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.

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A282132 Imaginary part of n!*Sum_{k=1..n} i^(k-1)/k, where i is sqrt(-1).

Original entry on oeis.org

0, 1, 3, 6, 30, 300, 2100, 11760, 105840, 1421280, 15634080, 147692160, 1919998080, 33106993920, 496604908800, 6638004172800, 112846070937600, 2386916704972800, 45351417394483200, 785383247480832000, 16493048197097472000, 413938002507853824000
Offset: 1

Views

Author

Daniel Suteu, Feb 06 2017

Keywords

Examples

			For n=5, a(5) = 30, which is the imaginary part of 5!*(1/1 + i/2 - 1/3 - i/4 + 1/5) = 104+30*i.

Links

Crossrefs

The corresponding real part is A281964.
Cf. A016655, A024167.

Programs

  • PARI
    a(n) = imag(n!*sum(k=1, n, I^(k-1)/k));

Formula

a(n) ~ log(sqrt(2)) * n!.
a(1) = 0, a(n+1) = a(n)*(n+1) + n!*sin(Pi*n/2).

A305307 Expansion of e.g.f. 1/(1 - log(1 + x)/(1 - x)).

Original entry on oeis.org

1, 1, 3, 17, 120, 1084, 11642, 146446, 2101656, 33958344, 609431232, 12033015840, 259163792016, 6047213451408, 151953760489008, 4091057804809104, 117485988199385088, 3584814699783432960, 115816462543697120640, 3949619921174717629056, 141780511159572486530304, 5344008726418981985707776
Offset: 0

Views

Author

Ilya Gutkovskiy, May 29 2018

Keywords

Comments

a(n)/n! is the invert transform of [1, 1 - 1/2, 1 - 1/2 + 1/3, 1 - 1/2 + 1/3 - 1/4, 1 - 1/2 + 1/3 - 1/4 + 1/5, ...].

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 120*x^4/4! + 1084*x^5/5! + 11642*x^6/6! + ...

Links

Crossrefs

Cf. A006252, A024167, A058312, A058313, A305306.

Programs

  • Maple
    g:= proc(n) g(n):= `if`(n=1, 0, g(n-1))-(-1)^n/n end:
b:= proc(n) option remember; `if`(n=0, 1,
      add(g(j)*b(n-j), j=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..20);  # Alois P. Heinz, May 29 2018
  • Mathematica
    nmax = 21; CoefficientList[Series[1/(1 - Log[1 + x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/(1 - Sum[Sum[(-1)^(j + 1)/j, {j, 1, k}] x^k , {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[((-1)^(k + 1) LerchPhi[-1, 1, k + 1] + Log[2]) a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 21}]

Formula

E.g.f.: 1/(1 - Sum_{k>=1} (A058313(k)/A058312(k))*x^k).
a(n) ~ n! * (2 - LambertW(exp(2))) / ((1 + 1/LambertW(exp(2))) * (LambertW(exp(2)) - 1)^(n+1)). - Vaclav Kotesovec, Aug 08 2021

A080958 a(n) = n!*(2/1 - 3/2 + 4/3 - ... + s*(n+1)/n), where s = (-1)^(n+1).

Original entry on oeis.org

2, 1, 11, 14, 214, 444, 8868, 25584, 633456, 2342880, 69317280, 312888960, 10773578880, 57424792320, 2256224544000, 13869128448000, 612385401600000, 4264876094976000, 209080119919104000, 1627055289796608000, 87692005265614848000, 754132445894209536000, 44321063722229403648000, 417405110861381271552000, 26566786216598757212160000
Offset: 1

Views

Author

Paul Barry, Mar 01 2003

Keywords

Links

Crossrefs

Cf. A024167.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( (x + (x+1)*Log(1+x))/(1-x^2) ))); // G. C. Greubel, May 09 2025
  • Maple
    f:= gfun:-rectoproc({-(n+1)*a(n+1) + a(n) + n^2*(n+2)*a(n-1)=0, a(1)=2,a(2)=1},a(n),remember):
map(f, [$1..30]); # Robert Israel, Dec 26 2018
  • Mathematica
    Rest[CoefficientList[Series[(x+(x+1)*Log[1+x])/(1-x^2), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 29 2013 *)
a[n_] := n!(Log[2] + Boole[OddQ[n]] - (-1)^n LerchPhi[-1, 1, 1 + n]);
Table[a[n], {n, 1, 20}] (* Peter Luschny, Dec 26 2018 *)
  • SageMath
    def A080958_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( x/(1-x^2) + log(1+x)/(1-x) ).egf_to_ogf().list()
a=A080958_list(51); print(a[1:40]) # G. C. Greubel, May 09 2025

