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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A260324 Triangle read by rows: T(n,k) = logarithmic polynomial A_k^(n)(x) evaluated at x=1.

Original entry on oeis.org

1, 0, 1, 1, -2, 2, 2, 9, -6, 6, 9, -28, 12, -24, 24, 44, 185, 100, 60, -120, 120, 265, -846, -690, -120, 360, -720, 720, 1854, 7777, 2478, 5250, -840, 2520, -5040, 5040, 14833, -47384, 33656, -40656, 1680, -6720, 20160, -40320, 40320, 133496, 559953, -347832, 181944, 359856, 15120, -60480, 181440, -362880, 362880
Offset: 1

Views

Author

N. J. A. Sloane, Jul 23 2015

Keywords

Examples

			Triangle begins:
1,
0,1,
1,-2,2,
2,9,-6,6,
9,-28,12,-24,24,
44,185,100,60,-120,120,
265,-846,-690,-120,360,-720,720,
...

Links

  • J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. Gives first 10 rows. [Annotated scanned copy]

Crossrefs

Rows, column sums give A000166, A002747, A002748, A002749.
Cf. A260322, A260323, A260325.

Programs

  • Maple
    A260324 := proc(n,r)
    if r = 0 then
        1 ;
    elif n > r+1 then
        0 ;
    else
        add( (-1)^(r-j*n+1)/(r-j*n+1)!,j=1..(r+1)/n) ;
        %*r! ;
    end if;
end proc:
for r from 0 to 20 do
    for n from 1 to r+1 do
        printf("%a,",A260324(n,r)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 24 2015
  • Mathematica
    T[n_, k_] := If[k == 0, 1, If[n > k + 1, 0, k! Sum[(-x)^(k - j n + 1)/(k - j n + 1)!, {j, 1, (k + 1)/n}]]];
Table[T[n, k] /. x -> 1, {k, 0, 9}, {n, 1, k + 1}] // Flatten (* Jean-François Alcover, Mar 30 2020 *)

A346397 Expansion of e.g.f. -log(1 - x) * exp(-2*x).

Original entry on oeis.org

0, 1, -3, 8, -18, 44, -80, 272, 112, 5280, 38464, 414336, 4573184, 55680000, 731374592, 10335551488, 156303374336, 2518984953856, 43099088904192, 780268881068032, 14902336355991552, 299452809651617792, 6315501510330286080, 139485953831281098752, 3219718099932087844864
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 15 2021

Keywords

Links

Crossrefs

Cf. A000023, A002741, A335111, A346398.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[-Log[1 - x] Exp[-2 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-2)^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 24}]
  • PARI
    a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(i-3)*v[i]+2*(i-1)*v[i-1]+(-2)^(i-1)); v; \\ Seiichi Manyama, May 27 2022

Formula

a(n) = n! * Sum_{k=0..n-1} (-2)^k / ((n-k) * k!).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002741(k).
a(0) = 0, a(1) = 1, a(n) = (n-3) * a(n-1) + 2 * (n-1) * a(n-2) + (-2)^(n-1). - Seiichi Manyama, May 27 2022
a(n) ~ exp(-2) * (n-1)!. - Vaclav Kotesovec, Jun 08 2022

A346398 Expansion of e.g.f. -log(1 - x) * exp(-3*x).

Original entry on oeis.org

0, 1, -5, 20, -72, 249, -825, 2736, -8568, 29385, -74709, 417636, 698544, 21853233, 244181223, 3608612208, 54277152624, 878859416817, 15072037479099, 273539358115092, 5235734703888648, 105419854939796937, 2227408664800976487, 49278475088626210704, 1139260699549648412856
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 15 2021

Keywords

Links

Crossrefs

Cf. A002741, A010843, A346397.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[-Log[1 - x] Exp[-3 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-3)^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 24}]
  • PARI
    a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(i-4)*v[i]+3*(i-1)*v[i-1]+(-3)^(i-1)); v; \\ Seiichi Manyama, May 27 2022

Formula

a(n) = n! * Sum_{k=0..n-1} (-3)^k / ((n-k) * k!).
a(0) = 0, a(1) = 1, a(n) = (n-4) * a(n-1) + 3 * (n-1) * a(n-2) + (-3)^(n-1). - Seiichi Manyama, May 27 2022
a(n) ~ exp(-3) * (n-1)!. - Vaclav Kotesovec, Jun 08 2022

A002748 Sum of logarithmic numbers.

