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.

A295792 Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(1/k).

Original entry on oeis.org

1, 2, 6, 28, 152, 1008, 7936, 70208, 689664, 7618816, 92013824, 1202362368, 17053410304, 258928934912, 4197838491648, 72840915607552, 1334630802489344, 25799982480556032, 527187369241870336, 11292834065764450304, 253498950169144590336, 5965951790211865772032, 146341359815078034538496
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 27 2017

Keywords

Comments

Convolution of A028342 and A168243. - Vaclav Kotesovec, Sep 07 2018

Links

Crossrefs

Cf. A001227, A028342, A028343, A054844, A156616, A168243, A206303, A294356.

Programs

  • Maple
    a:=series(mul(((1+x^k)/(1-x^k))^(1/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 - x^k))^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: exp(2*Sum_{k>=1} A001227(k)*x^k/k).
E.g.f.: exp(Sum_{k>=1} A054844(k)*x^k/k).

A300188 a(n) = n! * [x^n] Product_{k>=1} 1/(1 + x^k)^(n/k).

Original entry on oeis.org

1, -1, 4, -39, 536, -9115, 185904, -4461877, 123647488, -3886461081, 136538590400, -5300491027711, 225313697972736, -10409021924850211, 519298241645107456, -27824560148201248125, 1593597443825288904704, -97153909607626767338353, 6281720886474120790582272
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 28 2018

Keywords

Examples

			The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} 1/(1 + x^k)^(n/k) begins:
n = 0: (1),  0,   0,     0,     0,       0,        0,  ...
n = 1:  1, (-1),  1,    -5,    23,    -119,      619,  ...
n = 2:  1,  -2,  (4),  -16,    92,    -568,     3856,  ...
n = 3:  1,  -3,   9,  (-39),  243,   -1737,    13671,  ...
n = 4:  1,  -4,  16,   -80,  (536),  -4256,    37504,  ...
n = 5:  1,  -5,  25,  -145,  1055,  (-9115),   88075,  ...
n = 6:  1,  -6,  36,  -240,  1908,  -17784,  (185904), ...

Links

Crossrefs

Cf. A048272, A281266, A294356, A299033, A299034, A300187.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Product[1/(1 + x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 18}]

Formula

a(n) = n! * [x^n] exp(-n*Sum_{k>=1} A048272(k)*x^k/k).
a(n) ~ (-1)^n * c * d^n * n^n, where d = 1.3587950730244927060955... and c = 0.6449711831436950784... - Vaclav Kotesovec, Sep 08 2018

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]!

A294465 Expansion of e.g.f. Product_{k>0} (1+k*x^k)^(-1/k).

Original entry on oeis.org

1, -1, 0, -6, 36, -180, 720, -7560, 236880, -3099600, 15120000, -194594400, 9989179200, -131935003200, 337815878400, -50154760656000, 2018231927712000, -27611162875296000, 363290246871552000, -12648028196067264000, 521752941995725440000
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2017

Keywords

Crossrefs

Cf. A076717, A294356, A294462, A294463, A294464.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, (1+k*x^k)^(1/k))))

Formula

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^k*j^(k-1)*x^(j*k)/k). - Ilya Gutkovskiy, May 28 2018
a(0) = 1 and a(n) = -(n-1)! * Sum_{k=1..n} A076717(k)*a(n-k)/(n-k)! for n > 0. - Seiichi Manyama, Jan 21 2025

A296048 Expansion of e.g.f. Product_{k>=1} ((1 - x^k)/(1 + x^k))^(1/k).

Original entry on oeis.org

1, -2, 2, -4, 32, -128, 496, -2336, 29312, -395776, 3194624, -21951488, 277270528, -4027191296, 38850203648, -739834458112, 19460560584704, -299971773661184, 3169121209090048, -51853341314514944, 1234704403684130816, -30653318499154788352, 658369600764729884672, -10809496145754051313664
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2017

Keywords

Crossrefs

Cf. A001227, A028342, A028343, A054844, A156616, A168243, A285675, A294356, A295792.

Programs

  • Maple
    a:=series(mul(((1-x^k)/(1+x^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 + x^k))^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[-2 Sum[Total[Mod[Divisors[k], 2] x^k]/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: exp(-2*Sum_{k>=1} A001227(k)*x^k/k).
E.g.f.: exp(-Sum_{k>=1} A054844(k)*x^k/k).
Showing 1-6 of 6 results.

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