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 1-4 of 4 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.

Crossrefs

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

A171206 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^6.

Original entry on oeis.org

1, 1, 12, 348, 19744, 2108784, 428817600, 169398274624, 131889504749568, 203937600707475456, 628561895904796999680, 3868208404121906515820544, 47571342639450113377565933568, 1169589733863427138021074362433536, 57499379103783344787572704263568097280, 5652994168279651703590653986228287051923456
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Crossrefs

Programs

  • Mathematica
    terms = 16; 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

a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_6>=0 and x_1+x_2+...+x_6=n-1} Product_{k=1..6} a(x_k). - Seiichi Manyama, Jul 08 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

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

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
Showing 1-4 of 4 results.