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 11-20 of 22 results. Next

A218309 E.g.f. A(x) satisfies A( x/(exp(3*x)*cosh(3*x)) ) = exp(4*x)*cosh(4*x).

Original entry on oeis.org

1, 4, 56, 1264, 40640, 1711744, 89533184, 5607463936, 409621790720, 34218229227520, 3219000547131392, 336858779869020160, 38823224436435845120, 4886982191317154529280, 667188807538423365632000, 98200163047169655115350016, 15501781660715229538766815232
Offset: 0

Views

Author

Paul D. Hanna, Oct 25 2012

Keywords

Comments

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

Examples

			E.g.f.: A(x) = 1 + 4*x + 56*x^2/2! + 1264*x^3/3! + 40640*x^4/4! + 1711744*x^5/5! +...
where
A(x) = cosh(2*x) + 4*5^0*cosh(5*x)*x + 4*8^1*cosh(8*x)*x^2/2! + 4*11^2*cosh(11*x)*x^3/3! + 4*14^3*cosh(14*x)*x^4/4! + 4*17^4*cosh(17*x)*x^5/5! +...

Crossrefs

Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218308, A218310.

Programs

  • PARI
    {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(3*X)*cosh(3*X)))); Egf=exp(4*R)*cosh(4*R); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,4*(3*k+4)^(k-1)*cosh((3*k+4)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f.: A(x) = Sum_{n>=0} 4*(3*n+4)^(n-1) * cosh((3*n+4)*x) * x^n/n!.
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( 4*x - 4/3 * LambertW(-3*x * exp(3*x)) ).
a(n) = 2 * Sum_{k=0..n} (3*k+4)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 2 * Sum_{k>=0} (3*k+4)^(k-1) * x^k/(1 - (3*k+4)*x)^(k+1). (End)

A218310 E.g.f. A(x) satisfies A( x/(exp(5*x)*cosh(5*x)) ) = exp(x)*cosh(x).

Original entry on oeis.org

1, 1, 12, 364, 17248, 1118816, 92306432, 9251542784, 1091729307648, 148280571406336, 22785577791987712, 3908379504145178624, 740274425760340901888, 153456630172316832628736, 34557831428406144298647552, 8401098284435734877893033984
Offset: 0

Views

Author

Paul D. Hanna, Oct 25 2012

Keywords

Comments

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

Examples

			E.g.f.: A(x) = 1 + x + 12*x^2/2! + 364*x^3/3! + 17248*x^4/4! + 1118816*x^5/5! +...
where
A(x) = cosh(x) + 6^0*cosh(6*x)*x + 11^1*cosh(11*x)*x^2/2! + 16^2*cosh(16*x)*x^3/3! + 21^3*cosh(21*x)*x^4/4! + 26^4*cosh(26*x)*x^5/5! +...

Crossrefs

Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218308, A218309.

Programs

  • PARI
    {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(5*X)*cosh(5*X)))); Egf=exp(R)*cosh(R); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,(5*k+1)^(k-1)*cosh((5*k+1)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f.: A(x) = Sum_{n>=0} (5*n+1)^(n-1) * cosh((5*n+1)*x) * x^n/n!.
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( x - 1/5 * LambertW(-5*x * exp(5*x)) ).
a(n) = 1/2 * Sum_{k=0..n} (5*k+1)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 1/2 * Sum_{k>=0} (5*k+1)^(k-1) * x^k/(1 - (5*k+1)*x)^(k+1). (End)

A214225 E.g.f. satisfies: A(x) = x/(1 - tanh(A(x))).

Original entry on oeis.org

1, 2, 12, 112, 1440, 23616, 471296, 11085824, 300349440, 9211187200, 315448860672, 11932326789120, 494098626904064, 22230301612703744, 1079857012109475840, 56326462301645307904, 3140024293968001892352, 186308007164786201591808, 11722541029509094870876160
Offset: 1

Views

Author

Paul D. Hanna, Jul 07 2012

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...
Related expansions:
A(x) = x + x*tanh(x) + d/dx x^2*tanh(x)^2/2! + d^2/dx^2 x^3*tanh(x)^3/3! + d^3/dx^3 x^4*tanh(x)^4/4! +...
log(A(x)/x) = tanh(x) + d/dx x*tanh(x)^2/2! + d^2/dx^2 x^2*tanh(x)^3/3! + d^3/dx^3 x^3*tanh(x)^4/4! +...
A(x)/x = 1 + x + 4*x^2/2! + 28*x^3/3! + 288*x^4/4! + 3936*x^5/5! + 67328*x^6/6! +...+ A201595(n)*x^n/n! +...
tanh(A(x)) = x + 2*x^2/2! + 10*x^3/3! + 88*x^4/4! + 1096*x^5/5! + 17616*x^6/6! + 346704*x^7/7! + 8072576*x^8/8! +...

