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.

A337592 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 2^(k-1) * a(n-k).

Original entry on oeis.org

1, 1, 4, 28, 312, 4936, 104128, 2806336, 93560064, 3765265408, 179415074304, 9964625629696, 636737424291840, 46303081167540224, 3796275000959266816, 348100339275620651008, 35448445862069986361344, 3984266642444252234153984, 491556877841462376382332928
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 02 2020

Keywords

Crossrefs

Cf. A004211, A337593, A337594, A337595, A337597.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[2 x]] - 1)/2], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp((BesselI(0,2*sqrt(2*x)) - 1) / 2).
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} 2^(n-1) * x^n / (n!)^2).

A337593 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 3^(k-1) * a(n-k).

Original entry on oeis.org

1, 1, 5, 42, 573, 11226, 294804, 9946791, 417064365, 21187915362, 1278636342660, 90195692894451, 7338668846348844, 681052861293535251, 71405270562056271741, 8388541745045127600597, 1096298129481068449931085, 158383969954582566159384786, 25153555538082783169267336764
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 02 2020

Keywords

Crossrefs

Cf. A004212, A337592, A337594, A337595, A337597.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[3 x]] - 1)/3], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp((BesselI(0,2*sqrt(3*x)) - 1) / 3).
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} 3^(n-1) * x^n / (n!)^2).

A337594 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 4^(k-1) * a(n-k).

Original entry on oeis.org

1, 1, 6, 58, 920, 21176, 654960, 26114768, 1298070912, 78359732608, 5630565514496, 473796572027648, 46060380961356800, 5114737212582603776, 642502387594286036992, 90542358999393528670208, 14209873001490130067095552, 2467784343879850163370295296, 471558856613839054976849608704
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 02 2020

Keywords

Crossrefs

Cf. A004213, A337592, A337593, A337595, A337597.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 4 Sqrt[x]] - 1)/4], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp((BesselI(0,4*sqrt(x)) - 1) / 4).
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} 4^(n-1) * x^n / (n!)^2).

A337595 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * 5^(k-1) * a(n-k).

Original entry on oeis.org

1, 1, 7, 76, 1359, 35620, 1256470, 57247765, 3259660095, 225795951580, 18644190437550, 1805220546542425, 202173130530484350, 25889773647793362425, 3754040522961719322325, 611181508958872398483625, 110903705593861290502897375, 22285223101687304853202923500, 4930523789420612133816212731750
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 02 2020

Keywords

Crossrefs

Cf. A005011, A337592, A337593, A337594, A337597.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k 5^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[Exp[(BesselI[0, 2 Sqrt[5 x]] - 1)/5], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp((BesselI(0,2*sqrt(5*x)) - 1) / 5).
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} 5^(n-1) * x^n / (n!)^2).

A333985 a(0) = 0; a(n) = 6^(n-1) + (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * 6^(k-1) * (n-k) * a(n-k).

Original entry on oeis.org

0, 1, 8, 102, 2448, 99576, 6070032, 517803840, 58901955840, 8614609282944, 1574889814326528, 351896788824053760, 94354291010501932032, 29899137879209196380160, 11053567519385396409446400, 4715135497874174650128617472, 2298676381054790419739595571200, 1270045124912998373344157769891840
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 04 2020

Keywords

Crossrefs

Cf. A102223, A333981, A333982, A333983, A333984, A337597.

Programs

  • Mathematica
    a[0] = 0; a[n_] := a[n] = 6^(n - 1) + (1/n) Sum[Binomial[n, k]^2 6^(k - 1) (n - k) a[n - k], {k, 1, n - 1}]; Table[a[n],{n, 0, 17}]
nmax = 17; CoefficientList[Series[-Log[(7 - BesselI[0, 2 Sqrt[6 x]])/6], {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = -log((7 - BesselI(0,2*sqrt(6*x))) / 6).
Showing 1-5 of 5 results.

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