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 1-10 of 10 results.

A352410 Expansion of e.g.f. LambertW( -x/(1-x) ) / (-x).

Original entry on oeis.org

1, 2, 9, 67, 717, 10141, 179353, 3816989, 95076537, 2714895433, 87457961421, 3138260371225, 124147801973605, 5368353187693757, 251928853285058433, 12752446755011776741, 692625349011401620209, 40178978855796929378065, 2479383850197948228950293
Offset: 0

Views

Author

Paul D. Hanna, Mar 15 2022

Keywords

Comments

An interesting property of this e.g.f. A(x) is that the sum of coefficients of x^k, k=0..n, in 1/A(x)^n equals zero, for n > 1.

Examples

			E.g.f.: A(x) = 1 + 2*x + 9*x^2/2! + 67*x^3/3! + 717*x^4/4! + 10141*x^5/5! + 179353*x^6/6! + 3816989*x^7/7! + ...
such that A(x) = exp(x*A(x)) / (1-x), where
exp(x*A(x)) = 1 + x + 5*x^2/2! + 40*x^3/3! + 449*x^4/4! + 6556*x^5/5! + 118507*x^6/6! + ... + A052868(n)*x^n/n! + ...
which equals LambertW(-x/(1-x)) * (1-x)/(-x).
Related table.
Another defining property of the e.g.f. A(x) is illustrated here.
The table of coefficients of x^k/k! in 1/A(x)^n begins:
n=1: [1,  -2,  -1,    -7,   -71,   -961, -16409, -339571, ...];
n=2: [1,  -4,   6,    -2,   -24,   -362,  -6644, -144538, ...];
n=3: [1,  -6,  21,   -33,    -3,    -63,  -1395,  -34275, ...];
n=4: [1,  -8,  44,  -148,   232,     -4,   -152,   -4876, ...];
n=5: [1, -10,  75,  -395,  1305,  -2045,     -5,    -355, ...];
n=6: [1, -12, 114,  -822,  4224, -13806,  21636,      -6, ...];
n=7: [1, -14, 161, -1477, 10381, -52507, 170401, -267043, -7, ...];
...
from which we can illustrate that the partial sum of coefficients of x^k, k=0..n, in 1/A(x)^n equals zero, for n > 1, as follows:
n=1:-1 = 1 +  -2;
n=2: 0 = 1 +  -4 +   6/2!;
n=3: 0 = 1 +  -6 +  21/2! +   -33/3!;
n=4: 0 = 1 +  -8 +  44/2! +  -148/3! +   232/4!;
n=5: 0 = 1 + -10 +  75/2! +  -395/3! +  1305/4! +  -2045/5!;
n=6: 0 = 1 + -12 + 114/2! +  -822/3! +  4224/4! + -13806/5! +  21636/6!;
n=7: 0 = 1 + -14 + 161/2! + -1477/3! + 10381/4! + -52507/5! + 170401/6! + -267043/7!;
...

Links

Crossrefs

Cf. A352411, A352412, A352448, A052868.
Cf. A102743, A108919, A331726.

Programs

  • Mathematica
    terms = 19; A[] = 0; Do[A[x] = Exp[x*A[x]]/(1-x) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Mar 24 2025 *)
With[{nn=20},CoefficientList[Series[LambertW[-x/(1-x)]/-x,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 24 2025 *)
  • PARI
    {a(n) = n!*polcoeff( (1/x)*serreverse( x/(exp(x +x^2*O(x^n)) + x) ),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(lambertw(-x/(1-x))/(-x))) \\ Michel Marcus, Mar 17 2022
  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(n, k)/k!); \\ Seiichi Manyama, Sep 24 2022

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) A(x) = LambertW( -x/(1-x) ) / (-x).
(2) A(x) = exp( x*A(x) ) / (1-x).
(3) A(x) = log( (1-x) * A(x) ) / x.
(4) A( x/(exp(x) + x) ) = exp(x) + x.
(5) A(x) = (1/x) * Series_Reversion( x/(exp(x) + x) ).
(6) Sum_{k=0..n} [x^k] 1/A(x)^n = 0, for n > 1.
(7) [x^(n+1)/(n+1)!] 1/A(x)^n = -n for n >= (-1).
a(n) ~ (1 + exp(1))^(n + 3/2) * n^(n-1) / exp(n + 1/2). - Vaclav Kotesovec, Mar 15 2022
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n,k)/k!. - Seiichi Manyama, Sep 24 2022

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

A360601 E.g.f. satisfies A(x) = exp(x*A(x)^2) / (1-x).

