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 11-20 of 36 results. Next

A113133 a(0) = a(1) = 1, a(2) = x, a(3) = 2x^2, a(n) = x*(n-1)*a(n-1) + Sum_{j=2..n-2} (j-1)*a(j)*a(n-j), n>=4 and for x = 6.

Original entry on oeis.org

1, 1, 6, 72, 1332, 33264, 1040256, 38926656, 1692061488, 83688313536, 4638320578944, 284692939944192, 19169186341398912, 1404935464314299904, 111348880778746460160, 9489756817594314049536, 865470841829802331976448
Offset: 0

Views

Author

Philippe Deléham and Paul D. Hanna, Oct 28 2005

Keywords

Examples

			a(2) = 6.
a(3) = 2*6^2 = 72.
a(4) = 6*3*72 + 1*6*6 = 1332.
a(5) = 6*4*1332 + 1*6*72 + 2*72*6 = 33264.
a(6) = 6*5*33264 + 1*6*1332 + 2*72*72 + 3*1332*6 = 1040256.
G.f.: A(x) = 1 + x + 6*x^2 + 72*x^3 + 1332*x^4 + 33264*x^5
+...
= x/series_reversion(x + x^2 + 7*x^3 + 91*x^4 + 1729*x^5
+...).

Crossrefs

Cf. A008542, A075834(x=1), A111088(x=2), A113130(x=3), A113131(x=4), A113132(x=5), A113134(x=7), A113135(x=8).

Programs

  • Mathematica
    x=6;a[0]=a[1]=1;a[2]=x;a[3]=2x^2;a[n_]:=a[n]=x*(n-1)*a[n-1]+Sum[(j-1)*a[j ]*a[n-j], {j, 2, n-2}];Table[a[n], {n, 0, 17}](Robert G. Wilson v)
  • PARI
    a(n)=Vec(x/serreverse(x*Ser(vector(n+1,k,if(k==1,1, prod(j=0,k-2,6*j+1))))))[n+1]
  • PARI
    a(n,x=6)=if(n<0,0,if(n==0 || n==1,1,if(n==2,x,if(n==3,2*x^2,x*(n-1)*a(n-1)+sum(j=2,n-2,(j-1)*a(j)*a(n-j))))))

Formula

a(n+1) = Sum{k, 0<=k<=n} 6^k*A113129(n, k).
G.f.: A(x) = x/series_reversion(x*G(x)) where G(x) = g.f. of sextuple factorial numbers (A008542).
G.f. satisfies: A(x*G(x)) = G(x) = g.f. of sextuple factorial numbers (A008542).

A209881 G.f. satisfies: A(x) = 1 + x*[d/dx 1/(1 - x*A(x))].

Original entry on oeis.org

1, 1, 4, 21, 136, 1030, 8856, 84861, 894928, 10291986, 128165720, 1718395602, 24686953968, 378444958060, 6167922926704, 106525443913245, 1943838547593888, 37375737467294362, 755393226726677976, 16011417246585359046, 355187993770520180400, 8230524179585799932820
Offset: 0

Views

Author

Paul D. Hanna, Mar 14 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 136*x^4 + 1030*x^5 + 8856*x^6 +...
The g.f. of A075834, G(x) = 1/(1 - x*A(x)), begins:
G(x) = 1 + x + 2*x^2 + 7*x^3 + 34*x^4 + 206*x^5 + 1476*x^6 +...
The logarithm of the g.f. of A075834 begins:
log(G(x)) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 136*x^5/5 + 1030*x^6/6 +...

Links

Crossrefs

Cf. A075834.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+x*deriv(1/(1-x*A+x*O(x^n)))); polcoeff(A, n)}
for(n=0,25,print1(a(n),", "))

Formula

a(n) = n*A075834(n+1) for n>=1. [corrected by Vaclav Kotesovec, Aug 24 2017]
Given g.f. A(x), the g.f. of A075834 = 1 + x/(1 - x*A(x)).
Forms the logarithmic derivative of A075834.
O.g.f. A(x) satisfies: [x^n] ( 1 + x/(1 - x*A(x)) )^(n+1) = (n+1)! for n>=0.
O.g.f. A(x) satisfies: [x^n] exp( n * Integral A(x) dx ) * (n + 1 - A(x)) = 0 for n > 0. - Paul D. Hanna, Jun 04 2018
a(n) ~ exp(-1) * n^2 * n!. - Vaclav Kotesovec, Aug 24 2017

A386443 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} k^2 * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 11, 120, 2166, 58642, 2231959, 113926332, 7522541374, 624529876412, 63711767096254, 7837308575551868, 1144321503810951264, 195687862794184808186, 38747465910056072904383, 8795888226933223095245628, 2269380895962602685279019270, 660399219910352767447886420340
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2025

Keywords

Crossrefs

Cf. A075834, A386444, A386445, A386446, A386447.
Cf. A385830, A386448, A386452.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, j^2*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x - x^2 * (d/dx A(x)) - x^3 * (d^2/dx^2 A(x)) ).

