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 21-28 of 28 results.

A293121 Expansion of e.g.f. exp(x^3/(1+x)).

Original entry on oeis.org

1, 0, 0, 6, -24, 120, -360, 0, 20160, -302400, 3628800, -39916800, 419126400, -4151347200, 36324288000, -207048441600, -1743565824000, 103742166528000, -2925529096089600, 69945932735078400, -1571249213614080000, 34354603773794304000, -741528257908838400000
Offset: 0

Views

Author

Seiichi Manyama, Sep 30 2017

Keywords

Links

Crossrefs

Column k=2 of A293133.
Cf. A111884, A293120.
Cf. A293118.

Programs

  • Mathematica
    CoefficientList[Series[E^(x^3/(1+x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(x^3/(1+x))))

Formula

E.g.f.: exp(x^3/(1+x)).
a(n) = (-1)^n * A293118(n).

A293573 E.g.f.: exp(x/(1 + x + x^2 + x^3 + x^4)).

Original entry on oeis.org

1, 1, -1, -5, 1, 41, 751, -461, -61375, -114479, 3197791, 51784811, 95861569, -13289759495, -88386566449, 1959369708451, 39596487031681, 144047673616289, -17707296608390465, -244965944457878309, 5109519579305499521, 149507697206554889161
Offset: 0

Views

Author

Seiichi Manyama, Oct 12 2017

Keywords

Crossrefs

Cf. A111884, A293571, A293572.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(1+x+x^2+x^3+x^4))))
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(5*k-4)-x^(5*k-3)))))

Formula

E.g.f.: Product_{k>0} exp(x^(5*k-4)) / exp(x^(5*k-3)).

A317278 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*k^n*n!/k!.

Original entry on oeis.org

1, 1, 2, -15, -164, 4245, 46386, -4901939, 39141656, 11707820361, -671114863610, -29398709945319, 7385525824325364, -307076643365636963, -73748845974115224262, 14299745046516639280005, -237996466462017367478864, -377740669670216316717155055, 75515477307532501838072029326
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 25 2018

Keywords

Comments

a(n) is the n-th term of the inverse Lah transform of the n-th powers.

Links

Crossrefs

Cf. A111884, A256467, A317277, A317279.

Programs

  • Magma
    [1]cat[(&+[(-1)^(n+j)*Binomial(n-1,j-1)*Binomial(n,j)*Factorial(n-j)*j^n: j in [0..n]]): n in [1..30]]; // G. C. Greubel, Mar 09 2021
  • Maple
    A317278:= n-> `if`(n=0,1, add((-1)^(n+j)*binomial(n-1,j-1)*binomial(n,j)*(n-j)!*j^n, j=0..n));
seq(A317278(n), n=0..30); # G. C. Greubel, Mar 09 2021
  • Mathematica
    Join[{1}, Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] k^n n!/k!, {k, n}], {n, 18}]]
Join[{1}, Table[n! SeriesCoefficient[Sum[k^n (x/(1 + x))^k/k!, {k, n}], {x, 0, n}], {n, 18}]]
  • PARI
    a(n) = if (n==0, 1, sum(k=0, n, (-1)^(n-k)*binomial(n-1, k-1)*k^n*n!/k!)); \\ Michel Marcus, Mar 10 2021; corrected Jun 13 2022
  • Sage
    [1]+[sum((-1)^(n+j)*binomial(n-1,j-1)*binomial(n,j)*factorial(n-j)*j^n for j in (0..n)) for n in (1..30)] # G. C. Greubel, Mar 09 2021

Formula

a(n) = n! * [x^n] Sum_{k>=0} k^n*(x/(1 + x))^k/k!.

A293589 E.g.f.: exp(x^2/(1 + x + x^2)).

Original entry on oeis.org

1, 0, 2, -6, 12, 0, -240, 2520, -18480, 60480, 937440, -21621600, 220207680, -311351040, -34490776320, 724669545600, -6625031212800, -49471604582400, 3116728731916800, -58942964451571200, 335128094882380800, 15732203147781120000, -600651799248659558400
Offset: 0

Views

Author

Seiichi Manyama, Oct 12 2017

Keywords

Links

Crossrefs

Cf. A111884, A293590.

