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 1-9 of 9 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

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

A361069 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^3) ).

Original entry on oeis.org

1, 1, -3, 40, -719, 18396, -598157, 23713726, -1108701519, 59735988424, -3644505746549, 248358786667674, -18697767289462967, 1541202721786228060, -138046868771541971373, 13351368704222195975206, -1386710317839048140282783, 153939247458296219191539984
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Links

Crossrefs

Cf. A052868, A361065, A361066, A361067, A361068.

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/((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 / ( (1-x)/(3*x) * LambertW(3*x/(1-x)) )^(1/3).
a(n) ~ (-1)^(n+1) * 3^(-3/2) * exp(-1/3) * (3 - exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Apr 22 2024

A361094 E.g.f. satisfies A(x) = exp( 1/(1 - x * A(x)^3) - 1 ).

Original entry on oeis.org

1, 1, 9, 166, 4717, 182136, 8911549, 528571408, 36864033945, 2956595372416, 268116203622961, 27128338649300736, 3029974270053623941, 370289278173654092800, 49150116757136815109733, 7041536364582774222616576, 1083004122024520209576760369
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Links

Crossrefs

Cf. A052873, A361093, A361095, A361096, A361097.
Cf. A212917, A361066.

Programs

  • Mathematica
    Table[n! * Sum[(3*n+1)^(k-1) * Binomial[n-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 02 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, (3*n+1)^(k-1)*binomial(n-1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (3*n+1)^(k-1) * binomial(n-1,n-k)/k!.
a(n) ~ (5 + sqrt(21))^n * n^(n-1) / (3^(3/4) * 7^(1/4) * 2^n * exp((3 - sqrt(21))/6 + (5 - sqrt(21))*n/2)). - Vaclav Kotesovec, Mar 02 2023

A361182 E.g.f. satisfies A(x) = exp( 3*x*A(x) ) / (1-x).

Original entry on oeis.org

1, 4, 41, 735, 19293, 672573, 29342241, 1540097541, 94579646553, 6656561754345, 528414534842949, 46716837535074897, 4552821617337191637, 484953672676323320109, 56056228305888242732841, 6988787950179969557086797, 934866118278080385555647025
Offset: 0

Views

Author

Seiichi Manyama, Mar 03 2023

Keywords

Links

Crossrefs

Cf. A352410, A352448.
Cf. A361066.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[3*x*A[x]]/(1 - 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, 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).
a(n) ~ (1 + 3*exp(1))^(n + 3/2) * n^(n-1) / (3^(3/2) * exp(n + 1/2)). - Vaclav Kotesovec, Mar 03 2023

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

A361212 E.g.f. satisfies A(x) = exp( 3*x*A(x) / (1-x) ).

Original entry on oeis.org

1, 3, 33, 612, 16353, 576108, 25306803, 1334701854, 82258866225, 5805344935368, 461848917299499, 40904277651802458, 3992219566916292873, 425766991650939828828, 49266876888419716251315, 6147944525591645916094182, 823045511075200872642258273
Offset: 0

Views

Author

Seiichi Manyama, Mar 04 2023

Keywords

Links

Crossrefs

Cf. A052868, A360939.
Cf. A361066, A361182.

Programs

  • PARI
    a(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) = 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)).

A365015 E.g.f. satisfies A(x) = exp( x*A(x)^3/(1 - x * A(x)) ).

Original entry on oeis.org

1, 1, 9, 154, 3997, 140216, 6217549, 333774064, 21051514425, 1526073116032, 125040978948241, 11428407889500416, 1152792683163827413, 127215353330004610048, 15246125111980753585365, 1971966282368187450198016, 273796236099258954747416689
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Links

Crossrefs

Cf. A361066, A361094, A365016.

Programs

  • Mathematica
    Array[#!*Sum[ (# + 2 k + 1)^(k - 1)*Binomial[# - 1, # - k]/k!, {k, 0, #}] &, 17, 0] (* Michael De Vlieger, Aug 18 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, (n+2*k+1)^(k-1)*binomial(n-1, n-k)/k!);

Formula

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

A365016 E.g.f. satisfies A(x) = exp( x*A(x)^3/(1 - x * A(x)^2) ).

Original entry on oeis.org

1, 1, 9, 160, 4345, 159796, 7434199, 418864426, 27732988609, 2110729489048, 181587635465671, 17426825999144926, 1845855944285411425, 213900244312057975348, 26919356609721984494311, 3656322063766897691641666, 533110345129065969043548289
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Links

Crossrefs

Cf. A361066, A361094, A365015.
Cf. A212722, A361093, A361143, A365012.

Programs

  • Mathematica
    Array[#!*Sum[ (2 # + k + 1)^(k - 1)*Binomial[# - 1, # - k]/k!, {k, 0, #}] &, 17, 0] (* Michael De Vlieger, Aug 18 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, (2*n+k+1)^(k-1)*binomial(n-1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (2*n+k+1)^(k-1) * binomial(n-1,n-k)/k!.
Showing 1-9 of 9 results.

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

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