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 1-8 of 8 results.

A137953 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^2)^3.

Original entry on oeis.org

1, 1, 3, 9, 34, 132, 546, 2327, 10191, 45534, 206788, 951723, 4429182, 20808186, 98550468, 470038119, 2255684699, 10883852112, 52769785320, 256960840946, 1256147650818, 6162349332204, 30328107189312, 149698391878458
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137952, A137954; A137957, A137962, A137969.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[3*(n-k), k]/(n-k)*Binomial[2*k, n-k-1],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec after Paul D. Hanna, Mar 25 2014 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^2)^3);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(3*(n-k),k)/(n-k)*binomial(2*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^3 where B(x) is the g.f. of A137952.
a(n) = Sum_{k=0..n-1} C(3*(n-k),k)/(n-k) * C(2*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
Recurrence: 5*n*(5*n-3)*(5*n-2)*(5*n+1)*(5*n+4)*(2948400*n^11 - 80922240*n^10 + 991552680*n^9 - 7191167904*n^8 + 34388915791*n^7 - 113938412552*n^6 + 266574560812*n^5 - 439214051186*n^4 + 497527715029*n^3 - 367402366838*n^2 + 158427508008*n - 30063700800)*a(n) = -240*(5*n-1)*(3402000*n^13 - 102564900*n^12 + 1682146080*n^11 - 16176231033*n^10 + 95359496344*n^9 - 359981654612*n^8 + 893831335718*n^7 - 1468770570635*n^6 + 1566970769558*n^5 - 1019176919948*n^4 + 331927521052*n^3 + 34505928*n^2 - 32180612832*n + 6541274880)*a(n-1) + 180*(884520000*n^16 - 26930232000*n^15 + 372745486800*n^14 - 3118060887120*n^13 + 17644263763548*n^12 - 71507400823524*n^11 + 214013670957835*n^10 - 480132169105811*n^9 + 810380315383846*n^8 - 1022562903644722*n^7 + 947982058983979*n^6 - 624324084479227*n^5 + 273663045967416*n^4 - 68343334466444*n^3 + 4273926176256*n^2 + 2065304121408*n - 381518968320)*a(n-2) + 72*(5890903200*n^16 - 188191699920*n^15 + 2743292998800*n^14 - 24248455085592*n^13 + 145518104758338*n^12 - 628264374415281*n^11 + 2014705595228766*n^10 - 4876859081303636*n^9 + 8950855221646414*n^8 - 12378944029917433*n^7 + 12665670452628658*n^6 - 9249292270917382*n^5 + 4496305419163048*n^4 - 1229711760456116*n^3 + 68797455703176*n^2 + 53468550934560*n - 10544040864000)*a(n-3) + 72*(5731689600*n^16 - 191702972160*n^15 + 2927459413440*n^14 - 27105381081216*n^13 + 170350803352728*n^12 - 770345146059408*n^11 + 2589617705669352*n^10 - 6581794624393248*n^9 + 12710327685293639*n^8 - 18531898603387194*n^7 + 20012311600272546*n^6 - 15421584075698196*n^5 + 7904537517669183*n^4 - 2290793383663938*n^3 + 159318295564312*n^2 + 94065554487360*n - 19593691084800)*a(n-4) + 72*(2*n-9)*(3*n-11)*(3*n-7)*(6*n-25)*(6*n-23)*(2948400*n^11 - 48489840*n^10 + 344492280*n^9 - 1422208584*n^8 + 3817772239*n^7 - 6909787807*n^6 + 8311308487*n^5 - 6272196721*n^4 + 2621759746*n^3 - 403021048*n^2 - 67705152*n + 22579200)*a(n-5). - Vaclav Kotesovec, Mar 25 2014
a(n) ~ sqrt(3*s*(s-1)*(3*s-2)/(5*s-3)) / (2*sqrt(Pi)*n^(3/2)*r^n), where s = 1.7888356349988794022183... is the root of the equation 216*(s-1)^2 = s*(5*s-6)^4, and r = 1/(s*(5*s-6)) = 0.189873988477346598... - Vaclav Kotesovec, Mar 25 2014

A138960 a(n) = smallest prime divisor of A138957(n).