Original entry on oeis.org

1, 2, 13, 166, 3265, 87306, 2957509, 121400350, 5857287937, 324884241874, 20370279663901, 1424790170536470, 109990236302275201, 9289460282062082266, 852049115732672006101, 84345608594930495005966, 8962937531710834906989313, 1017655033307013508626619554
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2023

Keywords

Links

Crossrefs

Cf. A352410, A360609.
Cf. A052750, A352448.
Cf. A370875.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(lambertw(-2*x/(1-x)^2)/(-2*x))))

Formula

E.g.f.: sqrt(LambertW( -2*x/(1-x)^2 ) / (-2*x)).
a(n) ~ sqrt(1 + 2*exp(-1) - sqrt(1 + 2*exp(-1))) * n^(n-1) / (2 * (sqrt(1 + 2*exp(-1)) - 1)^(3/2) * exp(2*n + 1/2) * (1 + exp(-1) - sqrt(1 + 2*exp(-1)))^n). - Vaclav Kotesovec, Mar 06 2023
a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n+k,n-k)/k!. - Seiichi Manyama, Mar 09 2024

A352411 E.g.f.: x / LambertW( x/(1-x) ).

Original entry on oeis.org

1, 0, -1, 1, -7, 31, -281, 2381, -28015, 346879, -5149009, 82769149, -1499707991, 29444151023, -632715633577, 14631547277101, -364321853163871, 9686058045625471, -274387229080161569, 8241211775883617405, -261766195805536280839, 8763341168691985628719
Offset: 0

Views

Author

Paul D. Hanna, Mar 15 2022

Keywords

Comments

An interesting property of this e.g.f. A(x) is that the sum of coefficients of x^k, k=0..n, in A(x)^n equals zero, for n > 1.

Examples

			E.g.f.: A(x) = 1 + 0*x - x^2/2! + x^3/3! - 7*x^4/4! + 31*x^5/5! - 281*x^6/6! + 2381*x^7/7! - 28015*x^8/8! + ...
such that A(x) = (1-x) * exp(x/A(x)), where
exp(x/A(x)) = 1 + x + x^2/2! + 4*x^3/3! + 9*x^4/4! + 76*x^5/5! + 175*x^6/6! + 3606*x^7/7! + 833*x^8/8! + ...
Related series.
The e.g.f. A(x) satisfies A( x/(exp(-x) + x) ) = 1/(exp(-x) + x), where
1/(exp(-x) + x) = 1 - x^2/2! + x^3/3! + 5*x^4/4! - 19*x^5/5! - 41*x^6/6! + 519*x^7/7! - 183*x^8/8! + ...
Related table.
Another defining property of the e.g.f. A(x) is illustrated here.
The table of coefficients of x^k/k! in A(x)^n begins:
n=1: [1, 0, -1, 1,  -7,   31, -281, 2381, -28015, ...];
n=2: [1, 0, -2, 2,  -8,   42, -332, 2970, -33392, ...];
n=3: [1, 0, -3, 3,  -3,   33, -243, 2397, -26631, ...];
n=4: [1, 0, -4, 4,   8,    4, -104, 1292, -15712, ...];
n=5: [1, 0, -5, 5,  25,  -45,   -5,  285,  -6095, ...];
n=6: [1, 0, -6, 6,  48, -114,  -36,    6,   -720, ...];
n=7: [1, 0, -7, 7,  77, -203, -287, 1085,     -7, ...];
n=8: [1, 0, -8, 8, 112, -312, -848, 4152,  -1856, 8, ...];
...
from which we can illustrate that the partial sum of coefficients of x^k, k=0..n, in A(x)^n equals zero, for n > 1, as follows:
n=1: 1 = 1 + 0;
n=2: 0 = 1 + 0 + -2/2!;
n=3: 0 = 1 + 0 + -3/2! + 3/3!;
n=4: 0 = 1 + 0 + -4/2! + 4/3! +   8/4!;
n=5: 0 = 1 + 0 + -5/2! + 5/3! +  25/4! +  -45/5!;
n=6: 0 = 1 + 0 + -6/2! + 6/3! +  48/4! + -114/5! +  -36/6!;
n=7: 0 = 1 + 0 + -7/2! + 7/3! +  77/4! + -203/5! + -287/6! + 1085/7!;
n=8: 0 = 1 + 0 + -8/2! + 8/3! + 112/4! + -312/5! + -848/6! + 4152/7! + -1856/8!;
...

