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.

A135867 G.f. satisfies A(x) = 1 + x*A(2*x)^2.

Original entry on oeis.org

1, 1, 4, 36, 640, 21888, 1451008, 188941312, 48768745472, 25069815595008, 25722272102744064, 52730972085034156032, 216091838647321476726784, 1770657164881170759078117376, 29013990909330956353981535748096
Offset: 0

Views

Author

Paul D. Hanna, Dec 02 2007

Keywords

Comments

Self-convolution equals A135868 such that 2^n*A135868(n) = a(n+1) for n >= 0.

Links

Crossrefs

Cf. A015083, A135868, A171192.
Cf. A171200, A171202, A171204, A171206, A171208, A171210, A343439.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1 + x*A[2*x]^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 04 2021 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,2*x)^2);polcoeff(A,n)}
  • PARI
    a(n)=if(n==0,1,2^(n-1)*sum(k=0,n-1,a(k)*a(n-k-1))) \\ Paul D. Hanna, Feb 09 2010

Formula

a(n) = 2^(n-1)*Sum_{k=0..n-1} a(k)*a(n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Feb 09 2010
a(n) ~ c * 2^(n*(n+1)/2), where c = 0.715337433614869740944075474484711589980951273610257702786245519231799678... - Vaclav Kotesovec, Nov 04 2021

A171211 G.f. satisfies: A(x) = (1 + x*A(2x))^8.

Original entry on oeis.org

1, 8, 156, 5944, 425286, 57811000, 15246040860, 7918843018760, 8165859019876353, 16781575370067304448, 68855523432488884833408, 564547878300963670909315840, 9253510119618208634494942344960
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Crossrefs

Cf. A135868, A171200-A171209, A171210.

Programs

  • Mathematica
    m = 13; A[] = 0; Do[A[x] = (1 + x A[2 x])^8 + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 07 2019 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=(1+x*subst(A, x, 2*x))^8); polcoeff(A, n)}

Formula

Self-convolution 8th power of A171210 where a(n) = A171210(n+1)/2^n for n>=0.

A171201 G.f. satisfies: A(x) = (1 + x*A(2x))^3.

Original entry on oeis.org

1, 3, 21, 289, 7566, 380424, 37361616, 7252471584, 2799853666176, 2155959119115264, 3315891500224031232, 10193070293871040606464, 62646640175842537242599936, 769927299959295414569740867584, 18923273743619678311418282019397632, 930154604531789703005691292148132511744
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Crossrefs

Cf. A135868, A171200, A171202-A171211.

Programs

  • Mathematica
    terms = 16; A[] = 0; Do[A[x] = (1 + x*A[2x])^3 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=(1+x*subst(A, x, 2*x))^3); polcoeff(A, n)}

Formula

Self-convolution cube of A171200 where a(n) = A171200(n+1)/2^n for n>=0.

Extensions

a(14)-a(15) from Stefano Spezia, Apr 02 2025

A171203 G.f. satisfies: A(x) = (1 + x*A(2x))^4.

Original entry on oeis.org

1, 4, 38, 708, 24961, 1682688, 220959136, 57266675520, 29497077110720, 30294634141775360, 62134850895148484608, 254691311135373319017472, 2087196424913845641682560512, 34202892422993270952623113994240, 1120863025258656246362522776511881216, 73460242428855296330451249854756580540416
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Crossrefs

Cf. A135868, A171200, A171201, A171202, A171204-A171211.

Programs

  • Mathematica
    terms = 16; A[] = 0; Do[A[x] = (1 + x*A[2x])^4 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=(1+x*subst(A, x, 2*x))^4); polcoeff(A, n)}

Formula

Self-convolution 4th power of A171202 where a(n) = A171202(n+1)/2^n for n>=0.

Extensions

a(14)-a(15) from Stefano Spezia, Apr 02 2025

A171205 G.f. satisfies: A(x) = (1 + x*A(2x))^5.

Original entry on oeis.org

1, 5, 60, 1410, 62505, 5284401, 868838010, 281703950040, 181448450339760, 232989133846286240, 597389845561440183360, 3061032714235774931187200, 31357237236616342838622807040, 642321739861948533960660029617920, 26312068694834430629292373404100369920, 2155589935049851254662487477552439610480640
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Crossrefs

Cf. A135868, A171200-A171203, A171204, A171206-A171211.

Programs

  • Mathematica
    terms = 16; A[] = 0; Do[A[x] = (1 + x*A[2x])^5 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=(1+x*subst(A, x, 2*x))^5); polcoeff(A, n)}

Formula

Self-convolution 5th power of A171204 where a(n) = A171204(n+1)/2^n for n>=0.

Extensions

a(14)-a(15) from Stefano Spezia, Apr 02 2025

A171207 G.f. satisfies: A(x) = (1 + x*A(2x))^6.

Original entry on oeis.org

1, 6, 87, 2468, 131799, 13400550, 2646848041, 1030386755856, 796631252763576, 1227659952939056640, 3777547269650299331856, 23228194648169000672639616, 285544368619000766118426358016, 7018967175754802830514246125923840, 345031382341287335424234252089128848384
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Crossrefs

Cf. A135868, A171200-A171205, A171206, A171208-A171211.

Programs

  • Mathematica
    terms = 15; A[] = 0; Do[A[x] = (1+x*A[2x])^6 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=(1+x*subst(A, x, 2*x))^6); polcoeff(A, n)}

Formula

Self-convolution 6th power of A171206 where a(n) = A171206(n+1)/2^n for n>=0.

Extensions

a(14) from Stefano Spezia, Apr 02 2025

A171209 G.f. satisfies: A(x) = (1 + x*A(2x))^7.

Original entry on oeis.org

1, 7, 119, 3955, 247093, 29355725, 6770018269, 3075928905505, 2774997766597238, 4989660046676105752, 17913062958150482828608, 128508635121001835101510976, 1843071985575998120371392747776, 52855626540938653363337299348546560, 3031270298538159379928340759759663584000
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Crossrefs

Cf. A135868, A171200-A171207, A171208, A171210-A171211.

Programs

  • Mathematica
    terms = 15; A[] = 0; Do[A[x] = (1+x*A[2x])^7 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Apr 02 2025 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=(1+x*subst(A, x, 2*x))^7); polcoeff(A, n)}

Formula

Self-convolution 7th power of A171208 where a(n) = A171208(n+1)/2^n for n>=0.

Extensions

a(13)-a(14) from Stefano Spezia, Apr 02 2025

A135870 G.f. A(x) = (1 + x*A(3x))^2.

Original entry on oeis.org

1, 2, 13, 246, 13554, 2210436, 1076561037, 1570714992558, 6871883750090694, 90179738434097457084, 3550105797469080332658354, 419263945026939228053105930844, 148543119421079567735277052592081220
Offset: 0

Views

Author

Paul D. Hanna, Dec 02 2007

Keywords

Comments

Self-convolution of A135869.

Crossrefs

Cf. A135868, A135869.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,3*x)^2);polcoeff(A^2,n)}
Showing 1-8 of 8 results.

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