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.

Previous Showing 21-28 of 28 results.

A352465 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(2*n,2*k)^2 * k * a(n-k).

Original entry on oeis.org

1, 1, 19, 1576, 356035, 172499176, 154989443170, 234120771123513, 553941959716031715, 1945912976888526218512, 9731900583801946493234794, 66990924607889809703423378253, 617312916540194845307221190273098, 7439659538258619452171059589120614701
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2022

Keywords

Crossrefs

Cf. A005046, A023998, A346220, A352466.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[2 n, 2 k]^2 k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 13}]
nmax = 26; Take[CoefficientList[Series[Exp[Sum[x^(2 k)/(2 k)!^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2, {1, -1, 2}]

Formula

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

A062995 Doubly exponentiate the Bessel function I(0,2*sqrt(z)).

Original entry on oeis.org

1, 1, 5, 49, 789, 18741, 612383, 26218956, 1419303189, 94531262917, 7582017897795, 719690829785016, 79691175192777855, 10170046938232956048, 1480481369981439216732, 243659154929530351237884, 44987315567879408248084629, 9254611189980167520327621253
Offset: 0

Views

Author

Karol A. Penson, Jun 28 2001

Keywords

Links

Crossrefs

Cf. A023998.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[E^(-1 + E^(-1 + BesselI[0, 2*Sqrt[x]])), {x, 0, nmax}], x] * Range[0, nmax]!^2 (* Vaclav Kotesovec, Jun 09 2019 *)

Formula

Hypergeometric generating function for a(n): exp(exp(BesselI(0, 2*sqrt(z))-1)-1) = Sum_{n>=0} a(n)*z^n/(n!)^2.

Extensions

More terms from Vaclav Kotesovec, Jun 09 2019

A335501 a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} binomial(n,k)^2 * (n-k) * a(k).

Original entry on oeis.org

1, 2, 10, 86, 1098, 19142, 431926, 12150518, 414474570, 16781350694, 792845706630, 43107783435158, 2666346336398454, 185796230244565462, 14464057604306584774, 1248919312238777955086, 118855834572748011228490, 12397162719421869533115622
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 12 2020

Keywords

Crossrefs

Cf. A001861, A023998.

Programs

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

Formula

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

A336609 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(1 / BesselJ(0,2*sqrt(x)) - 1).

Original entry on oeis.org

1, 1, 5, 52, 917, 24396, 909002, 45062697, 2862532213, 226403027044, 21794813189810, 2507115921526437, 339421509956163362, 53393907140415300317, 9653668439939308357991, 1987242385193691443059527, 461955240782446199029195253
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 27 2020

Keywords

Crossrefs

Cf. A000275, A002190, A023998.

Programs

  • Mathematica
    nmax = 16; CoefficientList[Series[Exp[1/BesselJ[0, 2 Sqrt[x]] - 1], {x, 0, nmax}], x] Range[0, nmax]!^2
A000275[0] = 1; A000275[n_] := A000275[n] = -Sum[(-1)^(n - k) Binomial[n, k]^2 A000275[k], {k, 0, n - 1}]; a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k A000275[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

Formula

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * A000275(k) * a(n-k).

A346224 a(n) = (n!)^2 * Sum_{k=0..floor(n/2)} 1 / ((n-2*k)! * 4^k * k!).

Original entry on oeis.org

1, 1, 3, 15, 114, 1170, 15570, 256410, 5103000, 119773080, 3264445800, 101784097800, 3591396824400, 141958074258000, 6236035482877200, 302218901402418000, 16060366291617648000, 930654556409161584000, 58524794739862410960000, 3976525824684785163792000
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 16 2021

Keywords

Crossrefs

Cf. A000085, A000898, A023998, A080599, A239840.

Programs

  • Mathematica
    Table[(n!)^2 Sum[1/((n - 2 k)! 4^k k!), {k, 0, Floor[n/2]}], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Exp[x + x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2
  • PARI
    a(n) = (n!)^2 * sum(k=0, n\2, 1/((n-2*k)!*4^k*k!)); \\ Michel Marcus, Jul 17 2021

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( x + x^2 / 4 ).
a(n) = n! * Sum_{k=0..n} Stirling1(n,k) * Bell(k) / 2^(n-k).
D-finite with recurrence a(0) = a(1) = 1; a(n) = n * a(n-1) + n * (n-1)^2 * a(n-2) / 2.
a(n) ~ sqrt(Pi) * n^((3*n + 1)/2) / (2^(n/2) * exp((3*n + 1)/2 - sqrt(2*n))). - Vaclav Kotesovec, Jul 17 2021

A352658 a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * binomial(k+1,2) * k * a(n-k).

Original entry on oeis.org

1, 1, 5, 39, 508, 9235, 224481, 6959932, 266492388, 12302514945, 671505310855, 42664357009186, 3114726872133570, 258452373177094213, 24149855477595375815, 2520813303733886387220, 291892618561012451083816, 37264133443594227118861233, 5216461719269145457350349359
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 25 2022

Keywords

Crossrefs

Cf. A000217, A002411, A023998, A279361, A336227, A337591.

Programs

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

Formula

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

A365106 Sum_{n>=0} a(n) * x^n / n!^2 = exp( Sum_{n>=1} prime(n) * x^n / n!^2 ).

Original entry on oeis.org

1, 2, 11, 107, 1577, 32201, 860460, 28921567, 1187475909, 58232016701, 3350187053856, 222857979706305, 16935374386652282, 1455271176236200143, 140181486948923188907, 15023106134895469195114, 1779460642743292348315607, 231607462899834684300774917, 32954119475274480307491604062, 5102159139278049158548905019487
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 21 2023

Keywords

Crossrefs

Cf. A007446, A023998.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[Sum[Prime[k] x^k/k!^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, k]^2 k Prime[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

Formula

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

A365107 Sum_{n>=0} a(n) * x^n / n!^2 = exp( Sum_{n>=1} x^prime(n) / prime(n)!^2 ).

Original entry on oeis.org

1, 0, 1, 1, 18, 101, 1550, 22492, 424536, 10283064, 272319552, 8959493401, 328044534576, 13799304374077, 657306569855728, 34694458662034731, 2048559070407831424, 132868259271772801185, 9463476338179250300352, 736376651361995115417850, 62178423492630241909006224, 5689134205956573233701281462
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 21 2023

Keywords

Crossrefs

Cf. A023998, A190476.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[Sum[x^Prime[k]/Prime[k]!^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Binomial[n, Prime[k]]^2 Prime[k] a[n - Prime[k]], {k, 1, PrimePi[n]}]; Table[a[n], {n, 0, 21}]

Formula

a(0) = 1; a(n) = (1/n) * Sum_{p <= n, p prime} binomial(n,p)^2 * p * a(n-p).
Previous Showing 21-28 of 28 results.

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