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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A088181 E.g.f.: 1/(1-sinh(x)-x).

Original entry on oeis.org

1, 2, 8, 49, 400, 4081, 49964, 713665, 11649936, 213946513, 4365604372, 97988751433, 2399366732888, 63647177397961, 1818219651069852, 55651486523443537, 1816920458689600288, 63026617425620098465
Offset: 0

Views

Author

Karol A. Penson, Sep 22 2003

Keywords

Crossrefs

Cf. A006154.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1-Sinh[x]-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 10 2014 *)

Formula

a(n) ~ n! / ((cosh(r)+1) * r^(n+1)), where r = 0.49007306848054777421543028443773199472978824305677445770838... is the root of the equation 2*(r-1)*exp(r)+exp(2*r) = 1. - Vaclav Kotesovec, Jun 08 2021

Extensions

First term prepended and offset changed by Harvey P. Dale, Apr 10 2014

A193474 Table read by rows: The coefficients of the polynomials P(n, x) = Sum{k=0..n} Sum{j=0..k} (-1)^j * 2^(-k) * binomial(k, j) * (k-2*j)^n * x^(n-k).

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 6, 0, 1, 0, 24, 0, 8, 0, 0, 120, 0, 60, 0, 1, 0, 720, 0, 480, 0, 32, 0, 0, 5040, 0, 4200, 0, 546, 0, 1, 0, 40320, 0, 40320, 0, 8064, 0, 128, 0, 0, 362880, 0, 423360, 0, 115920, 0, 4920, 0, 1, 0, 3628800, 0, 4838400, 0, 1693440, 0, 130560, 0, 512, 0, 0
Offset: 1

Views

Author

Peter Luschny, Aug 01 2011

Keywords

Comments

See A196776 for a row reversed form of this triangle. - Peter Bala, Oct 06 2011

Examples

			The sequence of polynomials P(n, x) begins:
[0]    1;
[1]    1;
[2]    2;
[3]    6 +      x^2;
[4]   24 +    8*x^2;
[5]  120 +   60*x^2 +     x^4;
[6]  720 +  480*x^2 +  32*x^4;
[7] 5040 + 4200*x^2 + 546*x^4 + x^6.

Crossrefs

Cf. A196776, A000142, A006154, A191277, A000111, A000828, A000557, A107403, A007289, A005990.

Programs

  • Maple
    A193474_polynom := proc(n,x) local k, j;
add(add((-1)^j*2^(-k)*binomial(k,j)*(k-2*j)^n*x^(n-k),j=0..k),k=0..n) end: seq(seq(coeff(A193474_polynom(n,x),x,i),i=0..n),n=0..10);
  • Mathematica
    p[n_, x_] := Sum[(-1)^j*2^(-k)*Binomial[k, j]*(k-2*j)^n*x^(n-k), {k, 0, n}, {j, 0, k}]; t[n_, k_] := Coefficient[p[n, x], x, k]; t[0, 0] = 1; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 27 2014 *)

Formula

P(n, 0) = A000142(n).
P(n, 1) = A006154(n).
P(n, 2) = A191277(n).
P(n, i) = A000111(n+1), where i is the imaginary unit.
P(n, i)*2^n = A000828(n+1).
P(n, 1/2)*2^n = A000557(n).
P(n, 1/3)*3^n = A107403(n).
P(n, i/2)*2^n = A007289(n).
G(m, x) = 1/(1 - m*sinh(x)) is the generating function of m^n*P(n, 1/m).
GI(m, x) = 1/(1 - m*sin(x)) is the generating function of m^n*P(n, i/m).
[x^2] P(n+1, x) = A005990(n).

A332257 E.g.f.: (1 - sinh(x)) / (1 - 2*sinh(x)).

Original entry on oeis.org

1, 1, 4, 25, 208, 2161, 26944, 391945, 6515968, 121866721, 2532496384, 57890223865, 1443611004928, 38999338931281, 1134616226381824, 35367467110007785, 1175946733416153088, 41543231955279099841, 1553948045857778827264, 61355543097139813855705
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 08 2020

Keywords

Links

Crossrefs

Cf. A000557, A000670, A002866, A006154, A050351, A331608.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[(1 - Sinh[x])/(1 - 2 Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    seq(n)={Vec(serlaplace((1 - sinh(x + O(x*x^n))) / (1 - 2*sinh(x + O(x*x^n)))))} \\ Andrew Howroyd, Feb 08 2020

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A006154(k) * a(n-k).
a(n) ~ n! / (2*sqrt(5) * log((1 + sqrt(5))/2)^(n+1)). - Vaclav Kotesovec, Feb 08 2020

A352253 Expansion of e.g.f. 1 / (1 - x * sinh(x) / 2) (even powers only).

Original entry on oeis.org

1, 1, 8, 153, 5492, 316625, 26774622, 3121729709, 479962730648, 94087054172673, 22904161764512570, 6778870099212235805, 2397161662661680925364, 998186321121004312238513, 483430830256916593106991782, 269435322393253822641626419725, 171224984800186115316322226731952
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

Crossrefs

Cf. A006153, A006154, A094088, A205571, A352251, A352254.