A386444 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} k^3 * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 19, 550, 36314, 4612644, 1005608259, 346940795318, 178328747938574, 130358697631572620, 130619605078238043630, 174116069712361545382300, 301220935342882714418320660, 662385014999576998657776303368, 1818909557774291764795223960949603, 6142458248209027135766781428841480918
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2025

Keywords

Crossrefs

Cf. A075834, A386443, A386445, A386446, A386447.
Cf. A385831.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, j^3*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x - x*Sum_{k=1..3} Stirling2(3,k) * x^k * (d^k/dx^k A(x)) ).

A386445 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} k^4 * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 35, 2904, 749262, 469791130, 609789812623, 1465325443822620, 6004904311876287022, 39410188505158004325524, 394180711528456847821432318, 5771988198703021102520933624372, 119699491661363792184803354859998664, 3418976586120192927373434641290957978490
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2025

Keywords

Crossrefs

Cf. A075834, A386443, A386444, A386446, A386447.
Cf. A385832.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, j^4*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x - x*Sum_{k=1..4} Stirling2(4,k) * x^k * (d^k/dx^k A(x)) ).

A386446 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} k^5 * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 67, 16414, 16840826, 52661283276, 409599480216723, 6884957718009061046, 225620064835937122627934, 13323090455565480199133495252, 1332335691963961772604470940370302, 214576660211223693770379106296061734124, 53393968668333658608864584261609697870131860
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2025

Keywords

Crossrefs

Cf. A075834, A386443, A386444, A386445, A386447.
Cf. A385833.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, j^5*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x - x*Sum_{k=1..5} Stirling2(5,k) * x^k * (d^k/dx^k A(x)) ).

A386447 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} k^6 * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 131, 95760, 392424606, 6132419429842, 286126426174265119, 33663060172069656177612, 8824636572155130972996888814, 4689791333849576329442118802082252, 4689800713441077274969296364554337253614, 8308277421310507219950890075481144453543272228
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2025

Keywords

Crossrefs

Cf. A075834, A386443, A386444, A386445, A386446.
Cf. A385834.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, j^6*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x - x*Sum_{k=1..6} Stirling2(6,k) * x^k * (d^k/dx^k A(x)) ).

A386448 G.f. A(x) satisfies A(x) = 1/(1 - x - x^3*A''(x)).

Original entry on oeis.org

1, 1, 1, 3, 23, 319, 6999, 223725, 9838405, 570440733, 42203958765, 3882243620535, 434771830226307, 58255737747374083, 9203989127308306571, 1693477639607917108953, 359008305377998952818761, 86878355403079952880852217, 23804317478591173659253678809, 7331644401028481860472940727371
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2025

Keywords

Crossrefs

Cf. A075834, A386449, A386450, A386451.
Cf. A385762.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, sum(k=1, 2, stirling(2, k, 1)*j^k)*v[j+1]*v[i-j])); v;

Formula

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

A386452 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} binomial(k+1,2) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 2, 9, 71, 856, 14639, 338086, 10167592, 386920264, 18200571057, 1037970049307, 70605576249333, 5649723531576365, 525507834721871564, 56235831305760575845, 6861362229615344431713, 946930149578851143467375, 146781656943702604491445861, 25394248429778915431816805711
Offset: 0

Views

Author

Seiichi Manyama, Jul 22 2025

Keywords

Crossrefs

Cf. A075834, A386453, A386454, A386455.
Cf. A385874.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, binomial(j+1, 2)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( 1 - x - x^2 * (d/dx A(x)) - x^3/2 * (d^2/dx^2 A(x)) ).

A090753 Coefficients of power series A(x) such that n-th term of A(x)^n = n!*n*x^(n-1), for n>0.

Original entry on oeis.org

1, 2, 2, 4, 16, 88, 600, 4800, 43680, 443296, 4949920, 60217408, 792134528, 11200176128, 169375195136, 2728019576832, 46626359376384, 842947307334144, 16073131554826752, 322403473258650624, 6786861273524305920
Offset: 0

Views

Author

Philippe Deléham, Feb 06 2004

Keywords

Comments

At n=4 the 4th term of A(x)^4 is 4!*4x^3 = 96*x^3, as demonstrated by A(x)^4 = 1 + 8*x + 32*x^2 + 96*x^3 + 296*x^4 + ... See also A075834.

Links

Crossrefs

Cf. A075834, A211207, A136633.

Programs

  • PARI
    a(n)=if(n<0,0,polcoeff(x/serreverse(sum(k=1,n+1,k!*x^k,x^2*O(x^n))),n)) /* Michael Somos, Feb 14 2004 */
  • PARI
    a(n)=local(A=1+x); for(i=1, n, A=1+2*sum(m=1, n, m^m*x^m/(A+m*x+x*O(x^n))^m)); polcoeff(A, n)
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 04 2013

Formula

a(n) = Sum_{j=2..(n-2)} (j-1)*a(j)*a(n-j) for n>=2, with a(0)=1, a(1)=2.
Sum_{j>=0} a(j)*A090238(n-1, k+j-1) = A090238(n, k).
G.f. satisfies: A(x) = 1 + 2*Sum_{n>=1} n^n * x^n / (A(x) + n*x)^n. - Paul D. Hanna, Feb 04 2013
a(n) ~ exp(-2) * n! * n. - Vaclav Kotesovec, Nov 23 2024
Previous Showing 11-20 of 36 results. Next

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

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