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.

Previous Showing 31-38 of 38 results.

A226349 Number of functions f:{1,2,...,n} -> {1,2,...,n} such that the 1 and the 2 are in the same component of the functional digraph representation of f.

Original entry on oeis.org

0, 0, 3, 20, 188, 2280, 33864, 595196, 12081600, 278122032, 7159299200, 203771364324, 6354217539072, 215429796291320, 7889813961243648, 310413633428119500, 13057068314325008384, 584737112800511959104, 27776659696045110558720, 1395009275793285886030772, 73854320834079368232960000
Offset: 0

Views

Author

Geoffrey Critzer, Jun 04 2013

Keywords

Examples

			a(3)=20 because there are 17 connected functions on [3] and (2,1,3), (1,1,3), (2,2,3) where the functions are represented by their values.

Programs

  • Mathematica
    nn=18; t=Sum[n^(n-1)x^n/n!, {n,1,nn+2}]; Join[{0,0}, Range[0,nn]! CoefficientList[Series[D[D[Log[1/(1-t)], x], x]/(1-t), {x,0,nn}], x]]
a[ n_] := If[ n < 2, 0, With[ {m = n - 2}, With[ {t = 1 + Sum[k^k x^k/k!, {k, m + 2}]}, m! SeriesCoefficient[ D[ Log[ t], {x, 2}] t, {x, 0, m} ]]]] (* Michael Somos, Jun 04 2013 *)
  • PARI
    {a(n) = local(A); if( n<2, 0, m = n-2; A = sum( k=0, m+2, k^k * x^k / k!, x^3 * O(x^m)); m! * polcoeff( log(A)'' * A, m))} /* Michael Somos, Jun 04 2013 */

Formula

E.g.f. is the double integral of A''(x)*B(x) dx^2 where A(x) is the e.g.f. for A001865 and B(x) is the e.g.f. for A000312.

A308458 Expansion of e.g.f. log(Sum_{k>=0} k^binomial(k,2) * x^k / k!).

Original entry on oeis.org

1, 1, 23, 3994, 9745169, 470126386536, 558542572785461515, 19342808645467142112096240, 22528399370853856386499346950471953, 999999999774716004550606847948627702867525440, 1890591424701781041871514584507296209311760279398415565711
Offset: 1

Views

Author

Seiichi Manyama, May 27 2019

Keywords

Examples

			E.g.f.: x + x^2/2! + 23*x^3/3! + 3994*x^4/4! + 9745169*x^5/5! + 470126386536*x^6/6! + 558542572785461515*x^7/7! + ... .

Links

Crossrefs

Cf. A001187, A001865, A003027, A054941.

Programs

  • PARI
    N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, k^binomial(k, 2)*x^k/k!))))

A332236 E.g.f.: -log(2 - 1 / (1 + LambertW(-x))).

Original entry on oeis.org

1, 5, 41, 466, 6769, 119736, 2497585, 60037328, 1634619969, 49733223040, 1672657257721, 61636181886720, 2470033974057649, 106970912288285696, 4979259164362745025, 247940951411958163456, 13152705012933836446465, 740578125097986605678592, 44115815578591964641401289
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 07 2020

Keywords

Crossrefs

Cf. A000312, A001865, A133297, A202477, A210661, A308863, A323673, A332237.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[-Log[2 - 1/(1 + LambertW[-x])], {x, 0, nmax}], x] Range[0, nmax]! // Rest
a[n_] := a[n] = n^n + (1/n) Sum[Binomial[n, k] (n - k)^(n - k) k a[k], {k, 1, n - 1}]; Table[a[n], {n, 1, 19}]

Formula

E.g.f.: -log(1 - Sum_{k>=1} k^k * x^k / k!).
a(n) = n^n + (1/n) * Sum_{k=1..n-1} binomial(n,k) * (n-k)^(n-k) * k * a(k).
a(n) ~ (n-1)! * 2^n * exp(n/2). - Vaclav Kotesovec, Feb 16 2020

A332237 E.g.f.: -log(1 + LambertW(-x) * (2 + LambertW(-x)) / 2).

