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 14 results. Next

A072597 Expansion of 1/(exp(-x) - x) as exponential generating function.

Original entry on oeis.org

1, 2, 7, 37, 261, 2301, 24343, 300455, 4238153, 67255273, 1185860331, 23000296155, 486655768525, 11155073325917, 275364320099807, 7282929854486431, 205462851526617489, 6158705454187353297, 195465061563672788947, 6548320737474275229347, 230922973019493881984021
Offset: 0

Views

Author

Michael Somos, Jun 23 2002

Keywords

Comments

Polynomials from A140749/A141412 are linked to Stirling1 (see A048594, A129841, A140749). See also P. Flajolet, X. Gourdon, B. Salvy in, available on Internet, RR-1857.pdf (preprint of unavailable Gazette des Mathematiciens 55, 1993, pp. 67-78; for graph 2 see also X. Gourdon RR-1852.pdf, pp. 64-65). What is the corresponding graph for A152650/A152656 = simplified A009998/A119502 linked, via A152818, to a(n), then Stirling2? - Paul Curtz, Dec 16 2008
Denominators in rational approximations of Lambert W(1). See Ramanujan, Notebooks, volume 2, page 22: "2. If e^{-x} = x, shew that the convergents to x are 1/2, 4/7, 21/37, 148/261, &c." Numerators in A006153. - Michael Somos, Jan 21 2019
Call an element g in a semigroup a group element if g^j = g for some j > 1. Then a(n) is the number of group elements in the semigroup of partial transformations of an n-set. Hence a(n) = Sum_{k=0..n} A154372(n,k)*k!. - Geoffrey Critzer, Nov 27 2021

Examples

			G.f. = 1 + 2*x + 7*x^2 + 37*x^3 + 261*x^4 + 2301*x^5 + 24343*x^6 + ...

References

  • O. Ganyushkin and V Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 70.
  • S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 22.

Links

Crossrefs

Cf. A000110, A006153, A030178, A089148.
Cf. A048993, A052820.
Cf. A006153, A154372.

Programs

  • Mathematica
    CoefficientList[Series[1/(Exp[-x]-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / (Exp[-x] - x), {x, 0, n}]]; (* Michael Somos, Jan 21 2019 *)
a[ n_] := If[ n < 0, 0, n! Sum[ (n - k + 1)^k / k!, {k, 0, n}]]; (* Michael Somos, Jan 21 2019 *)
  • PARI
    {a(n) = if( n<0, 0, n! * polcoeff( 1 / (exp(-x + x * O(x^n)) - x), n))};
  • PARI
    {a(n) = if( n<0, 0, n! * sum(k=0, n, (n-k+1)^k / k!))}; /* Michael Somos, Jan 21 2019 */

Formula

E.g.f.: 1 / (exp(-x) - x).
a(n) = n!*Sum_{k=0..n} (n-k+1)^k/k!. - Vladeta Jovovic, Aug 31 2003
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*A052820(k). - Vladeta Jovovic, Apr 12 2004
Recurrence: a(n+1) = 1 + Sum_{j=1..n} binomial(n, j)*a(j)*j. - Jon Perry, Apr 25 2005
E.g.f.: 1/(Q(0) - x) where Q(k) = 1 - x/(2*k+1 - x*(2*k+1)/(x - (2*k+2)/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 04 2013
a(n) ~ n!/((1+c)*c^(n+1)), where c = A030178 = LambertW(1) = 0.5671432904... - Vaclav Kotesovec, Jun 26 2013
O.g.f.: Sum_{k>=0} k!*x^k/(1 - (k + 1)*x)^(k+1). - Ilya Gutkovskiy, Oct 09 2018
a(n) = A006153(n+1)/(n+1). - Seiichi Manyama, Nov 05 2024

A367845 Expansion of e.g.f. 1/(1 - x + log(1 - 2*x)).

Original entry on oeis.org

1, 3, 22, 250, 3816, 72968, 1675568, 44901456, 1375306368, 47392683648, 1814635323648, 76430014409472, 3511792144942080, 174806087920727040, 9370642040786049024, 538202280800536799232, 32972397141008692445184, 2146270648672407967137792
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Cf. A052820, A367846, A367847.
Cf. A367835, A367828.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 2^j*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 2^k * (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ n! * 2^(n+1) / ((1/LambertW(1/(2*exp(1/2))) - 1 - 2*LambertW(1/(2*exp(1/2)))) * (1 - 2*LambertW(1/(2*exp(1/2))))^n). - Vaclav Kotesovec, Dec 02 2023

A367846 Expansion of e.g.f. 1/(1 - x + log(1 - 3*x)).

Original entry on oeis.org

1, 4, 41, 654, 14028, 377112, 12177126, 458916588, 19769059944, 958125646080, 51597765220608, 3056601306206016, 197532472461453072, 13829353660386169344, 1042679226974498229456, 84229294995413626072608, 7257792124889497549663488
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Cf. A052820, A367845, A367847.
Cf. A367836.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 3^j*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 3^k * (k-1)! * binomial(n,k) * a(n-k).

