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 19 results. Next

A362771 E.g.f. satisfies A(x) = exp( x * (1+x) * A(x) ).

Original entry on oeis.org

1, 1, 5, 34, 353, 4756, 80107, 1617358, 38145473, 1029745576, 31326858611, 1060716408874, 39571357618465, 1612919873514028, 71321521181852411, 3400790769764598886, 173950205958460627073, 9501239617356541012432, 551961456374529522954595
Offset: 0

Views

Author

Seiichi Manyama, May 02 2023

Keywords

Links

Crossrefs

Cf. A052868, A125500.

Programs

  • Mathematica
    nmax = 20; 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
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(1+x)))))

Formula

E.g.f.: exp( -LambertW(-x * (1+x)) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(k,n-k)/k!.
a(n) ~ sqrt(2 + 8*exp(-1) - 2*sqrt(1 + 4*exp(-1))) * 2^(n-1) * n^(n-1) / ((sqrt(1 + 4*exp(-1)) - 1)^n * exp(n - 3/2)). - Vaclav Kotesovec, May 03 2023

A161630 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)) ).

Original entry on oeis.org

1, 1, 3, 19, 181, 2321, 37501, 731935, 16758393, 440525377, 13077834841, 432796650551, 15799794395749, 630773263606513, 27339525297079269, 1278550150117141231, 64171287394646697841, 3440711053857464325377
Offset: 0

Views

Author

Paul D. Hanna, Jun 17 2009

Keywords

Examples

			E.g.f: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...
log(A(x))/x = 1 + x*A(x) + x^2*A(x)^2 + x^3*A(x)^3 + x^4*A(x)^4 +...

Links

Crossrefs

Cf. A161633 (e.g.f. = log(A(x))/x).
Cf. A212722, A212917, A245265, A125500.

Programs

  • Mathematica
    Table[Sum[n! * (n-k+1)^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 10 2014 *)
  • PARI
    {a(n,m=1)=if(n==0,1,sum(k=0,n,n!/k!*m*(n-k+m)^(k-1)*binomial(n-1,n-k)))}
  • PARI
    {a(n,m=1)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(x/(1-x*A)));n!*polcoeff(A^m,n)}

Formula

a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(n-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(n-k+m)^(k-1)/k! * C(n-1,n-k).
E.g.f. satisfies: A(x) = exp(x) * A(x)^(x*A(x)). - Paul D. Hanna, Aug 02 2013
a(n) ~ n^(n-1) * (1+2*c)^(n+1/2) / (sqrt(1+c) * 2^(2*n+2) * exp(n) * c^(2*n+3/2)), where c = LambertW(1/2) = 0.351733711249195826... (see A202356). - Vaclav Kotesovec, Jan 10 2014

A143768 E.g.f. satisfies A(x) = exp(x + x^2*A(x)^2).

Original entry on oeis.org

1, 1, 3, 19, 169, 2041, 30811, 560827, 11957905, 292399345, 8069068531, 248093713891, 8411093625529, 311750189715433, 12541478207183563, 544268121894899851, 25345579186001847841, 1260715969618060192225
Offset: 0

Views

Author

Paul D. Hanna, Aug 31 2008

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 169*x^4/4! + 2041*x^5/5! + ...
A(x)^2 = 1 + 2*x + 8*x^2/2! + 56*x^3/3! + 544*x^4/4! + 6912*x^5/5! + ...
log(A(x)) = x + x^2 + 2*x^3 + 8*x^4/2! + 56*x^5/3! + 544*x^6/4! + ...

Links

Crossrefs

Cf. A047974, A088695, A125500 (variant).

Programs

  • Mathematica
    CoefficientList[Series[(LambertW[-2*x^2*E^(2*x)]/(-2*x^2))^(1/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x+x^2*A^2));(n+0)!*polcoeff(A,n)}
  • PARI
    {a(n)=local(A=sum(m=0,n,1*(2*m+1)^(m-1)*(x^2)^m*exp((2*m+1)*x+x*O(x^n))/m!));n!*polcoeff(A,n)}
  • PARI
    a(n,m=1)=n!*sum(k=0,n,m*(2*(n-k)+m)^(k-1)/k!*binomial(k,n-k)) \\ Paul D. Hanna, Jul 11 2009

Formula

Expansion of [LambertW(-2*x^2*exp(2x))/(-2*x^2)]^(1/2).
E.g.f.: A(x) = Sum_{m>=0} (2n+1)^(n-1) * exp((2n+1)*x) * x^(2n)/n! .
From Paul D. Hanna, Jul 11 2009: (Start)
a(n) = n! * Sum_{k=0..n} C(k,n-k) * (2*(n-k)+1)^(k-1)/k!.
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = n! * Sum_{k=0..n} C(k,n-k) * m*(2*(n-k)+m)^(k-1)/k!.
...
If log(A(x)) = Sum_{n>=1} L(n)*x^n/n!, then
L(n) = n! * Sum_{k=0..n} (2*(n-k))^(k-1)/k! * C(k,n-k). (End)
a(n) ~ sqrt(1+LambertW(1/sqrt(2*exp(1)))) * n^(n-1) / (2*exp(n) * (LambertW(1/sqrt(2*exp(1))))^(n+1)). - Vaclav Kotesovec, Jul 09 2013

A161631 E.g.f. satisfies A(x) = 1 + x*exp(x*A(x)).

