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 1-8 of 8 results.

A361065 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^2 ).

Original entry on oeis.org

1, 1, 7, 85, 1521, 36421, 1097743, 39968601, 1707558401, 83777885929, 4643185678551, 286930307457949, 19562851003118833, 1458832806486727725, 118121195050068075167, 10320576944751955718881, 967863775658734350214017, 96970880819175875321264209
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Links

Crossrefs

Cf. A052868, A361066, A361067, A361068, A361069.

Programs

  • Maple
    A361065 := proc(n)
    add((2*k+1)^(k-1)*binomial(n-1,n-k)/k!,k=0..n) ;
    %*n! ;
end proc:
seq(A361065(n),n=0..10) ; # R. J. Mathar, Mar 02 2023
  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(x/(1 - x))*A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n, (2*k+1)^(k-1)*binomial(n-1, n-k)/k!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x/(1-x))/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(-(1-x)/(2*x)*lambertw(-2*x/(1-x)))))

Formula

a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( -LambertW(-2*x/(1-x))/2 ).
E.g.f.: sqrt( -(1-x)/(2*x) * LambertW(-2*x/(1-x)) ).
a(n) ~ (1 + 2*exp(1))^(n + 1/2) * n^(n-1) / (2^(3/2) * exp(n)). - Vaclav Kotesovec, Mar 02 2023

A361066 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^3 ).

Original entry on oeis.org

1, 1, 9, 148, 3673, 123276, 5234599, 269262022, 16279709793, 1131627199816, 88926737901031, 7796168316687906, 754414052156289265, 79872584117422215484, 9184299004593618881655, 1139822558262829096519726, 151857077047173825979147969
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Links

Crossrefs

Cf. A052868, A361065, A361067, A361068, A361069.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(x/(1 - x))*A[x]^3] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n, (3*k+1)^(k-1)*binomial(n-1, n-k)/k!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-3*x/(1-x))/3)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((-(1-x)/(3*x)*lambertw(-3*x/(1-x)))^(1/3)))

Formula

a(n) = n! * Sum_{k=0..n} (3*k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( -LambertW(-3*x/(1-x))/3 ).
E.g.f.: ( -(1-x)/(3*x) * LambertW(-3*x/(1-x)) )^(1/3).
a(n) ~ (1 + 3*exp(1))^(n + 1/2) * n^(n-1) / (3^(3/2) * exp(n + 1/6)). - Vaclav Kotesovec, Mar 02 2023

A361068 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^2) ).

Original entry on oeis.org

1, 1, -1, 13, -127, 2101, -41801, 1030177, -29820127, 995977801, -37660751569, 1590847310581, -74242656468575, 3793664894534269, -210656932372422745, 12630986901470435401, -813335155262348743231, 55977540398642247218449
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Links

Crossrefs

Cf. A052868, A361065, A361066, A361067, A361069.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x/((1 - x)*A[x]^2)] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n, (-2*k+1)^(k-1)*binomial(n-1, n-k)/k!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*x/(1-x))/2)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/sqrt((1-x)/(2*x)*lambertw(2*x/(1-x)))))

Formula

a(n) = n! * Sum_{k=0..n} (-2*k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( LambertW(2*x/(1-x))/2 ).
E.g.f.: 1 / sqrt( (1-x)/(2*x) * LambertW(2*x/(1-x)) ).
a(n) ~ (-1)^(n+1) * 2^(-3/2) * exp(-1/2) * (2 - exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Apr 22 2024

A361067 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)) ).

Original entry on oeis.org

1, 1, 1, 4, 9, 76, 175, 3606, 833, 354376, -1605249, 65111410, -718371071, 20105327100, -351241054177, 9362931464446, -214514949732735, 6039303900168976, -165679758877120001, 5093296357218337386, -159900268661169533119, 5405435526807425433220
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Links

Crossrefs

Cf. A052868, A361065, A361066, A361068, A361069.

