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 21-29 of 29 results.

A368837 a(n) = n! * (n+2)! * Sum_{k=0..n} 1/(k! * (k+2)!).

Original entry on oeis.org

1, 4, 33, 496, 11905, 416676, 20000449, 1260028288, 100802263041, 9979424041060, 1197530884927201, 171246916544589744, 28769481979491076993, 5610048986000760013636, 1256650972864170243054465, 320445998080363411978888576
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2024

Keywords

Crossrefs

Cf. A099597, A006040, A228229, A368838.

Programs

  • PARI
    a(n) = n!*(n+2)!*sum(k=0, n, 1/(k!*(k+2)!));

Formula

a(n) = n*(n+2)*a(n-1) + 1.
a(n) ~ BesselI(2,2) * n! * (n+2)!. - Vaclav Kotesovec, Jan 09 2024

A368838 a(n) = n! * (n+3)! * Sum_{k=0..n} 1/(k! * (k+3)!).

Original entry on oeis.org

1, 5, 51, 919, 25733, 1029321, 55583335, 3890833451, 342393343689, 36978481118413, 4807202545393691, 740309191990628415, 133255654558313114701, 27717176148129127857809, 6596687923254732430158543, 1781105739278777756142806611, 541456144740748437867413209745
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2024

Keywords

Crossrefs

Cf. A099597, A006040, A228229, A368837.

Programs

  • PARI
    a(n) = n!*(n+3)!*sum(k=0, n, 1/(k!*(k+3)!));

Formula

a(n) = n*(n+3)*a(n-1) + 1.
a(n) ~ BesselI(3,2) * n! * (n+3)!. - Vaclav Kotesovec, Jan 09 2024

A074703 a(n) = n^2*a(n-1)+1, a(1)=0.

Original entry on oeis.org

0, 1, 10, 161, 4026, 144937, 7101914, 454522497, 36816322258, 3681632225801, 445477499321922, 64148759902356769, 10841140423498293962, 2124863523005665616553, 478094292676274763724426
Offset: 1

Views

Author

Vladeta Jovovic, Sep 03 2002

Keywords

Links

Crossrefs

Cf. A006040, A073701.

Programs

  • Mathematica
    nxt[{n_,a_}]:={n+1,(n+1)^2 a+1}; Transpose[NestList[nxt,{1,0},20]][[2]] (* Harvey P. Dale, Dec 11 2013 *)
  • PARI
    a(n)=round((besseli(0,2)-2)*n!^2) \\ Charles R Greathouse IV, Feb 19 2014

Formula

a(n) = round(n!^2*(BesselI(0, 2)-2)).

A228513 a(n) = Sum_{k=0..n} 2^k*(n!/k!)^2.

Original entry on oeis.org

1, 3, 16, 152, 2448, 61232, 2204416, 108016512, 6913057024, 559957619456, 55995761946624, 6775487195543552, 975670156158275584, 164888256390748581888, 32318098252586722066432, 7271572106832012464979968, 1861522459348995191034937344, 537979990751859610209097023488
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 24 2013

Keywords

Comments

Generally, Sum_{k=0..n} x^k*(n!/k!)^2 is asymptotic to BesselI(0,2*sqrt(x))*(n!)^2

Crossrefs

Cf. A000522, A006040, A217284.

Programs

  • Mathematica
    Table[(n!)^2*Sum[2^j/(j!)^2, {j, 0, n}], {n, 0, 20}]
Total/@Table[2^k (n!/k!)^2,{n,0,20},{k,0,n}] (* Harvey P. Dale, Jun 10 2018 *)

Formula

a(n) = (n^2+2)*a(n-1) - 2*(n-1)^2*a(n-2).
a(n) ~ 2*Pi*BesselI(0,2*sqrt(2)) * n^(2*n+1)/exp(2*n).

A295610 a(n) = Sum_{k=0..n} (n!/(n - k)!)^k.

Original entry on oeis.org

1, 2, 7, 256, 345749, 25090776406, 139507578065088907, 82622801516492599819822772, 6985137485409222182920705065038896201, 109110989095384931538566720095053550173384985449034, 395940975233113726268241745444050219538058574725338743701918216111
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 24 2017

Keywords

Links

Crossrefs

Cf. A000522, A006040, A036740, A217284, A219206.

