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.

A009416 Expansion of e.g.f. log(1+x)*cosh(x).

Original entry on oeis.org

0, 1, -1, 5, -12, 49, -225, 1301, -8848, 69441, -616185, 6095429, -66475596, 792143793, -10238194057, 142633278997, -2130537219840, 33964965659649, -575573407212753, 10331348137881349, -195807849389362540, 3907452790559751857, -81892400673047263761, 1798373345567005989781, -41294565798306731368272, 989581607891834678407745
Offset: 0

Views

Author

R. H. Hardin

Keywords

Crossrefs

Cf. A009410.

Programs

  • Mathematica
    CoefficientList[Series[Cosh[x]*Log[1 + x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 23 2015 *)

Formula

a(n) ~ (n-1)! * (-1)^(n+1) * cosh(1). - Vaclav Kotesovec, Jan 23 2015
a(n) = -(-1)^n * Sum_{k=0..floor((n-1)/2)} binomial(n,2*k) * (n-2*k-1)!. - Ilya Gutkovskiy, Apr 10 2022

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

A177699 Expansion of e.g.f. log(1+x) * sinh(x).

Original entry on oeis.org

0, 0, 2, -3, 12, -40, 190, -1071, 7224, -56232, 495898, -4880755, 53005700, -629398848, 8110146070, -112690225935, 1679413757168, -26719024870576, 451969255722162, -8099650628337987, 153288815339260796, -3054957193416951480, 63949589015139119598, -1402819397613793354063
Offset: 0

Views

Author

Michel Lagneau, May 11 2010

Keywords

Examples

			log(1+x) * sinh(x) = x^2 -x^3/2 +x^4/2 -x^5/3 +19*x^6/72 -17*x^7/80 +...

References

  • L. Comtet and M. Fiolet, Sur les dérivées successives d'une fonction implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251.

Crossrefs

Cf. A002104, A009410, A009416, A381016.

Programs

  • Maple
    A177699 := proc(n)
        log(1+x)*sinh(x) ;
        coeftayl(%,x=0,n)*n! ;
end proc;
seq(A177699(n),n=0..20) ; # R. J. Mathar, Nov 07 2011
  • Mathematica
    Table[n ((-1)^n HypergeometricPFQ[{1, 1, 1 - n}, {2}, -1] + HypergeometricPFQ[{1, 1, 1 - n}, {2}, 1])/2, {n, 1, 20}] (* Benedict W. J. Irwin, May 30 2016 *)
  • PARI
    a(n) = (-1)^n*sum(k=1, n\2, (n-2*k)!*binomial(n, 2*k-1)); \\ Seiichi Manyama, Feb 12 2025

Formula

a(n) = n*((-1)^n*3F1(1,1,1-n;2;-1)+3F1(1,1,1-n;2;1))/2, n>0. - Benedict W. J. Irwin, May 30 2016
a(n) ~ (-1)^n * (n-1)! * sinh(1). - Vaclav Kotesovec, May 30 2016
a(n) = (-1)^n * Sum_{k=1..floor(n/2)} (n-2*k)! * binomial(n,2*k-1). - Seiichi Manyama, Feb 12 2025

A351880 Expansion of e.g.f. 1 / (1 - x)^cos(x).

Original entry on oeis.org

1, 1, 2, 3, 6, 15, 105, 924, 8204, 73461, 700005, 7323976, 84472146, 1064285027, 14492861747, 211738655608, 3302847261448, 54800458320345, 963864555797385, 17914985159719376, 350861004976886190, 7221748369472388727, 155853930324297011719, 3519121773604369318856
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 23 2022

Keywords

Crossrefs

Cf. A007118, A007119, A009192, A009410, A298374, A351881.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 - x)^Cos[x], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(1/(1-x)^cos(x))) \\ Michel Marcus, Feb 23 2022

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} (-1)^k * binomial(n-1,k-1) * A009410(k) * a(n-k).
a(n) ~ n! * n^(cos(1)-1) / Gamma(cos(1)). - Vaclav Kotesovec, Feb 23 2022

A302610 Expansion of e.g.f. -log(1 - x)*arcsinh(x).

Original entry on oeis.org

0, 0, 2, 3, 4, 20, 158, 819, 3624, 33984, 427482, 3819915, 29665260, 404822340, 6948032310, 88407058635, 991515848400, 17715286764000, 383952670412850, 6349179054589875, 93532380775766100, 2063197602667372500, 53913667654307868750, 1098018631195048591875
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2018

Keywords

Examples

			-log(1 - x)*arcsinh(x) = 2*x^2/2! + 3*x^3/3! + 4*x^4/4! + 20*x^5/5! + 158*x^6/6! + 819*x^7/7! + 3624*x^8/8! + ...

Crossrefs

Cf. A009410, A009416, A009429, A009435, A012572, A104150, A177699, A177700, A296435, A296727, A302611.

Programs

  • Maple
    a:=series(-log(1-x)*arcsinh(x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[-Log[1 - x] ArcSinh[x], {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: -log(1 - x)*log(x + sqrt(1 + x^2)).

A302611 Expansion of e.g.f. -log(1 - x)*arctanh(x).

Original entry on oeis.org

0, 0, 2, 3, 16, 50, 368, 1764, 16896, 109584, 1297152, 10628640, 149944320, 1486442880, 24349317120, 283465647360, 5287713177600, 70734282393600, 1480103564083200, 22376988058521600, 519000166327910400, 8752948036761600000, 222845873874075648000, 4148476779335454720000
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2018

Keywords

Examples

			-log(1 - x)*arctanh(x) = 2*x^2/2! + 3*x^3/3! + 16*x^4/4! + 50*x^5/5! + 368*x^6/6! + 1764*x^7/7! + 16896*x^8/8! + ...

Crossrefs

Cf. A005359, A009410, A009416, A009429, A009435, A012697, A081358, A104150, A177699, A177700, A202139, A302610.

Programs

  • Maple
    a:=series(-log(1-x)*arctanh(x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019
  • Mathematica
    nmax = 23; CoefficientList[Series[-Log[1 - x] ArcTanh[x], {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    x='x+O('x^99); concat([0, 0], Vec(serlaplace(log(1-x)*log((1-x)/(1+x))/2))) \\ Altug Alkan, Apr 10 2018

Formula

E.g.f.: log(1 - x)*log((1 - x)/(1 + x))/2.

A381016 Expansion of e.g.f. -log(1-x) * sin(x).

Original entry on oeis.org

0, 0, 2, 3, 4, 20, 110, 651, 4520, 36000, 322618, 3213595, 35226860, 421419492, 5463436134, 76301056755, 1142009233872, 18236159031584, 309463272791538, 5561354285804115, 105510576441518164, 2107380222724155540, 44200537412519181278, 971311172969442165883
Offset: 0

Views

Author

Seiichi Manyama, Feb 12 2025

Keywords

Crossrefs

Cf. A002104, A009410, A009416, A177699.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(-log(1-x)*sin(x))))
  • PARI
    a(n) = -sum(k=1, n\2, (-1)^k*(n-2*k)!*binomial(n, 2*k-1));

Formula

a(n) = -Sum_{k=1..floor(n/2)} (-1)^k * (n-2*k)! * binomial(n,2*k-1).
Showing 1-6 of 6 results.

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