Links

Crossrefs

Cf. A201595, A202617, A214222, A214223, A214224.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x-x*Tanh[x],{x,0,20}],x],x]*Range[0,20]!] (* Vaclav Kotesovec, Sep 17 2013 *)
Flatten[{1,Table[1/2*Sum[Binomial[n,k]*k^(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Sep 17 2013 *)
  • PARI
    {a(n)=(1/2)*sum(k=0,n,binomial(n,k)*k^(n-1))}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {a(n)=(n-1)!*polcoeff(x/(1 - tanh(x+x*O(x^n)))^n,n)}
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-x*tanh(x+x*O(x^n))), n)}
  • PARI
    {a(n)=n!*polcoeff(sum(k=1, n, k^(k-1)*cosh(k*x +x*O(x^n))*x^k/k! +x*O(x^n)), n)} \\ Paul D. Hanna, Nov 20 2012
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*tanh(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*tanh(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}

Formula

E.g.f. A(x) satisfies:
(1) A(x - x*tanh(x)) = x.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*tanh(x)^n/n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*tanh(x)^n/n! ).
(4) A(x) = Sum_{n>=1} n^(n-1) * cosh(n*x) * x^n / n!. - Paul D. Hanna, Nov 20 2012
(5) A(x) = log(G(x)) where G(x) = exp(x*(1+G(x)^2)/2) is the e.g.f. of A202617. - Paul D. Hanna, Nov 20 2012
a(n) = n*A201595(n-1).
a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*k^(n-1).
a(n) = (n-1)! * [x^n] x/(1 - tanh(x))^n.
a(n) = A038049(n)/2. - R. J. Mathar, Peter Bala, Mar 24 2013
a(n) ~ 1/2 * n^(n-1) * sqrt((1+LambertW(1/exp(1)))) / (exp(1)*LambertW(1/exp(1)))^n. - Vaclav Kotesovec, Sep 17 2013

A074932 Row sums of unsigned triangle A075513.

Original entry on oeis.org

1, 3, 18, 170, 2200, 36232, 725200, 17095248, 463936896, 14246942336, 488428297984, 18491942300416, 766293946203136, 34498781924766720, 1676731077272217600, 87501958444207351808, 4880017252828686155776
Offset: 1

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Examples

			E.g.f.: A(x) = x + 3*x^2/2! + 18*x^3/3! + 170*x^4/4! + 2200*x^5/5! +...
where exp(A(x)) = 1 + x + 4*x^2/2! + 28*x^3/3! + 288*x^4/4! + 3936*x^5/5! + 67328*x^6/6! +...+ A201595(n)*x^n/n! +...

Links

Crossrefs

Cf. A201595.

Programs

  • Mathematica
    Rest[CoefficientList[Series[Log[x-LambertW[-x*Exp[x]]]-Log[2*x], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Dec 04 2012 *)
a[n_] := Sum[Binomial[n-1, k]*(k+1)^(n-1), {k, 0, n-1}]; Table[a[n], {n, 1, 17}] (* Jean-François Alcover, Jul 09 2013, after Paul D. Hanna *)
  • PARI
    {a(n)=sum(k=0,n-1,binomial(n-1,k)*(k+1)^(n-1))} \\ Paul D. Hanna, Aug 02 2012
  • PARI
    {a(n)=local(A201595=serreverse(x-x*tanh(x+x^2*O(x^n)))/x);n!*polcoeff(log(A201595), n)} \\ Paul D. Hanna, Aug 02 2012
  • PARI
    {a(n) = my(A); if( n<0, 0, A = O(x); for(k=1, n, A = log( (1 + exp( 2*x * exp(A))) / 2 )); n! * polcoeff(A, n))}; /* Michael Somos, Apr 10 2018 */

Formula

a(n) = sum(|A075513(n, m)|, m=0..n-1) = sum(binomial(n-1, m)*(m+1)^(n-1), m=0..n-1), n>=1.
E.g.f.: log(G(x)) where G(x) = (1 + exp(2*x*G(x)))/2 is the e.g.f. of A201595. - Paul D. Hanna, Aug 02 2012
E.g.f: log(x-LambertW(-x*exp(x)))-log(2*x). - Vaclav Kotesovec, Dec 04 2012
a(n) ~ n!/(sqrt(2*Pi*(1+LambertW(exp(-1))))*n^(3/2)*LambertW(exp(-1))^n). - Vaclav Kotesovec, Dec 04 2012
a(n) = A072034(n)/n. - Vladimir Reshetnikov, Nov 09 2016
O.g.f.: Sum_{k>=1} k^(k-1)*x^k/(1 - k*x)^k. - Ilya Gutkovskiy, Oct 09 2018

A201594 E.g.f. satisfies: A(x) = 1/(1 - tan( x*A(x) )).