Programs

  • Maple
    rec:= (n+3)*(n+2)*(n+1)*n*b(n)+(2*n+1)*(n+3)*(n+2)*b(n+1)+(3*n+4)*(n+3)*b(n+2)+2*(n+3)*b(n+3)+b(n+4)=0:
f:= gfun:-rectoproc({rec,b(0)=1,b(1)=0,b(2)=2,b(3)=-6},b(n),remember):
map(f, [$0..30]); # Robert Israel, Oct 27 2019
  • Mathematica
    CoefficientList[Series[E^(x^2/(1 + x + x^2)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 13 2017 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(x^2/(1+x+x^2))))
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(3*k-1)-x^(3*k)))))

Formula

E.g.f.: Product_{k>0} exp(x^(3*k-1)) / exp(x^(3*k)).
(n+3)*(n+2)*(n+1)*n*a(n)+(2*n+1)*(n+3)*(n+2)*a(n+1)+(3*n+4)*(n+3)*a(n+2)+2*(n+3)*a(n+3)+a(n+4)=0. - Robert Israel, Oct 27 2019

A317364 Expansion of e.g.f. exp(2*x/(1 + x)).

Original entry on oeis.org

1, 2, 0, -4, 16, -48, 64, 800, -12288, 127232, -1150976, 9266688, -58726400, 68777984, 7510646784, -207794409472, 4241007640576, -77359570944000, 1321952191971328, -21274345818161152, 313768799799607296, -3838962981483839488, 21775623343518515200, 859024717017756205056
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 26 2018

Keywords

Comments

Inverse Lah transform of the powers of 2 (A000079).

Links

Crossrefs

Cf. A000079, A052897, A111884.
Cf. A001263, A008297, A086915.

Programs

  • Magma
    [n eq 0 select 1 else (-1)^n*Factorial(n)*Evaluate(LaguerrePolynomial(n, -1), 2): n in [0..25]]; // G. C. Greubel, Feb 23 2021
  • Maple
    a:= proc(n) option remember; add((-1)^(n-k)*
      n!/k!*binomial(n-1, k-1)*2^k, k=0..n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2018
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[2 x/(1 + x)], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Product[Exp[-2 (-x)^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n-k) Binomial[n-1, k-1] 2^k n!/k!, {k, 0, n}], {n, 0, 23}]
Join[{1}, Table[2 (-1)^(n+1) n! Hypergeometric1F1[1-n, 2, 2], {n, 23}]]
  • PARI
    a(n) = if (n==0, 1, (-1)^n*n!*pollaguerre(n, -1, 2)); \\ Michel Marcus, Feb 23 2021
  • Sage
    [1 if n==0 else (-1)^n*factorial(n)*gen_laguerre(n, -1, 2) for n in (0..25)] # G. C. Greubel, Feb 23 2021

Formula

E.g.f.: Product_{k>=1} exp(-2*(-x)^k).
a(n) = 2*(-1)^(n+1) * n! * Hypergeometric1F1([1-n], [2], 2).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n-1,k-1)*2^k*n!/k!.
(n^2 + n)*a(n) + 2*n*a(n+1) + a(n+2) = 0. - Robert Israel, Aug 18 2019
From G. C. Greubel, Feb 23 2021: (Start)
a(n) = (-1)^n * n! * Laguerre(n, -1, 2) for n > 0 with a(0) = 1.
a(n) = Sum_{k=0..n} (-1)^(n-k) * A086915(n, k).
a(n) = (-1)^n * Sum_{k=0..n} 2^k * A008297(n, k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * (n-k+1)! * A001263(n, k). (End)

A318223 Expansion of e.g.f. exp(x/(1 + 2*x)).