Original entry on oeis.org

1, 2, 8, 49, 409, 4356, 56734, 877094, 15742521, 322454800, 7434673036, 190792267128, 5398552673617, 167087263076384, 5617979017621650, 203987454978218416, 7957053981454827601, 331920300203780633856, 14746208516909980554736, 695208730205550274544000
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 07 2020

Keywords

Crossrefs

Cf. A000272, A001858, A001865, A057817, A133297, A218688, A323673, A332236.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[-Log[1 + LambertW[-x] (2 + LambertW[-x])/2], {x, 0, nmax}], x] Range[0, nmax]! // Rest
a[n_] := a[n] = n^(n - 2) + (1/n) Sum[Binomial[n, k] (n - k)^(n - k - 2) k a[k], {k, 1, n - 1}]; Table[a[n], {n, 1, 20}]

Formula

E.g.f.: -log(1 - Sum_{k>=1} k^(k-2) * x^k / k!).
a(n) = n^(n-2) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * (n-k)^(n-k-2) * k * a(k).
a(n) ~ 2 * n^(n-2). - Vaclav Kotesovec, Feb 16 2020

A347993 a(n) = n! * Sum_{k=1..n} (-1)^(k+1) * n^(n-k) / (n-k)!.

Original entry on oeis.org

1, 2, 15, 136, 1645, 24336, 426979, 8658560, 199234809, 5128019200, 145969492471, 4552809182208, 154404454932325, 5656950010320896, 222655633595044875, 9369696305273798656, 419790650812640438641, 19950175280765680680960, 1002394352017754098219999, 53092232229227200348160000
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 23 2021

Keywords

Crossrefs

Cf. A000169, A000312, A001865, A063169, A133297, A134095, A277458, A347994.

Programs

  • Mathematica
    Table[n! Sum[(-1)^(k + 1) n^(n - k)/(n - k)!, {k, 1, n}], {n, 1, 20}]
nmax = 20; CoefficientList[Series[-LambertW[-x]/(1 - LambertW[-x]^2), {x, 0, nmax}], x] Range[0, nmax]! // Rest
  • PARI
    a(n) = n! * sum(k=1, n, (-1)^(k+1)*n^(n-k)/(n-k)!); \\ Michel Marcus, Sep 23 2021

Formula

E.g.f.: -LambertW(-x) / (1 - LambertW(-x)^2).
a(n) = n * A133297(n).

A134558 Array read by antidiagonals, a(n,k) = gamma(n+1,k)*e^k, where gamma(n,k) is the upper incomplete gamma function and e is the exponential constant 2.71828...

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 5, 3, 1, 24, 16, 10, 4, 1, 120, 65, 38, 17, 5, 1, 720, 326, 168, 78, 26, 6, 1, 5040, 1957, 872, 393, 142, 37, 7, 1, 40320, 13700, 5296, 2208, 824, 236, 50, 8, 1, 362880, 109601, 37200, 13977, 5144, 1569, 366, 65, 9, 1, 3628800, 986410, 297856
Offset: 0

Views

Author

Ross La Haye, Jan 22 2008

Keywords

Examples

			Square array begins:
    1,    1,    1,     1,     1,     1,      1, ...
    1,    2,    3,     4,     5,     6,      7, ...
    2,    5,   10,    17,    26,    37,     50, ...
    6,   16,   38,    78,   142,   236,    366, ...
   24,   65,  168,   393,   824,  1569,   2760, ...
  120,  326,  872,  2208,  5144, 10970,  21576, ...
  720, 1957, 5296, 13977, 34960, 81445, 176112, ...

Links

Crossrefs

Cf. a(n, 0) = A000142(n); a(n, 1) = A000522(n); a(n, 2) = A010842(n); a(n, 3) = A053486(n); a(n, 4) = A053487(n); a(n, 5) = A080954(n); a(n, 6) = A108869(n); a(1, k) = A000027(k+1); a(2, k) = A002522(k+1); a(n, n) = A063170(n); a(n, n+1) = A001865(n+1); a(n, n+2) = A001863(n+2).
Another version: A089258.
A transposed version: A080955.
Cf. A001113.

Programs

  • Mathematica
    T[n_,k_] := Gamma[n+1, k]*E^k; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] //Flatten (* Amiram Eldar, Jun 27 2020 *)

Formula

a(n,k) = gamma(n+1,k)*e^k = Sum_{m=0..n} m!*binomial(n,m)*k^(n-m).
a(n,k) = n*a(n-1,k) + k^n for n,k > 0.
E.g.f. (by columns) is e^(kx)/(1-x).
a(n,k) = the binomial transform by columns of a(n,k-1).
Conjecture: a(n,k) is the permanent of the n X n matrix with k+1 on the main diagonal and 1 elsewhere.