Crossrefs

Cf. A352410, A352412, A352448.

Programs

  • PARI
    {a(n) = n!*polcoeff( x/serreverse( x/(exp(-x  +x^2*O(x^n)) + x) ),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(x/lambertw(x/(1-x)))) \\ Michel Marcus, Mar 17 2022

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) A(x) = x / LambertW( x/(1-x) ).
(2) A(x) = (1-x) * exp( x/A(x) ).
(3) A(x) = x / log( A(x)/(1-x) ).
(4) A( x/(exp(-x) + x) ) = 1/(exp(-x) + x).
(5) A(x) = x / Series_Reversion( x/(exp(-x) + x) ).
(6) Sum_{k=0..n} [x^k] A(x)^n = 0, for n > 1.
(7) [x^(n+1)/(n+1)!] A(x)^n = (-1)^n * n for n >= (-1).
a(n) ~ (-1)^(n+1) * exp(-1) * (1 - exp(-1))^(n - 1/2) * n^(n-1). - Vaclav Kotesovec, Mar 15 2022

A352412 E.g.f.: 2*x / LambertW( 2*x/(1-x) ).

Original entry on oeis.org

1, 1, -4, 20, -224, 3392, -67232, 1629728, -46799104, 1552143104, -58386807296, 2455954797056, -114222622662656, 5819845970653184, -322384671892123648, 19290013218140254208, -1239886482366130946048, 85200320552417960394752
Offset: 0

Views

Author

Paul D. Hanna, Mar 15 2022

Keywords

Comments

An interesting property of this e.g.f. A(x) is that the sum of coefficients of x^k, k=0..n, in A(x)^n equals zero, for n > 1.

Examples

			E.g.f.: A(x) = 1 + x - 4*x^2/2! + 20*x^3/3! - 224*x^4/4! + 3392*x^5/5! - 67232*x^6/6! + 1629728*x^7/7! - 46799104*x^8/8! + ...
such that A(x) = (1-x) * exp(2*x/A(x)), where
exp(2*x/A(x)) = 1 + 2*x + 20*x^3/3! - 144*x^4/4! + 2672*x^5/5! - 51200*x^6/6! + 1271328*x^7/7! - 36628480*x^8/8! + ...
Related series.
The e.g.f. A(x) satisfies A( x/(exp(-2*x) + x) ) = 1/(exp(-2*x) + x), where
1/(exp(-2*x) + x) = 1 + x - 2*x^2/2! - 10*x^3/3! + 24*x^4/4! + 312*x^5/5! - 560*x^6/6! + ... + A336958(n)*(-x)^n/n! + ...
Related table.
Another defining property of the e.g.f. A(x) is illustrated here.
The table of coefficients of x^k/k! in A(x)^n begins:
n=1: [1, 1, -4,   20,  -224, 3392, -67232, 1629728, ...];
n=2: [1, 2, -6,   16,  -192, 2944, -58880, 1434752, ...];
n=3: [1, 3, -6,   -6,   -48, 1296, -29664,  776544, ...];
n=4: [1, 4, -4,  -40,    88,  128,  -7424,  263936, ...];
n=5: [1, 5,  0,  -80,   120,  280,   -320,   38720, ...];
n=6: [1, 6,  6, -120,   -24, 1872,  -3312,     768, ...];
n=7: [1, 7, 14, -154,  -392, 4424,  -3920,  -22288, ...];
...
from which we can illustrate that the partial sum of coefficients of x^k, k=0..n, in A(x)^n equals zero, for n > 1, as follows:
n=1: 2 = 1 + 1;
n=2: 0 = 1 + 2 + -6/2!;
n=3: 0 = 1 + 3 + -6/2! +   -6/3!;
n=4: 0 = 1 + 4 + -4/2! +  -40/3! +   88/4!;
n=5: 0 = 1 + 5 +  0/2! +  -80/3! +  120/4! +  280/5!;
n=6: 0 = 1 + 6 +  6/2! + -120/3! +  -24/4! + 1872/5! + -3312/6!;
n=7: 0 = 1 + 7 + 14/2! + -154/3! + -392/4! + 4424/5! + -3920/6! + -22288/7!;
...

Crossrefs

Cf. A352410, A352411, A352448, A336958.

