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 1-6 of 6 results.

A348858 G.f. A(x) satisfies: A(x) = 1 / ((1 - x) * (1 - x * A(3*x))).

Original entry on oeis.org

1, 2, 9, 103, 3101, 261192, 64285189, 47059492688, 103060910397021, 676492249628112382, 13317427360663454672669, 786420726604930579016189223, 139314431838014895142151741877241, 74037818920801629179455290512454633872, 118040419689979917511971388549088825283510249
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 02 2021

Keywords

Crossrefs

Cf. A007317, A015084, A348857, A348859.

Programs

  • Mathematica
    nmax = 14; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[3 x])) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = 1 + Sum[3^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]

Formula

a(n) = 1 + Sum_{k=0..n-1} 3^k * a(k) * a(n-k-1).
a(n) ~ c * 3^(n*(n-1)/2), where c = 4.508135635010167805309616576501854361005320931661829410476785686203732753... - 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

A386298 a(n) = 1 + Sum_{k=0..n-1} 2^k * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 7, 52, 721, 17594, 754063, 58139188, 8321310193, 2272187953346, 1206524396886823, 1260788083530821380, 2611061273843639666401, 10760136322351992470924570, 88437432027319862460463145551, 1451522912694521425631922482171812, 47608493474799808182534348919785356065
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2025

Keywords

Crossrefs

Cf. A054687, A386299, A386300.
Cf. A345104, A348857.

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, 2^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(2*x).

A348903 G.f. A(x) satisfies: A(x) = 1 / (1 - 2*x - x * A(2*x)).

Original entry on oeis.org

1, 3, 15, 123, 1623, 35427, 1349727, 94653195, 12690736167, 3325408581747, 1722610175806383, 1774299723226774683, 3644417103927252697335, 14949404433893216347632003, 122555228634241017164802041343, 2008680242472430855727593100321067
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 03 2021

Keywords

Crossrefs

Cf. A348857, A348860, A348875, A348901, A348904.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1/(1 - 2*x - x*A[2*x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

a(n) ~ c * 2^(n*(n-1)/2), where c = 6*Product_{j>=1} (2^j+1)/(2^j-1) = 49.5359276146695003932648450...
a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=0..n-1} 2^k * a(k) * a(n-k-1). - Ilya Gutkovskiy, Nov 03 2021

A376126 a(n) = 1 + Sum_{k=0..n-1} (k+1)^2 * a(k) * a(n-k-1).

Original entry on oeis.org

1, 2, 11, 127, 2446, 69766, 2758136, 144660949, 9742594594, 820936644105, 84720182796109, 10518406503248799, 1547501158599695053, 266325797207877471893, 53021169588315247959918, 12092601263851930181412631, 3132724961425012148414534036, 914922971820526793968713748093
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 11 2024

Keywords

Crossrefs

Cf. A007317, A321087, A348857, A376095, A376127.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[(k + 1)^2 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; A[] = 0; Do[A[x] = 1/((1 - x) (1 - x A[x] - 3 x^2 A'[x] - x^3 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) - 3 * x^2 * A'(x) - x^3 * 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 1-6 of 6 results.

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