766425234531, -411010168576899605911, -7970128344774479644991
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 28 2017

Keywords

Examples

			E.g.f.: Sum_{n>=0} a(n)*x^n/n! = ((1 + x)*(1 + x^3)^(1/3)*(1 + x^5)^(1/5)* ...)/((1 + x^2)^(1/2)*(1 + x^4)^(1/4)*(1 + x^6)^(1/6)* ...) = 1 + x - x^2/2! - x^3/3! + 11*x^4/4! + 19*x^5/5! - 311*x^6/6! - 1919*x^7/7! + ...

Crossrefs

Cf. A028342, A168243, A206303, A284467, A294356, A295792, A295833.

Programs

  • Maple
    a:=series(mul((1+x^k)^((-1)^(k+1)/k),k=1..100),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[Product[(1 + x^k)^((-1)^(k+1)/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

A318968 Expansion of exp(Sum_{k>=1} ( Sum_{d|k, d odd} d^k ) * x^k/k).

Original entry on oeis.org

1, 1, 1, 10, 10, 635, 797, 118446, 124071, 43174194, 45404910, 25982930761, 26443958420, 23324558686914, 23640266984002, 29216576615057082, 29447535265299613, 48690644491136860817, 48980258924147884960, 104176334607664412086539, 104636388540330684649083, 278323070872780066332365486
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 06 2018

Keywords

Crossrefs

Cf. A000009, A023881, A206303, A262811, A292919, A318969.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[Sum[Sum[Mod[d, 2] d^k, {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 21; CoefficientList[Series[Product[1/(1 - (2 k - 1)^(2 k - 1) x^(2 k - 1))^(1/(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Mod[d, 2] d^k, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]

Formula

G.f.: Product_{k>=1} 1/(1 - (2*k - 1)^(2*k-1)*x^(2*k-1))^(1/(2*k-1)).

A327927 Expansion of e.g.f. exp(Sum_{i>=1} Sum_{j=1..i} x^(i*j) / (i*j)).

Original entry on oeis.org

1, 1, 2, 6, 30, 150, 1020, 7140, 63420, 611100, 6625080, 72875880, 977213160, 12876743880, 190951160400, 2975661889200, 51767677962000, 886225654314000, 17136230971860000, 329530590793404000, 7035395004749311200, 151961029211943151200
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 30 2019

Keywords

Crossrefs

Cf. A000005, A028260, A028342, A038548, A206303, A327940.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[Sum[Ceiling[DivisorSigma[0, k]/2] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Ceiling[DivisorSigma[0, k]/2] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
nmax = 20; CoefficientList[Series[Exp[Sum[-(x^(k*(1 + k))*LerchPhi[x^k, 1, 1 + k] + Log[1 - x^k])/k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 06 2019 *)

Formula

E.g.f.: exp(Sum_{k>=1} ceiling(A000005(k)/2) * x^k / k).
E.g.f.: exp(Sum_{k>=1} A038548(k) * x^k / k).
E.g.f.: Product_{k>=1} 1 / (1 - x^A028260(k))^(1/A028260(k)).

A327940 Expansion of e.g.f. exp(Sum_{i>=1} Sum_{j=1..i-1} x^(i*j) / (i*j)).

Original entry on oeis.org

1, 0, 1, 2, 9, 44, 385, 1854, 23233, 153656, 2151441, 18787130, 338487721, 3165541092, 60609811249, 835202858294, 14913805143105, 228441779869424, 5319673396479073, 81040768940877426, 2153026504862728201, 39759334398324543260, 988919906784578473761
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 30 2019

Keywords

Crossrefs

Cf. A000005, A026424, A028342, A056924, A206303, A327927.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[Sum[Floor[DivisorSigma[0, k]/2] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Floor[DivisorSigma[0, k]/2] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

Formula

E.g.f.: exp(Sum_{k>=1} floor(A000005(k)/2) * x^k / k).
E.g.f.: exp(Sum_{k>=1} A056924(k) * x^k / k).
E.g.f.: Product_{k>=1} 1 / (1 - x^A026424(k))^(1/A026424(k)).

A327941 Expansion of e.g.f. exp(Sum_{i>=1} Sum_{j>=2} x^(i*j) / (i*j)).

Original entry on oeis.org

1, 0, 1, 2, 15, 44, 595, 2274, 36673, 247400, 3660921, 29194010, 632617711, 5289743172, 117393123835, 1525153361354, 32315717350785, 433901475732944, 11698737221494513, 168831340268759730, 4894554062081828431, 87212857278031619420, 2398463635663863045411
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 30 2019

Keywords

Crossrefs

Cf. A000005, A028342, A032741, A206303.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[Sum[(DivisorSigma[0, k] - 1) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[(DivisorSigma[0, k] - 1) a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

Formula

E.g.f.: exp(Sum_{k>=1} (A000005(k) - 1) * x^k / k).
E.g.f.: exp(Sum_{k>=1} A032741(k) * x^k / k).
E.g.f.: Product_{k>=2} 1 / (1 - x^k)^(1/k).
Previous Showing 11-17 of 17 results.

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