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 1-7 of 7 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

A171202 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^4.

Original entry on oeis.org

1, 1, 8, 152, 5664, 399376, 53846016, 14141384704, 7330134466560, 7551251740344320, 15510852680588984320, 63626087316632048238592, 521607805205244557347782656, 8549156556447111748331767857152, 280190094729160875643888549840814080, 18364219805837823940403573170370661842944
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

a(0) = 1; a(n) = 2^(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

A171204 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^5.

Original entry on oeis.org

1, 1, 10, 240, 11280, 1000080, 169100832, 55605632640, 36058105605120, 46450803286978560, 119290436529298554880, 611727201854914747760640, 6268994998754867059071385600, 128439243721180540266999017635840, 5261899692949082390205726962630000640, 431096933496167311430326245852780460769280
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]^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

a(0) = 1; a(n) = 2^(n-1) * Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} a(i) * a(j) * a(k) * a(l) * a(m). - Seiichi Manyama, Jul 08 2025

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

A171208 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^7.

Original entry on oeis.org

1, 1, 14, 476, 31640, 3953488, 939383200, 433281169216, 393718899904640, 710399428248892928, 2554705943898166145024, 18342976469146094416494592, 263185684727811758287894478848, 7549222852919288301041224694890496, 432993292623369448352459156263293419520
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

a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_7>=0 and x_1+x_2+...+x_7=n-1} Product_{k=1..7} a(x_k). - Seiichi Manyama, Jul 08 2025

A171210 G.f. A(x) satisfies A(x) = 1 + x*A(2*x)^8.

Original entry on oeis.org

1, 1, 16, 624, 47552, 6804576, 1849952000, 975746615040, 1013611906401280, 2090459909088346368, 8592166589474459877376, 70508055994868618069409792, 1156194054760373598022278840320, 37902377449956182566891283844956160, 2484501232375923934830943089632156319744
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]^8 + 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)^8); polcoeff(A, n)}

Formula

a(0) = 1; a(n) = 2^(n-1) * Sum_{x_1, x_2, ..., x_8>=0 and x_1+x_2+...+x_8=n-1} Product_{k=1..8} a(x_k). - Seiichi Manyama, Jul 08 2025
Showing 1-7 of 7 results.