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 127, 2, 3, 857, 3, 3, 18503, 3, 3, 43, 3, 3, 17, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 7, 3, 3, 1051, 3, 3, 67103, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Artur Jasinski, Apr 04 2008

Keywords

Comments

For largest prime divisors see A138961.

Crossrefs

Cf. A104759, A000422, A116504, A007908, A116505, A104759, A138789, A138790, A138961.

Programs

  • Mathematica
    a = {}; b = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[Length[w] - k + 1]]], {k, 1, Length[w]}]; p = FromDigits[a]; AppendTo[b, First[First[FactorInteger[p]]]], {n, 1, 38}]; b
A137957[n_] := FromDigits[Flatten[Reverse /@ IntegerDigits[Range[n]]]];
Table[FactorInteger[A137957[n]][[1, 1]], {n, 39}] (* Robert Price, May 10 2019 *)

Extensions

a(39)-a(69) from Robert Price, May 10 2019

A137956 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^2)^4.

Original entry on oeis.org

1, 1, 4, 14, 64, 301, 1500, 7738, 40948, 221278, 1215284, 6765148, 38083556, 216431253, 1240048740, 7155236960, 41542685352, 242513393884, 1422608044604, 8381507029660, 49574494112992, 294260899150492, 1752288415205896
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137955, A137957; A137958, A137964, A137971.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[4*(n-k),k]/(n-k)*Binomial[2*k,n-k-1],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Sep 18 2013 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^2)^4);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(4*(n-k),k)/(n-k)*binomial(2*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^4 where B(x) is the g.f. of A137955.
a(n) = Sum_{k=0..n-1} C(4*(n-k),k)/(n-k) * C(2*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(4*s*(1-s)*(2-3*s) / ((28*s - 16)*Pi)) / (n^(3/2) * r^n), where r = 0.1569043698639381952962655091205241634381480571697... and s = 1.683635070625292013962854364673077567156937629734... are real roots of the system of equations s = 1 + r*(1 + r*s^2)^4, 8 * r^2 * s * (1 + r*s^2)^3 = 1. - Vaclav Kotesovec, Nov 22 2017

A137962 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^3.

Original entry on oeis.org

1, 1, 3, 18, 106, 720, 5085, 37493, 284331, 2204973, 17404720, 139369905, 1129411314, 9244823986, 76326154857, 634847759955, 5314684735045, 44746683774474, 378652035541761, 3218705637379698, 27471657413667780
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137963, A137961; A137953, A137957, A137969.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^5)^3);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(3*(n-k),k)/(n-k)*binomial(5*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^3 where B(x) is the g.f. of A137963.
a(n) = Sum_{k=0..n-1} C(3*(n-k),k)/(n-k) * C(5*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(3*s*(1-s)*(5-6*s) / ((140*s - 120)*Pi)) / (n^(3/2) * r^n), where r = 0.1085884782751570249717333800652227343328635496829... and s = 1.301018963559115613510052458264916439485131890857... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^3, 15 * r^2 * s^4 * (1 + r*s^5)^2 = 1. - Vaclav Kotesovec, Nov 22 2017

A137969 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^6)^3.

Original entry on oeis.org

1, 1, 3, 21, 136, 1032, 8139, 66975, 567417, 4915386, 43350639, 387889254, 3512655498, 32133132074, 296496163113, 2756279003712, 25790064341592, 242699145598212, 2295564345035100, 21811226043019788, 208084639385653938
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Links

Crossrefs

Cf. A137970, A137968; A137953, A137957, A137962.