Programs

  • Mathematica
    Table[Sum[(n!/(n - k)!)^k, {k, 0, n}], {n, 0, 10}]
Table[Sum[(Gamma[n + 1]/Gamma[k + 1])^(n - k), {k, 0, n}], {n, 0, 10}]
Table[Sum[(Binomial[n, k] k!)^k, {k, 0, n}], {n, 0, 10}]
  • PARI
    a(n) = sum(k=0, n, (n!/(n - k)!)^k); \\ Michel Marcus, Nov 25 2017

Formula

a(n) = Sum_{k=0..n} A219206(n,k)*A036740(k).
a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n^2 + n/2) / exp(n^2 - 1/12). - Vaclav Kotesovec, Nov 25 2017

A336241 a(n) = (n!)^2 * Sum_{d|n} 1 / (d!)^2.

Original entry on oeis.org

1, 5, 37, 721, 14401, 662401, 25401601, 2034950401, 135339724801, 16461151257601, 1593350922240001, 293575350020198401, 38775788043632640001, 9500068369885892198401, 1757631343928533032960001, 547963926586675321282560001, 126513546505547170185216000001
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 13 2020

Keywords

Crossrefs

Cf. A006040, A057625, A336242.

Programs

  • Mathematica
    Table[(n!)^2 Sum[1/(d!)^2, {d, Divisors[n]}], {n, 1, 17}]
nmax = 17; CoefficientList[Series[Sum[(BesselI[0, 2 x^(k/2)] - 1), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2 // Rest
  • PARI
    a(n) = n!^2*sumdiv(n, d, 1/d!^2); \\ Michel Marcus, Jul 13 2020

Formula

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

A336809 a(n) = (n!)^2 * Sum_{k=0..n} (k+1) / ((n-k)!)^2.

Original entry on oeis.org

1, 3, 21, 271, 5649, 174051, 7447573, 422836191, 30767443521, 2792343036259, 309252314731701, 41051709426337743, 6434479982900111761, 1175819833620882461571, 247785659825802622964469, 59649892258930263778729951, 16268290830606063971956320513
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 27 2021

Keywords

Crossrefs

Cf. A001339, A006040.

Programs

  • Mathematica
    Table[n!^2 Sum[(k + 1)/(n - k)!^2, {k, 0, n}], {n, 0, 16}]
nmax = 16; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - x)^2, {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = BesselI(0,2*sqrt(x)) / (1 - x)^2.
a(n) ~ BesselI(0,2) * n!^2 * n. - Vaclav Kotesovec, Jul 11 2025

A340789 a(n) = (n!)^2 * Sum_{k=1..n} (-1)^(k+1) / (k!)^2.

Original entry on oeis.org

0, 1, 3, 28, 447, 11176, 402335, 19714416, 1261722623, 102199532464, 10219953246399, 1236614342814280, 178072465365256319, 30094246646728317912, 5898472342758750310751, 1327156277120718819918976, 339752006942904017899257855, 98188330006499261172885520096
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 23 2021

Keywords

Crossrefs

Cf. A002467, A006040, A066998, A073701.

Programs

  • Mathematica
    Table[n!^2 Sum[(-1)^(k + 1)/k!^2, {k, 1, n}], {n, 0, 17}]
nmax = 17; CoefficientList[Series[(1 - BesselJ[0, 2 Sqrt[x]])/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!^2

Formula

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

A346410 a(n) = (n!)^2 * Sum_{k=0..n-1} 1 / ((n-k) * k!)^2.

Original entry on oeis.org

0, 1, 5, 22, 152, 2001, 45097, 1527506, 71864928, 4466430513, 353828600029, 34770661312190, 4148422395161464, 590479899466175681, 98824492409739430401, 19209838771051338898234, 4291488438323868507946880, 1091819942877526843993466529, 313819508664449992611846900981
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 15 2021

Keywords

Crossrefs

Cf. A002104, A006040, A066998, A193563, A336291, A346411.

Programs

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

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = polylog(2,x) * BesselI(0,2*sqrt(x)).
Previous Showing 21-29 of 29 results.

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

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