Original entry on oeis.org

1, 2, 3, 26, 13, 1074, -1457, 61802, 7929, 4218722, -6385349, 934344762, -5065189307, 141111736466, 235257551943, 23219206152074, -97011062913167, 11887164842925762, -91890238533000461, 4819930221202545242, -14547510704199530499, 1184314832978574919922
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002750.

Programs

  • Maple
    A002748 := proc(n) local f1,f2 ; f1 := add(numtheory[sigma](i)*x^(i-1),i=1..n+1) ; f2 := add((-x)^i/i!,i=0..n+1) ; n!*coeftayl(f1*f2,x=0,n) ; end: seq(A002748(n),n=0..25) ; # R. J. Mathar, Oct 22 2007
  • Mathematica
    f1[n_] := Sum[DivisorSigma[1, i]*x^(i-1), {i, 1, n+1}]; f2[n_] := Sum[(-x)^i/i!, {i, 0, n+1}] ; a[n_] := n!*SeriesCoefficient[f1[n]*f2[n], {x, 0, n}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jan 17 2014, after R. J. Mathar *)

Formula

E.g.f.: F(x)/exp(x)/x where F(x) is o.g.f. for A000203(). - Vladeta Jovovic, Feb 09 2003

Extensions

More terms from Jeffrey Shallit
More terms from R. J. Mathar, Oct 22 2007

A002749 Sum of logarithmic numbers.

Original entry on oeis.org

1, 1, 1, 11, -7, 389, -1031, 19039, -24431, 1023497, -4044079, 225738611, -1711460279, 29974303501, 4656373513, 3798866053319, -34131041040991, 2131052083901969, -23678368533941471, 832900320313739227, -4752766287768240359, 148482851420849206421
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Programs

  • Maple
    A002749 := proc(r)
    add(A260324(n,r),n=1..r+1) ;
end proc:
seq(A002749(r),r=0..25) ; # R. J. Mathar, Jul 24 2015
  • Mathematica
    m = 22;
F[x_] = Sum[DivisorSigma[0, n] x^n , {n, 1, m}];
CoefficientList[F[x]/(x E^x) + O[x]^m, x] Range[0, m-1]! (* Jean-François Alcover, Mar 30 2020 *)

Formula

E.g.f.: F(x)/exp(x)/x where F(x) is o.g.f. for A000005(). - Vladeta Jovovic, Feb 09 2003

Extensions

Corrected and extended by Jeffrey Shallit

A291484 Expansion of e.g.f. arctanh(x)*exp(x).

Original entry on oeis.org

0, 1, 2, 5, 12, 49, 190, 1301, 7224, 69441, 495898, 6095429, 53005700, 792143793, 8110146070, 142633278997, 1679413757168, 33964965659649, 451969255722162, 10331348137881349, 153288815339260796, 3907452790559751857, 63949589015139119598, 1798373345567005989781, 32179694275204166066728
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 24 2017

Keywords

Examples

			E.g.f.: A(x) = x/1! + 2*x^2/2! + 5*x^3/3! + 12*x^4/4! + 49*x^5/5! + ...

Crossrefs

Cf. A002104, A002741, A009739, A009832, A010050, A012709, A087208 (first differences), A279927.

Programs

  • Maple
    a:=series(arctanh(x)*exp(x),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019
  • Mathematica
    nmax = 24; Range[0, nmax]! CoefficientList[Series[ArcTanh[x] Exp[x], {x, 0, nmax}], x]
nmax = 24; Range[0, nmax]! CoefficientList[Series[Log[(1 + x)/(1 - x)] Exp[x]/2, {x, 0, nmax}], x]
nmax = 24; Range[0, nmax]! CoefficientList[Series[Sum[x^(2 k + 1)/(2 k + 1), {k, 0, Infinity}] Exp[x], {x, 0, nmax}], x]
Table[Sum[Binomial[n+1,2k+1](n-2k)/(n+1) (2 k)!, {k,0,n/2}],{n,0,12}] (* Emanuele Munarini, Dec 16 2017 *)
  • Maxima
    makelist(sum(binomial(n+1,2*k+1)*(n-2*k)/(n+1)*(2*k)!,k,0,floor(n/2)),n,0,12); /* Emanuele Munarini, Dec 16 2017 */
  • PARI
    first(n) = x='x+O('x^n); Vec(serlaplace(atanh(x)*exp(x)), -n) \\ Iain Fox, Dec 16 2017

Formula

E.g.f.: log((1 + x)/(1 - x))*exp(x)/2.
From Emanuele Munarini, Dec 16 2017: (Start)
a(n) = Sum_{k=0..n/2} binomial(n+1,2*k+1)*((n-2*k)/(n+1))*(2*k)!.
a(n+3) - a(n+2) - (n+1)*(n+2)*a(n+1) + (n+1)*(n+2)*a(n) = 1.
a(n+4) - 2*a(n+3) - (n^2+5*n+5)*a(n+2) + 2*(n+2)^2*a(n+1) - (n+1)*(n+2)*a(n) = 0.
(End)
a(n) ~ (n-1)! * (exp(1) - (-1)^n * exp(-1))/2. - Vaclav Kotesovec, Dec 16 2017

A336292 a(n) = (n!)^2 * Sum_{k=1..n} (-1)^(n-k) / (k * ((n-k)!)^2).

