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 1-3 of 3 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

A352448 Expansion of e.g.f. LambertW( -2*x/(1-x) ) / (-2*x).

Original entry on oeis.org

1, 3, 22, 278, 5128, 125592, 3850000, 142013328, 6129705088, 303238991744, 16920975718144, 1051612647426816, 72045481821580288, 5394849460316820480, 438392509692455286784, 38424395486908104071168, 3613476161122656804438016
Offset: 0

Views

Author

Paul D. Hanna, Mar 16 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 + 3*x + 22*x^2/2! + 278*x^3/3! + 5128*x^4/4! + 125592*x^5/5! + 3850000*x^6/6! + 142013328*x^7/7! + ...
such that A(x) = exp( 2*x*A(x) ) / (1-x), where
exp( 2*x*A(x) ) = 1 + 2*x + 16*x^2/2! + 212*x^3/3! + 4016*x^4/4! + 99952*x^5/5! + 3096448*x^6/6! + 115063328*x^7/7! + ...
Related table.
Another interesting 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,  -3,  -4,   -44,  -736,  -16832, -491168, ...];
n=2: [1,  -6,  10,   -16,  -320,   -8064, -249344, ...];
n=3: [1,  -9,  42,   -78,   -48,   -1776,  -66528, ...];
n=4: [1, -12,  92,  -392,   728,    -128,   -8960, ...];
n=5: [1, -15, 160, -1120,  4600,   -8520,    -320, ...];
n=6: [1, -18, 246, -2424, 16104,  -64752,  119952, ...];
...
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:-2 = 1 +  -3;
n=2: 0 = 1 +  -6 +  10/2!;
n=3: 0 = 1 +  -9 +  42/2! +   -78/3!;
n=4: 0 = 1 + -12 +  92/2! +  -392/3! +   728/4!;
n=5: 0 = 1 + -15 + 160/2! + -1120/3! +  4600/4! +   -8520/5!;
n=6: 0 = 1 + -18 + 246/2! + -2424/3! + 16104/4! +  -64752/5! +  119952/6!;
...

Links

Crossrefs

Cf. A352410, A352411, A352412.
Cf. A361068.

Programs

  • Mathematica
    terms = 17; A[] = 0; Do[A[x] = Exp[2x*A[x]]/(1-x) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Mar 24 2025 *)
  • PARI
    {a(n) = n!*polcoeff( (1/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(lambertw(-2*x/(1-x))/(-2*x))) \\ Michel Marcus, Mar 17 2022
  • PARI
    a(n) = n!*sum(k=0, n, 2^k*(k+1)^(k-1)*binomial(n, k)/k!); \\ Seiichi Manyama, Mar 03 2023

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) A(x) = LambertW( -2*x/(1-x) ) / (-2*x).
(2) A(x) = exp( 2*x*A(x) ) / (1-x).
(3) A(x) = log( (1-x) * A(x) ) / (2*x).
(4) A( x/(exp(2*x) + x) ) = exp(2*x) + x.
(5) A(x) = (1/x) * Series_Reversion( x/(exp(2*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 = -2^(n+1) * n for n >= (-1).
a(n) ~ (1 + 2*exp(1))^(n + 3/2) * n^(n-1) / (2^(3/2) * exp(n + 1/2)). - Vaclav Kotesovec, Mar 18 2022
a(n) = n! * Sum_{k=0..n} 2^k * (k+1)^(k-1) * binomial(n,k)/k!. - Seiichi Manyama, Mar 03 2023

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
Showing 1-3 of 3 results.

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