A367847 Expansion of e.g.f. 1/(1 - x + log(1 - 4*x)).

Original entry on oeis.org

1, 5, 66, 1358, 37592, 1304536, 54384080, 2646247152, 147186205056, 9210766696320, 640472632680192, 48989958019395840, 4087959251421060096, 369547591764702870528, 35976590549993421907968, 3752609987262290143082496, 417518648351593243448279040
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Cf. A052820, A367845, A367846.
Cf. A367837.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 4^j*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 4^k * (k-1)! * binomial(n,k) * a(n-k).

A367851 Expansion of e.g.f. 1/(1 - x + log(1 - 2*x)/2).

Original entry on oeis.org

1, 2, 10, 80, 872, 11984, 198416, 3840192, 85031040, 2119385856, 58714881792, 1789646610432, 59515302478848, 2144299161348096, 83204666280609792, 3459286210445942784, 153413140701637804032, 7228914528043587796992, 360670654712328998289408
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Cf. A052820, A367852, A367853.
Cf. A227917, A355110, A367845.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 2^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 2^(k-1) * (k-1)! * binomial(n,k) * a(n-k).

A367852 Expansion of e.g.f. 1/(1 - x + log(1 - 3*x)/3).

Original entry on oeis.org

1, 2, 11, 102, 1320, 21804, 436986, 10283580, 277697304, 8458929792, 286825214592, 10712216384352, 436859348261904, 19313926491051360, 920053448561989296, 46977842202096405024, 2559387620091962391552, 148187802162935002975488
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Cf. A052820, A367851, A367853.
Cf. A352069, A355111, A367846.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 3^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 3^(k-1) * (k-1)! * binomial(n,k) * a(n-k).

A367853 Expansion of e.g.f. 1/(1 - x + log(1 - 4*x)/4).

Original entry on oeis.org

1, 2, 12, 128, 1952, 38464, 926336, 26323968, 861419520, 31882358784, 1316275003392, 59954841649152, 2985997926727680, 161401148097036288, 9408988894966579200, 588381964243109412864, 39285329204482179858432, 2789234068575581984784384
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Cf. A052820, A367851, A367852.
Cf. A352071, A355112, A367847.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 4^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 4^(k-1) * (k-1)! * binomial(n,k) * a(n-k).

A343685 a(0) = 1; a(n) = 2 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 3, 19, 182, 2328, 37234, 714674, 16004064, 409587144, 11792756640, 377261048592, 13275818803488, 509646721402032, 21195285059025648, 949279217570464944, 45552467588773815744, 2331624264279599225088, 126804353256754734370176, 7301857349340031590836352, 443826900013575494233057536
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2021

Keywords

Crossrefs

Cf. A007840, A010842, A052820, A343686, A343687.

Programs

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

Formula

E.g.f.: 1 / (1 - 2*x + log(1 - x)).
a(n) ~ n! / ((2/c + 1 - c) * (1 - c/2)^n), where c = LambertW(2*exp(1)) = 1.3748225281836233816178373171119... - Vaclav Kotesovec, Apr 26 2021

A343686 a(0) = 1; a(n) = 3 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 4, 33, 410, 6796, 140824, 3501782, 101589732, 3368237928, 125634319104, 5206805098752, 237370661584704, 11805144854303760, 636030155604374400, 36903603627294958416, 2294156656214759133024, 152126925169297299197184, 10718105879980375520103936, 799564645068022035991527552
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2021

Keywords

Crossrefs

Cf. A007840, A052820, A053486, A343685, A343687.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = 3 n a[n - 1] + Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(1 - 3 x + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1 / (1 - 3*x + log(1 - x)).
a(n) ~ n! / ((3/c + 2 - c) * (1 - c/3)^n), where c = LambertW(3*exp(2)) = 2.2761339297716461777892556270138... - Vaclav Kotesovec, Apr 26 2021

A343687 a(0) = 1; a(n) = 4 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 5, 51, 782, 15992, 408814, 12541010, 448834728, 18358297416, 844755218400, 43190363326992, 2429044756967520, 149029669269441456, 9905401062535389072, 709016063545908259248, 54375505616232613595904, 4448148376192382963462400, 386619861956492109750650496, 35580548688887294090357622912
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2021

Keywords

Crossrefs

Cf. A007840, A052820, A053487, A343685, A343686.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = 4 n a[n - 1] + Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(1 - 4 x + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1 / (1 - 4*x + log(1 - x)).
a(n) ~ n! / ((4/c + 3 - c) * (1 - c/4)^n), where c = LambertW(4*exp(3)) = 3.2176447220005493578369738... - Vaclav Kotesovec, Apr 26 2021
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