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.

A348857 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(2*x))).

Original entry on oeis.org

1, 2, 7, 44, 481, 9254, 326395, 21927776, 2874607189, 744650622170, 383510575423471, 393869218949592212, 807827718206737362889, 3311287802485779192925838, 27136007596894473408507305443, 444677773080105539125038867872456, 14572535437424416878539776253365375549
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 02 2021

Keywords

Crossrefs

Cf. A007317, A015083, A348858, A348859.

Programs

  • Mathematica
    nmax = 16; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[2 x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + Sum[2^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]

Formula

a(n) = 1 + Sum_{k=0..n-1} 2^k * a(k) * a(n-k-1).
a(n) ~ c * 2^(n*(n-1)/2), where c = 10.96416094535958612421479005398505892527943513193882801485045169159164... - Vaclav Kotesovec, Nov 02 2021

A348859 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(4*x))).

Original entry on oeis.org

1, 2, 11, 204, 13701, 3550838, 3646912991, 14948746703872, 244965160945456921, 16054771878797715999594, 4208710286900635084866205491, 4413165224136772109314051383922356, 18510169791808150609141704979384516863021, 310549172324407121253872529077196811473762678750
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 02 2021

Keywords

Crossrefs

Cf. A007317, A015085, A348857, A348858.

Programs

  • Mathematica
    nmax = 13; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[4 x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + Sum[4^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]

Formula

a(n) = 1 + Sum_{k=0..n-1} 4^k * a(k) * a(n-k-1).
a(n) ~ c * 2^(n*(n-1)), where c = 3.399782064170449155365557063612838469541502782488369640092639686931819... - Vaclav Kotesovec, Nov 02 2021

A386299 a(n) = 1 + Sum_{k=0..n-1} 3^k * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 9, 115, 3869, 349233, 88835413, 65934937157, 145194342935565, 955092851917410169, 18817250316042492760133, 1111535058740789497290819885, 196930668231818953760620540315069, 104661954649505883286587026252584631249, 166867787421063078832424708621648215185207669
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2025

Keywords

Crossrefs

Cf. A054687, A386298, A386300.
Cf. A345105, A348858.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=1+sum(j=0, i-1, 3^j*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

E.g.f. A(x) satisfies A'(x) = exp(x) + A(x) * A(3*x).

A376127 a(n) = 1 + Sum_{k=0..n-1} (k+1)^3 * a(k) * a(n-k-1).

Original entry on oeis.org

1, 2, 19, 565, 38056, 4886164, 1071397370, 370880032881, 191040201050842, 139853547948358801, 140279102716474353325, 187136598610376840549341, 323937672908434382002891895, 712668454800648677607151322833, 1957709831409075714559805601326566, 6613164804688226108094777888275765585
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 11 2024

Keywords

Crossrefs

Cf. A007317, A321087, A348858, A376096, A376126.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[(k + 1)^3 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
nmax = 15; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x] - 7 x^2 A'[x] - 6 x^3 A''[x] - x^4 A'''[x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 / ( (1 - x) * (1 - x * A(x) - 7 * x^2 * A'(x) - 6 * x^3 * A''(x) - x^4 * A'''(x)) ).

A386301 a(n) = 1 + Sum_{k=0..n-1} (-1)^k * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 1, -1, 1, 8, 1, -28, 1, 134, 1, -649, 1, 3320, 1, -17497, 1, 94526, 1, -520507, 1, 2910896, 1, -16487794, 1, 94393106, 1, -545337199, 1, 3175320608, 1, -18615098836, 1, 109783526822, 1, -650884962907, 1, 3877184797784, 1, -23193307022860, 1, 139271612505362
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2025

Keywords

Crossrefs

Cf. A007317, A348857, A348858, A348859.
Cf. A064641, A386300.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=1+sum(j=0, i-1, (-1)^j*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies:
(1) A(x) = 1/( (1-x) * (1-x*A(-x)) ).
(2) A(x)*A(-x) = B(-x^2), where B(x) is the g.f. of A064641.
(3) A(x) = 1/(1-x) + 2*x/(1+x^2 + sqrt(1+6*x^2-3*x^4)).
a(2*n) = 1 and a(2*n+1) = 1 + (-1)^n * A064641(n) for n >= 0.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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