Original entry on oeis.org

0, 1, -2, 3, 8, 305, 10734, 502747, 30344992, 2307890097, 216571514030, 24619605092291, 3337294343698248, 532148381719443073, 98646472269855762238, 21041945289232131607995, 5118447176652195630775424, 1408601897794844346184122017, 435481794298015565250651718302
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 16 2020

Keywords

Crossrefs

Cf. A002741, A009940, A336291.

Programs

  • Mathematica
    Table[(n!)^2 Sum[(-1)^(n - k)/(k ((n - k)!)^2), {k, 1, n}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[-Log[1 - x] BesselJ[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2
  • PARI
    a(n) = (n!)^2 * sum(k=1, n, (-1)^(n-k) / (k * ((n-k)!)^2)); \\ Michel Marcus, Jul 17 2020

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = -log(1 - x) * BesselJ(0,2*sqrt(x)).

A002751 Sum of logarithmic numbers.

Original entry on oeis.org

1, 3, 9, 37, 153, 951, 5473, 42729, 353937, 3455083, 30071001, 426685293, 4707929449, 59350096287, 882391484913, 15177204356401, 205119866263713, 4040196156574419, 64262949875511337, 1408785031894483893, 29514546353782633401, 593055713389216814983
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Programs

  • Maple
    [seq(numtheory[tau](n),n=0..40)] ;gfun[listtoseries](%,x,'ogf') ; %/x*exp(x) ; taylor(%,x=0,40) ; eg := gfun[seriestolist](%,'ogf') ; seq( op(i,eg)*(i-1)!, i=1..nops(eg)) ; # R. J. Mathar, Jul 08 2011

Formula

E.g.f.: F(x)/x*exp(x) where F(x) is o.g.f. for A000005(). - Vladeta Jovovic, Feb 09 2003

Extensions

More terms from Jeffrey Shallit

A302581 a(n) = n! * [x^n] -exp(-n*x)*log(1 - x).

Original entry on oeis.org

0, 1, -3, 20, -186, 2249, -33360, 586172, -11901008, 274098393, -7060189120, 201092672604, -6275340884736, 212915635727313, -7803567334571008, 307245946117223700, -12933084380738398208, 579587518114690731601, -27550568677612746940416, 1384553892443352890245636
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2018

Keywords

Links

Crossrefs

Cf. A002104, A002741, A104150, A134095, A190314.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[-Exp[-n x] Log[1 - x], {x, 0, n}], {n, 0, 19}]
Table[Sum[(-n)^(n - k) (k - 1)! Binomial[n, k], {k, 1, n}], {n, 0, 19}]
nmax = 20; CoefficientList[Series[-Log[1 - LambertW[x]]/(1 + LambertW[x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 09 2019 *)

Formula

a(n) = Sum_{k=1..n} (-n)^(n-k)*(k-1)!*binomial(n,k).
E.g.f.: -log(1 - LambertW(x))/(1 + LambertW(x)). - Vaclav Kotesovec, Jun 09 2019
a(n) ~ -(-1)^n * log(2) * n^n. - Vaclav Kotesovec, Jun 09 2019

A346409 a(n) = (n!)^2 * Sum_{k=0..n-1} (-1)^k / ((n-k)^2 * k!).

Original entry on oeis.org

0, 1, -3, 13, -52, 476, 1344, 156192, 6935424, 470168064, 38948065920, 3979380286080, 489922581219840, 71586095491054080, 12249193741572372480, 2426646293132502067200, 551096248249459158220800, 142236660450422499604070400, 41404182857569072540171468800
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 15 2021

Keywords

Crossrefs

Cf. A002741, A009940, A082491, A142999, A346405.

Programs

  • Mathematica
    Table[(n!)^2 Sum[(-1)^k/((n - k)^2 k!), {k, 0, n - 1}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[PolyLog[2, x] Exp[-x], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = polylog(2,x) * exp(-x).
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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