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 1-5 of 5 results.

A171200 G.f. satisfies A(x) = 1 + x*A(2x)^3.

Original entry on oeis.org

1, 1, 6, 84, 2312, 121056, 12173568, 2391143424, 928316362752, 716762538541056, 1103851068987015168, 3395472896229407981568, 20875407961847891162038272, 256600638160251032545689337856, 6307244441266548036155317187248128
Offset: 0

Views

Author

Paul D. Hanna, Dec 05 2009

Keywords

Crossrefs

Programs

  • Mathematica
    m = 15; A[] = 0; Do[A[x] = 1 + x A[2x]^3 + 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)^3); polcoeff(A, n)}

Formula

a(0) = 1; a(n) = 2^(n-1) * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1). - Ilya Gutkovskiy, Nov 03 2021

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

A143047 G.f. A(x) satisfies A(x) = 1 + x*A(-x)^4.

Original entry on oeis.org

1, 1, -4, -10, 84, 265, -2604, -8900, 94692, 337940, -3767312, -13812674, 158785964, 593029550, -6967201736, -26372738120, 314904180100, 1204230041900, -14560722724912, -56130528427400, 685514219386576, 2659770565898729, -32749512944380172
Offset: 0

Views

Author

Paul D. Hanna, Jul 19 2008

Keywords

Examples

			A(x) = 1 + x - 4*x^2 - 10*x^3 + 84*x^4 + 265*x^5 - 2604*x^6 - 8900*x^7 +...
A(x)^4 = 1 + 4*x - 10*x^2 - 84*x^3 + 265*x^4 + 2604*x^5 - 8900*x^6 -...
A(x)^5 = 1 + 5*x - 10*x^2 - 120*x^3 + 265*x^4 + 3906*x^5 - 8900*x^6 -...
Note that a bisection of A^5 equals a bisection of A^4.
		

Crossrefs

Programs

  • PARI
    a(n)=local(A=x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,-x)^4);polcoeff(A,n)

Formula

G.f. satisfies: A(x) = 1 + x*(1 - x*A(x)^4)^4.
G.f. satisfies: [A(x)^5 + A(-x)^5]/2 = [A(x)^4 + A(-x)^4]/2.
a(0) = 1; a(n) = (-1)^(n-1) * Sum_{i, j, k, l>=0 and i+j+k+l=n-1} a(i) * a(j) * a(k) * a(l). - Seiichi Manyama, Jul 08 2025

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

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

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
Showing 1-5 of 5 results.