Extensions

More terms from Amiram Eldar, Jun 27 2020

A211398 E.g.f.: Sum_{n>=0} log( Sum_{k>=0} (n+k)^k*x^k/k! )^n / n!.

Original entry on oeis.org

1, 2, 14, 153, 2262, 42120, 945823, 24870937, 749702348, 25490350284, 965219424913, 40287663094503, 1837912330721162, 90988147658574582, 4858595700832370019, 278371180944699911227, 17034913752075240500920, 1108950617553341656112312, 76523438074756638449399565
Offset: 0

Views

Author

Paul D. Hanna, Feb 10 2013

Keywords

Comments

Compare to the trivial identities:
(1) Sum_{n>=0} log( Sum_{k>=0} n^k*x^k/k! )^n/n! = Sum_{n>=0} n^n*x^n/n!;
(2) Sum_{n>=0} log( Sum_{k>=0} k^k*x^k/k! )^n/n! = Sum_{n>=0} n^n*x^n/n!.

Examples

			E.g.f.: A(x) = 1 + 2*x + 14*x^2/2! + 153*x^3/3! + 2262*x^4/4! + 42120*x^5/5! +...
such that
A(x) = 1 + log(F(x,1)) + log(F(x,2))^2/2! + log(F(x,3))^3/3! + log(F(x,4))^4/4! +...
where
F(x,n) = 1 + (n+1)*x + (n+2)^2*x^2/2! + (n+3)^3*x^3/3! + (n+4)^4*x^4/4! + (n+5)^5*x^5/5! +...+ (n+k)^k*x^k/k! +...
Also,
A(x) = 1 + G(x,1) + G(x,2)^2/2! + G(x,3)^3/3! + G(x,4)^4/4! +...+ G(x,n)^n/n! +...
where G(x,n) = log( (LambertW(-x)/(-x))^n / (1+LambertW(-x)) ):
G(x,n) = (n+1)*x + (2*n+3)*x^2/2! + (9*n+17)*x^3/3! + (64*n+142)*x^4/4! + (625*n+1569)*x^5/5! +...+ (k^(k-1)*n + A001865(k))*x^k/k! +...
Related expansion:
Sum_{n>=0} n^n*log(LambertW(-x)/(-x))^n/n! = 1/(1+LambertW(LambertW(-x)));
1/(1+LambertW(LambertW(-x))) = 1 + x + 6*x^2/2! + 60*x^3/3! + 836*x^4/4! +...

Crossrefs

Cf. A000169, A001865.

Programs

  • PARI
    {a(n)=n!*polcoeff(sum(m=0,n,log(sum(k=0,n,(m+k)^k*x^k/k! +x*O(x^n)))^m/m!),n)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f.: Sum_{n>=0} log( (LambertW(-x)/(-x))^n / (1+LambertW(-x)) )^n / n!.
E.g.f.: Sum_{n>=0} [ Sum_{k>=1} (k^(k-1)*n + A001865(k))*x^k/k! ]^n / n!.

A219530 Number of functions f:{1,2,...,n}->{1,2,...,n} such that each component of f is a function on an interval of {1,2,...,n}.

Original entry on oeis.org

1, 1, 4, 24, 195, 2046, 26752, 422546, 7849611, 167781117, 4054557471, 109246333917, 3245641491658, 105366022410057, 3709933487122164, 140791348680766521, 5728108758307500165, 248696925989154108462, 11476424805495560002162, 560894026563924188981599, 28941826672247857117927894
Offset: 0

Views

Author

Geoffrey Critzer, Nov 21 2012

Keywords

Comments

Here, a component of f is a weakly connected component of its functional digraph.

Examples

			a(3)=24 because there are 27 functions f:{1,2,3}->{1,2,3} but three of these are not counted: 1->3 2->2 3->3; 1->3 2->2 3->1; 1->1 2->2 3->1.

Programs

  • Mathematica
    nn=20; t= Sum[n^(n-1)x^n/n!, {n,1,nn}]; a=Range[0,nn]! CoefficientList[Series[Log[1/(1-t)], {x,0,nn}], x]; b=Sum[a[[i]]x^(i-1), {i,1,nn+1}]; CoefficientList[Series[1/(1-b), {x,0,nn}], x]

Formula

G.f.: 1/ (1 - A(x)) where A(x) is the o.g.f. for A001865.
Previous Showing 31-38 of 38 results.

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