Formula

a(n) = n!*Sum_{j=1..n} (-1)^(j+1)*(j+1)/j.
E.g.f.: (x + (x+1)*log(1+x))/(1-x^2). - Vladeta Jovovic, Mar 03 2003
Conjecture: -(n+1)*a(n+1) + a(n) + n^2*(n+2)*a(n-1) = 0. - R. J. Mathar, Sep 27 2012, corrected for offset 1 by Robert Israel, Dec 26 2018
Conjecture verified, using the differential equation (x^3-x)*g''(x) + (5*x^2-1)*g'(x) + (3*x+1)*g(x) + 2 = 0 satisfied by the e.g.f. - Robert Israel, Dec 26 2018
a(n) ~ n! * (log(2) + 1/2 - 1/2*(-1)^n). - Vaclav Kotesovec, Sep 29 2013
a(n) = n!*(log(2) + (n mod 2) - (-1)^n*LerchPhi(-1, 1, n+1)). - Peter Luschny, Dec 26 2018
a(n) = n!*((1-(-1)^n)/2 + H(n) - H(floor(n/2))), where H(n) is the n-th harmonic number. - G. C. Greubel, May 09 2025

A346845 E.g.f.: log(1 + x) / (1 - x)^3.

Original entry on oeis.org

1, 5, 29, 186, 1374, 11352, 105048, 1070640, 11978640, 145558080, 1914027840, 27035890560, 408891369600, 6585851059200, 112656894336000, 2038285492992000, 38915729475840000, 781515776369664000, 16475855040820224000, 363685261902133248000, 8391522945839007744000
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 05 2021

Keywords

Crossrefs

Cf. A000217, A001711, A024167, A109792, A346846, A346847.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Log[1 + x]/(1 - x)^3, {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! Sum[(-1)^(k + 1) Binomial[n - k + 2, 2]/k , {k, 1, n}], {n, 1, 21}]
Table[n!*(((-1)^n*(2*n + 5) - 4*n - 5)/8 + (n+1)*(n+2)*(Log[2] - (-1)^n * LerchPhi[-1, 1, 1 + n])/2), {n, 1, 21}] // Simplify (* Vaclav Kotesovec, Aug 06 2021 *)
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(log(1+x)/(1-x)^3)) \\ Michel Marcus, Aug 06 2021

Formula

a(n) = n! * Sum_{k=1..n} (-1)^(k+1) * binomial(n-k+2,2) / k.
a(n) ~ log(2) * n^2 * n! / 2. - Vaclav Kotesovec, Aug 06 2021

A346846 E.g.f.: log(1 + x) / (1 - x)^4.

Original entry on oeis.org

1, 7, 50, 386, 3304, 31176, 323280, 3656880, 44890560, 594463680, 8453128320, 128473430400, 2079045964800, 35692494566400, 648044312832000, 12406994498304000, 249834635947008000, 5278539223415808000, 116768100285720576000, 2699047267616544768000, 65071515565786447872000
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 05 2021

Keywords

Crossrefs

Cf. A000292, A001716, A024167, A109792, A346845, A346847.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Log[1 + x]/(1 - x)^4, {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! Sum[(-1)^(k + 1) Binomial[n - k + 3, 3]/k , {k, 1, n}], {n, 1, 21}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(log(1+x)/(1-x)^4)) \\ Michel Marcus, Aug 06 2021

Formula

a(n) = n! * Sum_{k=1..n} (-1)^(k+1) * binomial(n-k+3,3) / k.
a(n) ~ log(2) * n^3 * n! / 6. - Vaclav Kotesovec, Aug 06 2021

A346847 E.g.f.: log(1 + x) / (1 - x)^5.

Original entry on oeis.org

1, 9, 77, 694, 6774, 71820, 826020, 10265040, 137275920, 1967222880, 30092580000, 489584390400, 8443643040000, 153903497126400, 2956596769728000, 59712542813952000, 1264947863784192000, 28047600771531264000, 649672514944814592000, 15692497566512836608000, 394613964462556016640000
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 05 2021

Keywords

Crossrefs

Cf. A000332, A001721, A024167, A109792, A346845, A346846.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Log[1 + x]/(1 - x)^5, {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! Sum[(-1)^(k + 1) Binomial[n - k + 4, 4]/k , {k, 1, n}], {n, 1, 21}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(log(1+x)/(1-x)^5)) \\ Michel Marcus, Aug 06 2021

Formula

a(n) = n! * Sum_{k=1..n} (-1)^(k+1) * binomial(n-k+4,4) / k.
a(n) ~ log(2) * n^4 * n! / 24. - Vaclav Kotesovec, Aug 06 2021

A383897 Expansion of e.g.f. log(1 + x)/(1 - 2*x).

Original entry on oeis.org

0, 1, 3, 20, 154, 1564, 18648, 261792, 4183632, 75345696, 1506551040, 33147751680, 795506123520, 20683638213120, 579135642946560, 17374156466688000, 555971699259648000, 18903058697617920000, 680509757426817024000, 25859377184592752640000, 1034374965738609696768000
Offset: 0

Views

Author

Seiichi Manyama, May 22 2025

Keywords

Links

Crossrefs

Cf. A024167, A384199.
Cf. A068102.