Original entry on oeis.org

1, 1, 4, 32, 384, 6176, 124928, 3049472, 87265280, 2865848320, 106258440192, 4391008927744, 200131590356992, 9973976451383296, 539604322034384896, 31496226303081709568, 1972926888464596598784, 132015791534989604028416, 9398128264859870497341440, 709248762402156849800413184
Offset: 0

Views

Author

Paul D. Hanna, Dec 02 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 32*x^3/3! + 384*x^4/4! + 6176*x^5/5! +...
The coefficients in the initial powers of G(x) = 1/(1 - tan(x)) begin:
G^1: [(1), 1, 2, 8, 40, 256, 1952, 17408, ..., A000828(n), ...];
G^2: [1,(2), 6, 28, 168, 1232, 10656, 106048, ...];
G^3: [1, 3,(12), 66, 456, 3768, 36192, 395616, ...];
G^4: [1, 4, 20,(128), 1000, 9184, 96800, 1150208, ...];
G^5: [1, 5, 30, 220,(1920), 19400, 222480, 2852320, ...];
G^6: [1, 6, 42, 348, 3360,(37056), 459312, 6317088, ...];
G^7: [1, 7, 56, 518, 5488, 65632, (874496), 12841808, ...];
G^8: [1, 8, 72, 736, 8496, 109568, 1562112, (24395776), ...]; ...
where coefficients in parenthesis form initial terms of this sequence:
[1/1, 2/2, 12/3, 128/4, 1920/5, 37056/6, 874496/7, 24395776/8, ...].

Crossrefs

Cf. A214224, A201595, A201128, A000828.

Programs

  • Mathematica
    CoefficientList[1/x*InverseSeries[Series[x*(1-Tan[x]), {x, 0, 21}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 12 2014 *)
  • PARI
    {a(n)=n!*polcoeff(1/x*serreverse(x-x*tan(x+x^2*O(x^n))),n)}
  • PARI
    {a(n)=n!*polcoeff(1/(1-tan(x+x*O(x^n)))^(n+1)/(n+1), n)}

Formula

E.g.f. A(x) satisfies: A( x*(1-tan(x)) ) = 1/(1-tan(x)).
E.g.f.: (1/x)*Series_Reversion( x*(1-tan(x)) ).
a(n) = [x^n/n!] 1/(1 - tan(x))^(n+1) / (n+1).
a(n) = A214224(n+1)/(n+1).
a(n) ~ n^(n-1) * ((t^2+1)/(t-1)^2)^(n+1/2) / (sqrt(2*(t+1)) * exp(n)), where t = 0.46733877379062994365... is the root of the equation t = tan((1-t)/(1+t^2)). - Vaclav Kotesovec, Jan 12 2014

A201628 E.g.f. satisfies: A(x) = 1/(1 - sinh(x*A(x))).

Original entry on oeis.org

1, 1, 4, 31, 360, 5601, 109568, 2586151, 71555200, 2271961825, 81441188352, 3253620672303, 143361363439616, 6907049546879041, 361245668908466176, 20383791705206338807, 1234336634416972726272, 79843983527411321710401, 5494767253686351671459840, 400863405346004202504321343
Offset: 0

Views

Author

Paul D. Hanna, Dec 03 2011

Keywords

Comments

The function 1/(1-sinh(x)) is the e.g.f. of A006154, where A006154(n) is the number of labeled ordered partitions of an n-set into odd parts.

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 31*x^3/3! + 360*x^4/4! + 5601*x^5/5! +...
The coefficients in initial powers of G(x) = 1/(1 - sinh(x)) begin:
G^1: [(1), 1, 2, 7, 32, 181, 1232, 9787, 88832, ..., A006154(n), ...];
G^2: [1,(2), 6, 26, 144, 962, 7536, 67706, ...];
G^3: [1, 3,(12), 63, 408, 3123, 27552, 275103, ...];
G^4: [1, 4, 20,(124), 920, 7924, 77600, 850924, ...];
G^5: [1, 5, 30, 215,(1800), 17225, 185280, 2211515, ...];
G^6: [1, 6, 42, 342, 3192,(33606), 393792, 5080662, ...];
G^7: [1, 7, 56, 511, 5264, 60487, (766976), 10634911, ...];
G^8: [1, 8, 72, 728, 8208, 102248, 1395072,(20689208), ...]; ...
where coefficients in parenthesis form initial terms of this sequence:
[1/1, 2/2, 12/3, 124/4, 1800/5, 33606/6, 766976/7, 20689208/8, ...].

Crossrefs

Cf. A214223, A201627, A201595, A006154.

