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 1-9 of 9 results.

A348607 Decimal expansion of BesselJ(1,2).

Original entry on oeis.org

5, 7, 6, 7, 2, 4, 8, 0, 7, 7, 5, 6, 8, 7, 3, 3, 8, 7, 2, 0, 2, 4, 4, 8, 2, 4, 2, 2, 6, 9, 1, 3, 7, 0, 8, 6, 9, 2, 0, 3, 0, 2, 6, 8, 9, 7, 1, 9, 6, 7, 5, 4, 4, 0, 1, 2, 1, 1, 3, 9, 0, 2, 0, 7, 6, 4, 0, 8, 7, 1, 1, 6, 2, 8, 9, 6, 1, 2, 1, 8, 4, 9, 4, 8, 3, 9, 9
Offset: 0

Views

Author

Dumitru Damian, Oct 25 2021

Keywords

Examples

			0.5767248077568733872...

Crossrefs

Cf. A010790, A054687, A058798, A058797, A133920, A245308, A296168, A301484, A318224.
Bessel function values: A334380 (J(0,1)), A091681 (J(0,2)), A334383 (J(0,sqrt(2))), this sequence (J(1,2)), A197036 (I(0,1)), A070910 (I(0,2)), A334381 (I(0,sqrt(2))), A096789 (I(1,2)).

Programs

  • Mathematica
    RealDigits[BesselJ[1, 2], 10, 100][[1]] (* Amiram Eldar, Oct 25 2021 *)
  • PARI
    besselj(1, 2) \\ Michel Marcus, Oct 25 2021
  • Sage
    bessel_J(1, 2).n(digits=100)

Formula

Equals Sum_{k>=0} (-1)^k/(k!*(k+1)!).

A386300 a(n) = 1 + Sum_{k=0..n-1} (-1)^k * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 1, -5, 1, 89, 1, -2603, 1, 133265, 1, -10449779, 1, 1161734969, 1, -173838018059, 1, 33692909616161, 1, -8210919096355811, 1, 2457354686029706057, 1, -886023132463389334523, 1, 378811390242562021309361, 1, -189490246977278296495043411, 1, 109640147405400620012620910681
Offset: 0

Views

Author

Seiichi Manyama, Jul 17 2025

Keywords

Crossrefs

Cf. A054687, A386298, A386299.
Cf. A309473, A386301.

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*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(-x).
a(2*n) = 1 for n >= 0.

A335441 a(n) = 1 + Sum_{k=1..n-1} binomial(n-2,k-1) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 2, 4, 11, 40, 176, 907, 5360, 35668, 263789, 2146390, 19054040, 183248581, 1897952690, 21061861828, 249309196559, 3135518918800, 41754612283244, 586922460056851, 8684272948653068, 134919751191875572, 2195942678525060093, 37365571515146318650
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 10 2020

Keywords

Links

Crossrefs

Cf. A000111, A006677, A052886, A054687, A080635, A332776.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[Binomial[n - 2, k - 1] a[k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
terms = 23; A[] = 0; Do[A[x] = Normal[Integrate[Integrate[Exp[x] + A[x] D[A[x], x], x], x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]!

Formula

E.g.f. A(x) satisfies: A''(x) = exp(x) + A(x) * A'(x).
From Vaclav Kotesovec, Jun 11 2020: (Start)
E.g.f.: (BesselY(0, sqrt(2))*(BesselJ(1, sqrt(2)*exp(x/2)) - sqrt(2)*exp(x/2)*BesselJ(0, sqrt(2)*exp(x/2))) + BesselJ(0, sqrt(2))*(sqrt(2)*exp(x/2)*BesselY(0, sqrt(2)*exp(x/2)) - BesselY(1, sqrt(2)*exp(x/2)))) / (BesselJ(1, sqrt(2)*exp(x/2))*BesselY(0, sqrt(2)) - BesselJ(0, sqrt(2))*BesselY(1, sqrt(2)*exp(x/2))).
a(n) ~ 2 * n! / r^(n+1), where r = 1.35169030867903432729790416904526340210784862703704233748118252928787... is the smallest real root of the equation BesselY(0, sqrt(2))*BesselJ(1, sqrt(2)*exp(r/2)) = BesselJ(0, sqrt(2))*BesselY(1, sqrt(2)*exp(r/2)). (End)