Original entry on oeis.org

1, 1, 2, 9, 52, 425, 4206, 50827, 713000, 11500785, 208833850, 4226139731, 94226705772, 2296472176297, 60727113115046, 1732020500240955, 52998549321251536, 1731977581804704737, 60205422811336194546
Offset: 0

Views

Author

Paul D. Hanna, Jun 18 2009

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 425*x^5/5! +...
exp(x*A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! +...

Links

Crossrefs

Cf. A125500, A364978, A364979.

Programs

  • Mathematica
    CoefficientList[Series[1-LambertW[-x^2*E^x]/x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
  • PARI
    {a(n,m=1)=n!*sum(k=0,n,m*binomial(n-k+m,k)/(n-k+m)*k^(n-k)/(n-k)!)}

Formula

E.g.f.: A(x) = 1 - LambertW(-x^2*exp(x))/x.
E.g.f.: A(x) = 1 + Sum_{n>=1} x^(2*n-1) * n^(n-1) * exp(n*x) / n!.
E.g.f.: A(x) = (1/x)*Series_Reversion(x/B(x)) where B(x) = 1 + x*exp(x)/B(x) = (1+sqrt(1+4*x*exp(x)))/2.
a(n) = n*A125500(n-1) for n>0, where exp(x*A(x)) = e.g.f. of A125500.
a(n) = n!*Sum_{k=0..n} C(n-k+1,k)/(n-k+1) * k^(n-k)/(n-k)!.
If A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then
a(n,m) = n!*Sum_{k=0..n} m*C(n-k+m,k)/(n-k+m) * k^(n-k)/(n-k)!.
a(n) ~ sqrt(1+LambertW(1/(2*exp(1/2)))) * n^(n-1) / (exp(n) * 2^(n+1/2) * (LambertW(1/(2*exp(1/2))))^(n+1)). - Vaclav Kotesovec, Jul 09 2013

A362377 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (k/2)^j * (j+1)^(n-j-1) / (j! * (n-2*j)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 7, 1, 1, 1, 4, 13, 34, 1, 1, 1, 5, 19, 85, 216, 1, 1, 1, 6, 25, 154, 701, 1696, 1, 1, 1, 7, 31, 241, 1456, 7261, 15898, 1, 1, 1, 8, 37, 346, 2481, 18136, 89125, 173468, 1, 1, 1, 9, 43, 469, 3776, 35761, 260002, 1277865, 2161036, 1
Offset: 0

Views

Author

Seiichi Manyama, Apr 20 2023

Keywords

Examples

			Square array begins:
  1,    1,    1,     1,     1,     1,     1, ...
  1,    1,    1,     1,     1,     1,     1, ...
  1,    2,    3,     4,     5,     6,     7, ...
  1,    7,   13,    19,    25,    31,    37, ...
  1,   34,   85,   154,   241,   346,   469, ...
  1,  216,  701,  1456,  2481,  3776,  5341, ...
  1, 1696, 7261, 18136, 35761, 61576, 97021, ...

Links

Crossrefs

Columns k=0..3 give A000012, A143740, A125500, A362380.
Cf. A362378, A362394.

Programs

  • PARI
    T(n, k) = n! * sum(j=0, n\2, (k/2)^j*(j+1)^(n-j-1)/(j!*(n-2*j)!));

Formula

E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x + k*x^2/2 * A_k(x)).
A_k(x) = exp(x - LambertW(-k*x^2/2 * exp(x))).
A_k(x) = -2 * LambertW(-k*x^2/2 * exp(x))/(k*x^2) for k > 0.

A362392 E.g.f. satisfies A(x) = exp(x + x^3 * A(x)).

Original entry on oeis.org

1, 1, 1, 7, 49, 241, 2041, 26041, 282913, 3449377, 57170161, 973059121, 16847893921, 343341027745, 7680743819113, 175958943331081, 4375517632543681, 118932887426911681, 3374685950589927649, 100735118425384221025, 3217474234925998764481
Offset: 0

Views

Author

Seiichi Manyama, Apr 20 2023

Keywords

Links

Crossrefs

Column k=6 of A362378.
Cf. A118395, A125500, A362393, A362430.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^3*exp(x)))))

Formula

E.g.f.: exp(x - LambertW(-x^3 * exp(x))) = -LambertW(-x^3 * exp(x))/x^3.
a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(n-2*k-1) / (k! * (n-3*k)!).