Programs

  • Magma
    [0] cat [n le 1 select 1  else  2*n * Self(n-1) - (-1)^n * Factorial(n-1): n in [1..20]]; // Vincenzo Librandi, May 23 2025
  • Mathematica
    a[n_]:= n! * Sum[(-1)^(k-1)*2^(n-k)/k,{k,1,n}];Table[a[n],{n,0,19}] (* Vincenzo Librandi, May 23 2025 *)
  • PARI
    a(n) = n!*sum(k=1, n, (-1)^(k-1)*2^(n-k)/k);

Formula

a(n) = n! * Sum_{k=1..n} (-1)^(k-1) * 2^(n-k)/k.
a(n) = 2 * n * a(n-1) - (-1)^n * (n-1)!.
a(n) = (n+1) * a(n-1) + 2 * (n-1)^2 * a(n-2).
a(n) ~ log(3/2) * 2^n * n!. - Vaclav Kotesovec, May 23 2025
a(n) = n!*((-1)^(n+1)/n + (-1)^n*LerchPhi(-1/2,1,n) + 2^n*log(3/2)) for n > 0. - Stefano Spezia, May 23 2025

A384199 Expansion of e.g.f. log(1 + x)/(1 - 3*x).

Original entry on oeis.org

0, 1, 5, 47, 558, 8394, 150972, 3171132, 76102128, 2054797776, 61643570400, 2034241452000, 73232652355200, 2856073920854400, 119955098448864000, 5397979517377171200, 259103015526429849600, 13214253812770712217600, 713569705533931031654400
Offset: 0

Views

Author

Seiichi Manyama, May 22 2025

Keywords

Links

Crossrefs

Cf. A024167, A383897.
Cf. A069015.

Programs

  • Magma
    [0] cat [n le 1 select 1  else 3 * n * Self(n-1) - (-1)^n * Factorial(n-1): n in [1..20]]; // Vincenzo Librandi, May 22 2025
  • Mathematica
    a[n_]:= n! * Sum[(-1)^(k-1)*3^(n-k)/k,{k,1,n}];Table[a[n],{n,0,18}] (* Vincenzo Librandi, May 22 2025 *)
  • PARI
    a(n) = n!*sum(k=1, n, (-1)^(k-1)*3^(n-k)/k);

Formula

a(n) = n! * Sum_{k=1..n} (-1)^(k-1) * 3^(n-k)/k.
a(n) = 3 * n * a(n-1) - (-1)^n * (n-1)!.
a(n) = (2*n+1) * a(n-1) + 3 * (n-1)^2 * a(n-2).
a(n) ~ log(4/3) * 3^n * n!. - Vaclav Kotesovec, May 23 2025

A073480 Triangle T(n,k) read by rows, where e.g.f. for T(n,k) is exp(x*y)*log(1+x)/(1-x).

Original entry on oeis.org

1, 1, 2, 5, 3, 3, 14, 20, 6, 4, 94, 70, 50, 10, 5, 444, 564, 210, 100, 15, 6, 3828, 3108, 1974, 490, 175, 21, 7, 25584, 30624, 12432, 5264, 980, 280, 28, 8, 270576, 230256, 137808, 37296, 11844, 1764, 420, 36, 9, 2342880, 2705760, 1151280, 459360, 93240
Offset: 1

Views

Author

Vladeta Jovovic, Aug 26 2002

Keywords

Crossrefs

Cf. A073107.

Programs

  • Maple
    G:=exp(x*y)*ln(1+x)/(1-x): Gser:=series(G,x=0,12): for n from 1 to 10 do P[n]:=n!*coeff(Gser,x^n) od: for n from 1 to 10 do seq(coeff(y*P[n],y^k),k=1..n) od; # Emeric Deutsch, Dec 14 2004

Formula

E.g.f. for k-th column is x^k/k!*log(1+x)/(1-x).
O.g.f. for n-th row is Sum_{i=0..n} binomial(n, i)*A024167(n-i)*y^i.

Extensions

More terms from Emeric Deutsch, Dec 14 2004

A073590 Expansion of e.g.f. exp(x) * log(1+x)/(1-x).

Original entry on oeis.org

1, 3, 11, 44, 229, 1339, 9603, 75200, 690009, 6779803, 75792507, 896040188, 11811267389, 163229695459, 2478388484947, 39203092296480, 673698509829233, 12002969025435603, 230288108992819819, 4563243145806294636
Offset: 1

Views

Author

Vladeta Jovovic, Aug 28 2002

Keywords

Comments

Row sums of A073480.

Links

Crossrefs

Cf. A024167, A073480.

Programs

  • Mathematica
    Rest[CoefficientList[Series[E^x*Log[1+x]/(1-x), {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jul 02 2015 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x)*log(1+x)/(1-x))) \\ Seiichi Manyama, Feb 20 2022
  • PARI
    a(n) = sum(k=1, n, k!*binomial(n, k)*sum(j=1, k, (-1)^(j+1)/j)); \\ Seiichi Manyama, Feb 20 2022

Formula

a(n) ~ n! * exp(1) * log(2). - Vaclav Kotesovec, Jul 02 2015
a(n) = Sum_{k=1..n} k! * binomial(n,k) * Sum_{j=1..k} (-1)^(j+1)/j. - Seiichi Manyama, Feb 20 2022
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