Original entry on oeis.org

1, 1, -3, 13, -71, 441, -2699, 9157, 206193, -8443151, 236126701, -6169406979, 161388751657, -4327824442967, 120012465557349, -3450029411174219, 102741264191105761, -3160671409312412703, 99982488984008583133, -3230094912866216253971, 105481073534842477321881, -3423260541695907002392679
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 21 2018

Keywords

Links

Crossrefs

Cf. A002866, A025168, A111884, A122803.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x/(1+2*x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Feb 07 2019
  • Maple
    seq(n!*coeff(series(exp(x/(1+2*x)),x=0,22),x,n),n=0..21); # Paolo P. Lava, Jan 09 2019
  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[x/(1 + 2 x)], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-2)^(n - k) Binomial[n - 1, k - 1] n!/k!, {k, 0, n}], {n, 0, 21}]
a[n_] := a[n] = Sum[(-2)^(k - 1) k! Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 21}]
Join[{1}, Table[(-2)^(n - 1) n! Hypergeometric1F1[1 - n, 2, 1/2], {n, 21}]]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(x/(1+2*x)))) \\ G. C. Greubel, Feb 07 2019
  • Sage
    m = 30; T = taylor(exp(x/(1+2*x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 07 2019

Formula

E.g.f.: Product_{k>=1} exp((-2)^(k-1)*x^k).
a(n) = Sum_{k=0..n} (-2)^(n-k)*binomial(n-1,k-1)*n!/k!.
a(0) = 1; a(n) = Sum_{k=1..n} (-2)^(k-1)*k!*binomial(n-1,k-1)*a(n-k).

A318224 a(n) = n! * [x^n] exp(x/(1 + n*x)).

Original entry on oeis.org

1, 1, -3, 37, -1007, 47901, -3514499, 367671697, -51952729023, 9529552851193, -2201241933756899, 625136460673954461, -214066473170125310063, 86976878219664125966677, -41368038169392401671082787, 22767783580493235411255966601, -14356419990032448099044028030719
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 21 2018

Keywords

Crossrefs

Cf. A111884, A293146, A317279, A318223.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[x/(1 + n x)], {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[(-n)^(n - k) Binomial[n - 1, k - 1] n!/k!, {k, n}], {n, 16}]]
Join[{1}, Table[(-1)^(n + 1) n^n (n - 1)! Hypergeometric1F1[1 - n, 2, 1/n], {n, 16}]]
Flatten[{1, Table[-(-1)^n * n^(n-1) * (n-1)! * LaguerreL[n-1, 1, 1/n], {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 21 2018 *)

Formula

a(n) = n! * [x^n] Product_{k>=1} exp((-n)^(k-1)*x^k).
a(n) = Sum_{k=0..n} (-n)^(n-k)*binomial(n-1,k-1)*n!/k!.
a(n) ~ -(-1)^n * c * n^(2*n - 1/2) / exp(n), where c = BesselJ(1,2) * sqrt(2*Pi) = 1.44563470980450699365002928132323794056211645203313522173628289... - Vaclav Kotesovec, Aug 21 2018

A293590 E.g.f.: exp(x^3/(1 + x + x^2 + x^3)).

Original entry on oeis.org

1, 0, 0, 6, -24, 0, 360, 0, -20160, 60480, 1814400, -19958400, -59875200, 2075673600, 21794572800, -860885625600, 3487131648000, 148203095040000, -524638956441600, -71973351075225600, 942749528168448000, 19878836825143296000, -437111394135736320000
Offset: 0

Views

Author

Seiichi Manyama, Oct 12 2017

Keywords

Crossrefs

Cf. A111884, A293589.

Programs

  • Mathematica
    CoefficientList[Series[E^(x^3/(1 + x + x^2 + x^3)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 13 2017 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(x^3/(1+x+x^2+x^3))))
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, exp(x^(4*k-1)-x^(4*k)))))

Formula

E.g.f.: Product_{k>0} exp(x^(4*k-1)) / exp(x^(4*k)).
Previous Showing 21-28 of 28 results.

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