A143740 E.g.f. satisfies A(x) = exp(x + x^2/2 * A(x)).

Original entry on oeis.org

1, 1, 2, 7, 34, 216, 1696, 15898, 173468, 2161036, 30282076, 471599316, 8082816160, 151218316120, 3066890630168, 67031194526416, 1570793031033616, 39290173530686544, 1044871388684004304, 29440090627527552976
Offset: 0

Views

Author

Paul D. Hanna, Aug 30 2008

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 34*x^4/4! + 216*x^5/5! + ...

Links

Crossrefs

Column k=1 of A362377.
Cf. A125500.

Programs

  • Mathematica
    CoefficientList[Series[-2*LambertW[-x^2*E^x/2]/x^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
  • Maxima
    a[n]:=(if n<2 then 1 else a[n-1]+sum((n-1)*(n-k)*binomial(n-2,k)*a[k]*a[n-2-k],k,0,n-2)/2);
makelist(a[n],n,0,100); /* Tani Akinari, Nov 01 2017 */
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x+x^2*A/2));(n+0)!*polcoeff(A,n)}
  • PARI
    {a(n)=local(A=sum(m=0,n,(m+1)^(m-1)*(x^2/2)^m*exp((m+1)*x+x*O(x^n))/m!)); n!*polcoeff(A,n)}
  • PARI
    {a(n)=local(Ex=exp(x+x*O(x^n)),W=Ex);for(k=0,n,W=exp(x*W)); n!*polcoeff(subst(W,x,x^2*Ex/2)*Ex,n)}

Formula

E.g.f.: A(x) = -2*LambertW( -x^2*exp(x)/2 )/x^2.
E.g.f.: A(x) = Sum_{n>=0} (n+1)^(n-1)*(x^2/2)^n*exp((n+1)*x)/n!.
a(n) ~ sqrt(1+LambertW(1/sqrt(2*exp(1)))) * n^(n-1) /(2^(n+1/2) * exp(n) * (LambertW(1/sqrt(2*exp(1))))^(n+2)). - Vaclav Kotesovec, Jul 09 2013
Recurrence: a(0)=1, a(1)=1, for n > 1, a(n) = a(n-1) + Sum_{k=0..n-2} (n-1)*(n-k)*binomial(n-2,k)*a(k)*a(n-2-k)/2. - Tani Akinari, Nov 01 2017
From Seiichi Manyama, Apr 20 2023: (Start)
E.g.f.: exp(x - LambertW(-x^2/2 * exp(x))).
a(n) = n! * Sum_{k=0..floor(n/2)} (1/2)^k * (k+1)^(n-k-1) / (k! * (n-2*k)!). (End)

A362393 E.g.f. satisfies A(x) = exp(x + x^4 * A(x)).

Original entry on oeis.org

1, 1, 1, 1, 25, 241, 1441, 6721, 87361, 1729729, 24816961, 270452161, 3705324481, 85344916801, 1992230175937, 38047293910081, 709217112938881, 17385498239168641, 514103858592923521, 14254662916125735553, 366807994235438359681, 10338786602768939575681
Offset: 0

Views

Author

Seiichi Manyama, Apr 20 2023

Keywords

Links

Crossrefs

Cf. A125500, A362392.
Cf. A362431.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^4*exp(x)))))

Formula

E.g.f.: exp(x - LambertW(-x^4 * exp(x))) = -LambertW(-x^4 * exp(x))/x^4.
a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(n-3*k-1) / (k! * (n-4*k)!).

A363354 E.g.f. satisfies A(x) = exp(x * (1 + x * A(x)^3)).

Original entry on oeis.org

1, 1, 3, 25, 277, 4221, 81421, 1891429, 51638217, 1618907257, 57332786041, 2264047223241, 98641443498973, 4700569138096885, 243213757144477029, 13579261873673960941, 813757288951509415441, 52098716516012891238129, 3548972379593741013388657
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2023

Keywords

Links

Crossrefs

Cf. A125500, A143768, A363529.
Cf. A212917.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-3*x^2*exp(3*x))/3)))

Formula

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

A363529 E.g.f. satisfies A(x) = exp(x * (1 + x * A(x)^4)).

Original entry on oeis.org

1, 1, 3, 31, 409, 7361, 170251, 4732351, 154694961, 5814634753, 246946119571, 11698927124831, 611660759515081, 34984757221103041, 2173041881789331099, 145669007565799127551, 10482025117382045382241, 805892200757926620144641
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2023

Keywords

Links

Crossrefs

Cf. A125500, A143768, A363354.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-4*x^2*exp(4*x))/4)))

Formula

E.g.f.: exp( x - LambertW(-4*x^2*exp(4*x))/4 ).
a(n) = n! * Sum_{k=0..n} (4*n-4*k+1)^(k-1) * binomial(k,n-k)/k!.
Showing 1-10 of 19 results. Next

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