Programs

  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^6)^3);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(3*(n-k),k)/(n-k)*binomial(6*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^3 where B(x) is the g.f. of A137970.
a(n) = Sum_{k=0..n-1} C(3*(n-k),k)/(n-k) * C(6*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(3*s*(1-s)*(6-7*s) / ((204*s - 180)*Pi)) / (n^(3/2) * r^n), where r = 0.0971328555591006631243189792661187629516513365080... and s = 1.254068189138542668013320901661524162625316815207... are real roots of the system of equations s = 1 + r*(1 + r*s^6)^3, 18 * r^2 * s^5 * (1 + r*s^6)^2 = 1. - Vaclav Kotesovec, Nov 22 2017

A138961 a(n) = largest prime divisor of A138957(n).

Original entry on oeis.org

1, 3, 41, 617, 823, 643, 9721, 14593, 3803, 14405693, 10939223, 4156374407, 2663693, 5603770631, 1221751714624799, 287108811653770498027, 74103167823547, 11843077531813991, 726216405947772436185983423, 769725127, 18274551225153265813469
Offset: 1

Views

Author

Artur Jasinski, Apr 04 2008

Keywords

Comments

For smallest prime divisors see A138960.

Links

Crossrefs

Cf. A104759, A000422, A116504, A007908, A116505, A104759, A138789, A138790, A138960.

Programs

  • Mathematica
    a = {}; b = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[Length[w] - k + 1]]], {k, 1, Length[w]}]; p = FromDigits[a]; AppendTo[b, First[Last[FactorInteger[p]]]], {n, 1, 18}]; b
A137957[n_] := FromDigits[Flatten[Reverse /@ IntegerDigits[Range[n]]]];
Table[First[Last[FactorInteger[A137957[n]]]], {n, 39}] (* Robert Price, May 10 2019 *)

A137958 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^3)^4.

Original entry on oeis.org

1, 1, 4, 18, 100, 587, 3660, 23640, 157076, 1066281, 7363620, 51568732, 365369868, 2614235293, 18862816112, 137096744232, 1002785827620, 7376023180645, 54525165453672, 404858512190316, 3018190533410664, 22581907465905018
Offset: 0

Views

Author

Paul D. Hanna, Feb 26 2008

Keywords

Crossrefs

Cf. A137957, A137959; A137956, A137964, A137971.

Programs

  • Mathematica
    Flatten[{1,Table[Sum[Binomial[4*(n-k),k]/(n-k)*Binomial[3*k,n-k-1],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Nov 22 2017 *)
  • PARI
    {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^3)^4);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,sum(k=0,n-1,binomial(4*(n-k),k)/(n-k)*binomial(3*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^4 where B(x) is the g.f. of A137957.
a(n) = Sum_{k=0..n-1} C(4*(n-k),k)/(n-k) * C(3*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(4*s*(1-s)*(3-4*s) / ((66*s - 48)*Pi)) / (n^(3/2) * r^n), where r = 0.1243879037293364492255197677726812528516871521834... and s = 1.442260525872978775674461288363175530136608288804... are real roots of the system of equations s = 1 + r*(1 + r*s^3)^4, 12 * r^2 * s^2 * (1 + r*s^3)^3 = 1. - Vaclav Kotesovec, Nov 22 2017

A378787 G.f. A(x) satisfies A(x) = ( 1 + x * (1 + x*A(x)^2)^3 )^2.

Original entry on oeis.org

1, 2, 7, 36, 197, 1184, 7425, 48308, 322521, 2198064, 15227850, 106924154, 759245463, 5442675080, 39335090088, 286296369000, 2096706604597, 15439417451928, 114243931954962, 849030345258990, 6334510149389409, 47428709540589036, 356261301882333885
Offset: 0

Views

Author

Seiichi Manyama, Dec 07 2024

Keywords

Crossrefs

Cf. A069271, A371607, A371609.
Cf. A137957.

Programs

  • PARI
    a(n, r=2, s=3, t=0, u=4) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r));

Formula

a(n) = Sum_{k=0..n} binomial(4*(n-k)+2,k) * binomial(3*k,n-k)/(2*(n-k)+1).
G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A137957.
Showing 1-8 of 8 results.

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