Programs

  • Mathematica
    nmax = 21; A[_] = 1;
Do[A[x_] = Exp[x/((1 - x)*A[x])] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n, (-k+1)^(k-1)*binomial(n-1, n-k)/k!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(x/(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/((1-x)/x*lambertw(x/(1-x)))))

Formula

a(n) = n! * Sum_{k=0..n} (-k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( LambertW(x/(1-x)) ).
E.g.f.: 1 / ( (1-x)/x * LambertW(x/(1-x)) ).
a(n) ~ -(-1)^n * exp(-1) * (1 - exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Mar 02 2023

A361092 E.g.f. satisfies A(x) = exp( x/(1 - x/A(x)^3) ).

Original entry on oeis.org

1, 1, 3, -5, -107, 1041, 20701, -440033, -8464455, 343190593, 5639857561, -423764450889, -4968055259771, 754544622295153, 3846355902999429, -1818148417882379729, 6637679490204153841, 5658469355898945338625, -84578525845602646639823
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Links

Crossrefs

Cf. A161630, A212722, A212917, A361090, A361091.
Cf. A361069, A361097.

Programs

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

Formula

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

A361194 E.g.f. satisfies A(x) = exp( -3*x*A(x) ) / (1-x).

Original entry on oeis.org

1, -2, 17, -237, 4893, -133683, 4567905, -187666587, 9017657433, -496470972951, 30824023641669, -2131090659947439, 162397790115179733, -13525005928296072915, 1222285110682680848169, -119135392516302191619507, 12458374493322416970025521
Offset: 0

Views

Author

Seiichi Manyama, Mar 03 2023

Keywords

Links

Crossrefs

Cf. A352410, A352448, A361182, A361193.
Cf. A361069.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (-3)^k * (k+1)^(k-1) * binomial(n,k)/k!.
E.g.f.: LambertW( 3*x/(1-x) ) / (3*x).

A361214 E.g.f. satisfies A(x) = exp( 3*x*A(x) / (1+x) ).

Original entry on oeis.org

1, 3, 21, 288, 5841, 158148, 5370003, 219641922, 10518990129, 577629889848, 35788733371179, 2470154920005798, 187970878034549001, 15636177199793409444, 1411635193678825868979, 137469669176542404342042, 14364540773583252035937633
Offset: 0

Views

Author

Seiichi Manyama, Mar 04 2023

Keywords

Links

Crossrefs

Cf. A335945, A361213.
Cf. A361069, A361194.

Programs

  • Mathematica
    Table[(-1)^n*n!*Sum[(-3)^k*(k + 1)^(k - 1)*Binomial[n - 1, n - k]/k!, {k, 0, n}], {n, 0, 20}] (* Wesley Ivan Hurt, May 25 2024 *)
  • PARI
    a(n) = (-1)^n*n!*sum(k=0, n, (-3)^k*(k+1)^(k-1)*binomial(n-1, n-k)/k!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-3*x/(1+x)))))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(-(1+x)/(3*x)*lambertw(-3*x/(1+x))))

Formula

a(n) = (-1)^n * n! * Sum_{k=0..n} (-3)^k * (k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp ( -LambertW(-3*x/(1+x)) ).
E.g.f.: -(1+x)/(3*x) * LambertW(-3*x/(1+x)).

A365040 E.g.f. satisfies A(x) = exp(x * (1 + x)/A(x)^3).

Original entry on oeis.org

1, 1, -3, 34, -623, 15636, -499277, 19382686, -886663647, 46716323752, -2786249779829, 185574001203834, -13652735530485647, 1099602989008154476, -96230900016000250269, 9092834662610587023286, -922622745817066477888703, 100054409045940667152740304
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2023

Keywords

Links

Crossrefs

Cf. A362771, A362773, A363478, A365038, A365039.
Cf. A361069.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(3*x*(1+x))/3)))

Formula

E.g.f.: exp( LambertW(3*x * (1+x))/3 ).
a(n) = n! * Sum_{k=0..n} (-3*k+1)^(k-1) * binomial(k,n-k)/k!.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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