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 1-6 of 6 results.

A125500 Expansion of -LambertW(-x^2*exp(x))/x^2.

Original entry on oeis.org

1, 1, 3, 13, 85, 701, 7261, 89125, 1277865, 20883385, 384194521, 7852225481, 176651705869, 4337650936789, 115468033349397, 3312409332578221, 101881034223806161, 3344745711740899697, 116747433680684736817
Offset: 0

Views

Author

Vladeta Jovovic, Dec 27 2006

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! +... - _Paul D. Hanna_, Aug 30 2008

Links

Crossrefs

Column k=2 of A362377.
Cf. A143740.

Programs

  • GAP
    List([0..30],n->Sum([0..n],k->Factorial(n)*(n-k+1)^(k-1)/Factorial(k)*Binomial(k,n-k))); # Muniru A Asiru, Feb 19 2018
  • Mathematica
    Table[Sum[n!*(n-k+1)^(k-1)/k!*Binomial[k,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 04 2013 *)
With[{nmax=30}, CoefficientList[Series[-LambertW[-x^2*Exp[x]]/x^2, {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Feb 19 2018 *)
  • 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)*Ex,n)} \\ Paul D. Hanna, Jan 02 2007
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x+x^2*A));n!*polcoeff(A,n)} \\ Paul D. Hanna, Aug 30 2008
  • PARI
    {a(n,m=1)=if(n==0,1,sum(k=0,n,n!/k!*m*(n-k+m)^(k-1)*binomial(k,n-k)))} \\ Paul D. Hanna, Jun 17 2009
  • 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)} \\ Paul D. Hanna, Jun 17 2009
  • PARI
    x='x+O('x^30); Vec(serlaplace(-lambertw(-x^2*exp(x))/x^2)) \\ G. C. Greubel, Feb 19 2018

Formula

E.g.f.: A(x) = exp(x + x^2*A(x)). [Paul D. Hanna, Aug 30 2008]
From Paul D. Hanna, Jun 17 2009: (Start)
a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(k,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(k,n-k). (End)
a(n) ~ sqrt((c+1)/2)/(2*c^2) * exp(n*(2*c-1)/2) * n^(n-1), where c = LambertW(exp(-1/2)/2) = 0.2388350311316... - Vaclav Kotesovec, Jan 04 2013
E.g.f.: exp(x - LambertW(-x^2 * exp(x))). - Seiichi Manyama, Apr 20 2023

Extensions

More terms from Paul D. Hanna, Jan 02 2007

A362378 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * (j+1)^(n-2*j-1) / (j! * (n-3*j)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 9, 1, 1, 1, 1, 4, 17, 41, 1, 1, 1, 1, 5, 25, 81, 191, 1, 1, 1, 1, 6, 33, 121, 441, 1191, 1, 1, 1, 1, 7, 41, 161, 751, 3641, 9353, 1, 1, 1, 1, 8, 49, 201, 1121, 7351, 33825, 77897, 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,   1,   1,   1,    1,    1,    1, ...
  1,   2,   3,   4,    5,    6,    7, ...
  1,   9,  17,  25,   33,   41,   49, ...
  1,  41,  81, 121,  161,  201,  241, ...
  1, 191, 441, 751, 1121, 1551, 2041, ...

Links

Crossrefs

Columns k=0..3 give A000012, A362381, A362390, A362391.
Cf. A362043, A362377.

Programs

  • PARI
    T(n, k) = n! * sum(j=0, n\3, (k/6)^j*(j+1)^(n-2*j-1)/(j!*(n-3*j)!));

Formula

E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x + k*x^3/6 * A_k(x)).
A_k(x) = exp(x - LambertW(-k*x^3/6 * exp(x))).
A_k(x) = -6 * LambertW(-k*x^3/6 * exp(x))/(k*x^3) for k > 0.

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)

A362380 E.g.f. satisfies A(x) = exp(x + 3*x^2/2 * A(x)).

Original entry on oeis.org

1, 1, 4, 19, 154, 1456, 18136, 260002, 4430812, 85170988, 1854422236, 44693165716, 1188169271488, 34434053438968, 1082632555160248, 36666259172292016, 1331754793762045456, 51622725829298301520, 2127683533625205288400
Offset: 0

Views

Author

Seiichi Manyama, Apr 20 2023

Keywords

Links

Crossrefs

Column k=3 of A362377.
Cf. A362397.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x + 3*x^2/2*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(x-lambertw(-3*x^2/2*exp(x)))))

Formula

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

A362394 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, 0, 1, 1, 1, -1, -5, 1, 1, 1, -2, -11, -14, 1, 1, 1, -3, -17, -11, 56, 1, 1, 1, -4, -23, 10, 381, 736, 1, 1, 1, -5, -29, 49, 976, 2461, 1114, 1, 1, 1, -6, -35, 106, 1841, 3736, -21083, -45156, 1, 1, 1, -7, -41, 181, 2976, 3121, -106910, -449623, -428660, 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,   0,   -1,   -2,   -3,   -4,    -5, ...
  1,  -5,  -11,  -17,  -23,  -29,   -35, ...
  1, -14,  -11,   10,   49,  106,   181, ...
  1,  56,  381,  976, 1841, 2976,  4381, ...
  1, 736, 2461, 3736, 3121, -824, -9539, ...

Links

Crossrefs

Columns k=0..3 give A000012, A362395, A362396, A362397.
Cf. A362277, A362377.

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.

A362483 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 * (2*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, 10, 1, 1, 1, 4, 19, 70, 1, 1, 1, 5, 28, 169, 646, 1, 1, 1, 6, 37, 298, 2041, 7576, 1, 1, 1, 7, 46, 457, 4186, 30811, 106744, 1, 1, 1, 8, 55, 646, 7081, 74116, 560827, 1761628, 1, 1, 1, 9, 64, 865, 10726, 141901, 1578340, 11957905, 33361948, 1
Offset: 0

Views

Author

Seiichi Manyama, Apr 21 2023

Keywords

Examples

			Square array begins:
  1,   1,    1,    1,    1,     1, ...
  1,   1,    1,    1,    1,     1, ...
  1,   2,    3,    4,    5,     6, ...
  1,  10,   19,   28,   37,    46, ...
  1,  70,  169,  298,  457,   646, ...
  1, 646, 2041, 4186, 7081, 10726, ...

Links

Crossrefs

Columns k=0..3 give A000012, A362474, A143768, A362475.
Cf. A362377, A362490.

Programs

  • PARI
    T(n, k) = n! * sum(j=0, n\2, (k/2)^j*(2*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)^2).
A_k(x) = exp(x - LambertW(-k*x^2 * exp(2*x))/2).
A_k(x) = sqrt( -LambertW(-k*x^2 * exp(2*x))/(k*x^2) ) for k > 0.
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.