Programs

  • PARI
    {a(n)=n!*polcoeff(1/x*serreverse(x*(1-sinh(x+x^2*O(x^n)))),n)}
  • PARI
    {a(n)=n!*polcoeff(1/(1 - sinh(x+x*O(x^n)))^(n+1)/(n+1), n)}

Formula

E.g.f. A(x) satisfies: A( x*(1 - sinh(x)) ) = 1/(1 - sinh(x)).
E.g.f.: (1/x)*Series_Reversion( x*(1 - sinh(x)) ).
a(n) = [x^n] 1/(1 - sinh(x))^(n+1) / (n+1).
a(n) = A214223(n+1)/(n+1).

A360548 E.g.f. satisfies A(x) = x * exp( 2*(x + A(x)) ).

Original entry on oeis.org

0, 1, 8, 96, 1792, 46080, 1511424, 60325888, 2837970944, 153778913280, 9432255692800, 646039266656256, 48874810528235520, 4047655951598092288, 364221261622538141696, 35384754572803304325120, 3691411033400626898796544, 411569264258973944034361344
Offset: 0

Views

Author

Seiichi Manyama, Feb 11 2023

Keywords

Links

Crossrefs

Cf. A100526, A216857, A360545.
Cf. A038049, A201595, A214225, A360547.

Programs

  • Maple
    A360548 := proc(n)
    add((2*k)^(n-1)*binomial(n,k),k=1..n) ;
end proc:
seq(A360548(n),n=0..60) ; # R. J. Mathar, Mar 12 2023
  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*x*exp(2*x))/2)))
  • PARI
    a(n) = sum(k=1, n, (2*k)^(n-1)*binomial(n, k));

Formula

E.g.f.: A(x) = (-1/2) * LambertW(-2*x * exp(2*x)).
a(n) = Sum_{k=1..n} (2*k)^(n-1) * binomial(n,k) = 4^(n-1) * A100526(n).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (LambertW(exp(-1))^n * exp(n)). - Vaclav Kotesovec, Feb 17 2023

A370907 Expansion of e.g.f. (1/x) * Series_Reversion( 3*x/(2 + exp(3*x)) ).

Original entry on oeis.org

1, 1, 5, 42, 519, 8526, 175329, 4338594, 125632035, 4169652390, 156101072373, 6508965708378, 299190004799679, 15031796956994286, 819581031710623017, 48199003176462356754, 3041324249730311069595, 204962505644116505863926
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2024

Keywords

Links

Crossrefs

Cf. A201595, A370908.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(3*x/(2+exp(3*x)))/x))
  • PARI
    a(n) = sum(k=0, n+1, 2^(n+1-k)*k^n*binomial(n+1, k))/(3*(n+1));

Formula

a(n) = 1/(3*(n+1)) * Sum_{k=0..n+1} 2^(n+1-k) * k^n * binomial(n+1,k).
a(n) = n! * Sum_{k=0..n} 3^(n-k) * Stirling2(n,k)/(n-k+1)!. - Seiichi Manyama, Nov 07 2024

A370908 Expansion of e.g.f. (1/x) * Series_Reversion( 4*x/(3 + exp(4*x)) ).

Original entry on oeis.org

1, 1, 6, 58, 824, 15576, 368560, 10494352, 349680000, 13354956160, 575343613184, 27606884967168, 1460295317318656, 84429863673895936, 5297505756426098688, 358520710389920598016, 26033795963713021116416, 2019060825791610516504576
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2024

Keywords

Links

Crossrefs

Cf. A201595, A370907.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(4*x/(3+exp(4*x)))/x))
  • PARI
    a(n) = sum(k=0, n+1, 3^(n+1-k)*k^n*binomial(n+1, k))/(4*(n+1));

Formula

a(n) = 1/(4*(n+1)) * Sum_{k=0..n+1} 3^(n+1-k) * k^n * binomial(n+1,k).
a(n) = n! * Sum_{k=0..n} 4^(n-k) * Stirling2(n,k)/(n-k+1)!. - Seiichi Manyama, Nov 07 2024

A370909 Expansion of e.g.f. (1/x) * Series_Reversion( 3*x/(1 + 2*exp(3*x)) ).

Original entry on oeis.org

1, 2, 14, 174, 3174, 76902, 2331630, 85048686, 3629630070, 177523551990, 9793095667326, 601667773414974, 40747538527887366, 3016185673617546822, 242280567558408368142, 20991011860150103490318, 1951271511259385883645846, 193723174296061459833879702
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2024

Keywords

Links

Crossrefs

Cf. A201595, A370910.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(3*x/(1+2*exp(3*x)))/x))
  • PARI
    a(n) = sum(k=0, n+1, 2^k*k^n*binomial(n+1, k))/(3*(n+1));

Formula

a(n) = 1/(3*(n+1)) * Sum_{k=0..n+1} 2^k * k^n * binomial(n+1,k).
a(n) = n! * Sum_{k=0..n} 2^k * 3^(n-k) * Stirling2(n,k)/(n-k+1)!. - Seiichi Manyama, Nov 07 2024
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