A345104 a(n) = 1 + 2 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 3, 13, 89, 825, 9601, 134185, 2188353, 40788745, 855303265, 19927758377, 510728051073, 14279388168137, 432505475357729, 14107767947949289, 493046896702987841, 18380057918926012809, 728005164671113691105, 30531323352522247757225, 1351567976217998536472833
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 08 2021

Keywords

Crossrefs

Cf. A000165, A052886, A054687, A345105.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + 2 Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; A[] = 1; Do[A[x] = Normal[Integrate[2 A[x]^2 + Exp[x], x] + O[x]^(nmax + 1)], nmax]; CoefficientList[A[x], x] Range[0, nmax]!

Formula

E.g.f. A(x) satisfies: A'(x) = 2 * A(x)^2 + exp(x).

A345105 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 4, 25, 247, 3283, 54661, 1092427, 25473037, 678837319, 20351864821, 677954261635, 24842157250117, 993040102321927, 43003754679356941, 2005536858420616963, 100211634039201328381, 5341144936822423446247, 302468060262966258380773, 18136282125753572653056355
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 08 2021

Keywords

Crossrefs

Cf. A032031, A054687, A345102, A345104.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + 3 Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; A[] = 1; Do[A[x] = Normal[Integrate[3 A[x]^2 + Exp[x], x] + O[x]^(nmax + 1)], nmax]; CoefficientList[A[x], x] Range[0, nmax]!

Formula

E.g.f. A(x) satisfies: A'(x) = 3 * A(x)^2 + exp(x).

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).

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).

A309473 a(n) = (-1)^n + Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

Original entry on oeis.org

1, 0, 1, 1, 3, 11, 43, 195, 1063, 6395, 42371, 311883, 2501159, 21672355, 202544323, 2028522067, 21658255431, 245738583307, 2952508103651, 37440976938875, 499785548010759, 7005210659040979, 102862231664567651, 1579045889274408259, 25294106622048460903
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 11 2020

Keywords

Crossrefs

Cf. A006677, A014304, A054687, A332776, A335441.

Programs

  • Mathematica
    a[n_] := a[n] = (-1)^n + Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
terms = 24; A[] = 1; Do[A[x] = Normal[Integrate[A[x]^2 - Exp[-x], x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]!

Formula

E.g.f. A(x) satisfies: A'(x) = A(x)^2 - exp(-x).
From Vaclav Kotesovec, Jun 11 2020: (Start)
E.g.f.: exp(-x/2) * (BesselI(1, 2*exp(-x/2)) * (BesselK(0, 2) + BesselK(1, 2)) + (BesselI(0, 2) - BesselI(1, 2)) * BesselK(1, 2*exp(-x/2))) / ((BesselI(1, 2) - BesselI(0, 2)) * BesselK(0, 2*exp(-x/2)) + BesselI(0, 2*exp(-x/2)) * (BesselK(0, 2) + BesselK(1, 2))).
a(n) ~ n! / r^(n+1), where r = 1.4982609322383959128764444062824740935658895762... is the real root of the equation (BesselI(0, 2) - BesselI(1, 2)) * BesselK(0, 2*exp(-r/2)) = (BesselK(0, 2) + BesselK(1, 2)) * BesselI(0, 2*exp(-r/2)). (End)

A337187 a(n) = 1 + Sum_{k=0..n-2} binomial(n-2,k) * a(k) * a(n-k-2).

Original entry on oeis.org

1, 1, 2, 3, 7, 19, 63, 229, 955, 4407, 22445, 124249, 746003, 4821287, 33394193, 246652725, 1935828995, 16086138151, 141100295557, 1302780182449, 12630092274099, 128275445380247, 1362029496267529, 15090795795916493, 174167341456580947, 2090520625244752407
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 29 2021

Keywords

Crossrefs

Cf. A007558, A054687, A090344.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + Sum[Binomial[n - 2, k] a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]

Formula

E.g.f. A(x) satisfies: A''(x) = exp(x) + A(x)^2.
Showing 1-9 of 9 results.

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