Programs

  • PARI
    {a(n) = n!*polcoeff( x/serreverse( x/(exp(-2*x  +x^2*O(x^n)) + x) ),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(2*x/lambertw(2*x/(1-x)))) \\ Michel Marcus, Mar 17 2022

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) A(x) = 2*x / LambertW( 2*x/(1-x) ).
(2) A(x) = (1-x) * exp( 2*x/A(x) ).
(3) A(x) = 2*x / log( A(x)/(1-x) ).
(4) A( x/(exp(-2*x) + x) ) = 1/(exp(-2*x) + x).
(5) A(x) = x / Series_Reversion( x/(exp(-2*x) + x) ).
(6) Sum_{k=0..n} [x^k] A(x)^n = 0, for n > 1.
(7) [x^(n+1)/(n+1)!] A(x)^n = -(-2)^(n+1) * n for n >= (-1).
a(n) ~ (-1)^(n+1) * exp(-1) * sqrt(2) * (2 - exp(-1))^(n - 1/2) * n^(n-1). - Vaclav Kotesovec, Mar 15 2022

A361193 E.g.f. satisfies A(x) = exp( -2*x*A(x) ) / (1-x).

Original entry on oeis.org

1, -1, 6, -50, 648, -10952, 232336, -5919664, 176435328, -6024464000, 231972167424, -9946181374208, 470038191434752, -24276240445152256, 1360508977539004416, -82233680186863536128, 5332689963474238341120, -369321737420738845638656
Offset: 0

Views

Author

Seiichi Manyama, Mar 03 2023

Keywords

Links

Crossrefs

Cf. A352410, A352448, A361182, A361194.
Cf. A361068.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (-2)^k * (k+1)^(k-1) * binomial(n,k)/k!.
E.g.f.: LambertW( 2*x/(1-x) ) / (2*x).

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).

A380828 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x) / (1 + x) ).

Original entry on oeis.org

1, 3, 26, 398, 8904, 264072, 9790192, 436382256, 22748241024, 1358633214080, 91503397265664, 6862436244211968, 567252637423922176, 51244493078278198272, 5023312927780022323200, 531082672018567209801728, 60239691905397303186849792, 7297357396264290237329473536
Offset: 0

Views

Author

Seiichi Manyama, Feb 05 2025

Keywords

Links

Crossrefs

Cf. A352448, A380808.
Cf. A088690, A380830.
Cf. A376093.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp(2*x*A(x)) / ( 1 - x*exp(2*x*A(x)) ).
a(n) = n! * Sum_{k=0..n} (2*n+2)^k * binomial(n,k)/(k+1)!.
a(n) = A376093(n+1)/(n+1).

A376098 Expansion of e.g.f. -LambertW(-2*x / (1 - x))/2.

Original entry on oeis.org

0, 1, 6, 66, 1112, 25640, 753552, 26950000, 1136106624, 55167345792, 3032389917440, 186130732899584, 12619351769121792, 936591263680543744, 75527892444435486720, 6575887645386829301760, 614790327790529665138688, 61429094739085165675446272
Offset: 0

Views

Author

Seiichi Manyama, Sep 10 2024

Keywords

Links

Crossrefs

Cf. A052871, A376099.
Cf. A352448, A376093.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = x * (A(x) + exp(2*A(x))).
E.g.f.: Series_Reversion( x / (x + exp(2*x)) ).
a(n) = n! * Sum_{k=1..n} (2*k)^(k-1) * binomial(n-1,k-1)/k!.
a(n) = n * A352448(n-1).
a(n) ~ (2 + exp(-1))^(n + 1/2) * n^(n-1) / 2^(3/2). - Vaclav Kotesovec, Sep 10 2024

A380808 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x) / (1 + x*exp(-x)) ).

Original entry on oeis.org

1, 3, 24, 335, 6812, 183397, 6168406, 249350285, 11785793352, 638146503593, 38960123581154, 2648475653518081, 198429466488527164, 16246940820392924189, 1443430758561178861758, 138305198841617791230533, 14217431594874334746229520, 1560842183273111251153540945
Offset: 0

Views

Author

Seiichi Manyama, Feb 04 2025

Keywords

Links

Crossrefs

Cf. A088690, A161633, A380826.
Cf. A352448, A380828.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp(2*x*A(x)) / ( 1 - x*exp(x*A(x)) ).
a(n) = n! * Sum_{k=0..n} (n+k+2)^k * binomial(n,k)/(k+1)!.
Showing 1-10 of 10 results.

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