Programs

  • Mathematica
    nmax = 32; Take[CoefficientList[Series[1/(1 - x Sinh[x]/2), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]
  • PARI
    my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sinh(x)/2)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n,2*k) * k * a(n-k).

A352470 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1)^2 * a(n-2*k-1).

Original entry on oeis.org

1, 1, 4, 37, 608, 15601, 576472, 28993693, 1904637184, 158352856129, 16253786050904, 2018684970206653, 298373110433984192, 51757706826973479697, 10412613242348421164400, 2404755328388872932588037, 631887117002962512609921024, 187441600433239155105076467457
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2022

Keywords

Crossrefs

Cf. A006154, A102221, A346220, A352467, A352471.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 2 k + 1]^2 a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[1/(1 - Sum[x^(2 k + 1)/(2 k + 1)!^2, {k, 0, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

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

A352471 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1)^3 * a(n-2*k-1).

Original entry on oeis.org

1, 1, 8, 217, 13952, 1752001, 380168432, 130996038265, 67377689108480, 49343690620021249, 49570079811804165008, 66280482720537078211945, 115058150837606807142692096, 253942526419333142443328522689, 700015299612132412448976873339008
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2022

Keywords

Crossrefs

Cf. A006154, A336195, A352468, A352470.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, 2 k + 1]^3 a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 14}]
nmax = 14; CoefficientList[Series[1/(1 - Sum[x^(2 k + 1)/(2 k + 1)!^3, {k, 0, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^3

Formula

Sum_{n>=0} a(n) * x^n / n!^3 = 1 / (1 - Sum_{n>=0} x^(2*n+1) / (2*n+1)!^3).

A381284 Expansion of e.g.f. 1/(1 - sinh(3*x) / 3).

Original entry on oeis.org

1, 1, 2, 15, 96, 741, 7632, 87795, 1149696, 17155881, 282880512, 5128464375, 101592631296, 2178698451021, 50314379323392, 1245198047833755, 32868161979088896, 921803465256094161, 27373850876851126272, 858044392807801699935, 28311289100161039466496
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Crossrefs

Cf. A006154, A191277.
Cf. A136630.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*3^(n-k)*a136630(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 9^k * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * 3^(n-k) * A136630(n,k).
a(n) ~ sqrt(Pi/5) * 3^(n+1) * n^(n + 1/2) / (arcsinh(3)^(n+1) * exp(n)). - Vaclav Kotesovec, Apr 19 2025

A009124 Expansion of e.g.f. cosh(log(1+sinh(x))).

Original entry on oeis.org

1, 0, 1, -3, 16, -90, 616, -4893, 44416, -453540, 5145856, -64222983, 874402816, -12897183390, 204862698496, -3486535686273, 63292764553216, -1220795601029640, 24931903045697536, -537463528325234763
Offset: 0

Views

Author

R. H. Hardin

Keywords

Links

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Cosh(Log(1+Sinh(x))))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 28 2018
  • Mathematica
    CoefficientList[Series[(1 + (1 + Sinh[x])^2)/(2*(1 + Sinh[x])), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 22 2015 *)
  • PARI
    x='x+O('x^30); Vec(serlaplace(cosh(log(1+sinh(x))))) \\ G. C. Greubel, Jul 28 2018

Formula

a(2*n) = A006154(2*n)/2, n>0. - Ralf Stephan, Apr 29 2004
a(n) ~ n! * (-1)^n / (2 * sqrt(2) * (log(1+sqrt(2)))^(n+1)). - Vaclav Kotesovec, Jan 22 2015

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

A009568 Expansion of e.g.f.: sinh(log(1+sinh(x))).

Original entry on oeis.org

0, 1, -1, 4, -16, 91, -616, 4894, -44416, 453541, -5145856, 64222984, -874402816, 12897183391, -204862698496, 3486535686274, -63292764553216, 1220795601029641, -24931903045697536, 537463528325234764, -12196043454064623616, 290588368323926512291, -7253387436695186046976
Offset: 0

Views

Author

R. H. Hardin

Keywords

Links

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Sinh[Log[1+Sinh[x]]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 08 2017 *)
  • PARI
    x='x+O('x^30); concat([0], Vec(serlaplace( sinh(log(1+sinh(x)))))) \\ G. C. Greubel, Jan 21 2018

Formula

a(2*n) = A006154(2*n)/2, n>0. - Ralf Stephan, Apr 29 2004

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997
Prior Mathematica program replaced and definition clarified by Harvey P. Dale, Dec 08 2017
Terms a(20) to a(22) added by G. C. Greubel, Jan 21 2018

A381428 E.g.f. A(x) satisfies A(x) = 1/( 1 - sinh(x) * A(x)^2 ).

Original entry on oeis.org

1, 1, 6, 73, 1344, 33481, 1054656, 40223233, 1802385024, 92827015921, 5403527705856, 350854589607193, 25142008355656704, 1971003462240791161, 167802783944207917056, 15417877986778302551953, 1520661128893781018640384, 160249491538400609431567201, 17969682580669053325124960256
Offset: 0

Views

Author

Seiichi Manyama, Feb 23 2025

Keywords

Crossrefs

Cf. A006154, A295254, A381429.
Cf. A136630.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(3*k+1,k)/(3*k+1)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(3*k+1,k)/(3*k+1) * A136630(n,k).
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