A096789 -id:A096789 - cp's OEIS frontend

A340103 a(n) = [x^n] Product_{k>=1} (1 + n^(k-1)*x^k).

1, 1, 2, 12, 80, 875, 10584, 170471, 2949120, 63772920, 1441000000, 38818444632, 1089573617664, 35185728919614, 1175820172477440, 44425722744140625, 1722925924631969792, 74364737115532234518, 3291298649632850485248, 159785357022861166517580, 7932051456000000000000000

Offset: 0

Ilya Gutkovskiy, Apr 24 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..385

Cf. A000009, A003056, A008289, A291698, A292305, A304961, A338697, A344094.

Table[SeriesCoefficient[Product[(1 + n^(k - 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Unprotect[Power]; 0^0 = 1; Table[Sum[Length[Select[IntegerPartitions[n, {k}], UnsameQ @@ # &]] n^(n - k), {k, 0, Floor[(Sqrt[8 n + 1] - 1)/2]}], {n, 0, 20}]
Join[{1}, Table[SeriesCoefficient[n*QPochhammer[-1/n, n*x]/(n+1), {x, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 09 2021 *)

a(n) = Sum_{k=0..A003056(n)} q(n,k) * n^(n-k), where q(n,k) is the number of partitions of n into k distinct parts.

a(n) ~ c * n^(n-1), where c = BesselI(1,2) = A096789 = 1.590636854637329... - Vaclav Kotesovec, May 09 2021

A048764 Largest factorial <= n.

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24

Offset: 1

Charles T. Le (charlestle(AT)yahoo.com)

J. Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3 (1999), 202-204.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Krassimir T. Atanassov, On Some of Smarandache's Problems.
Vassia K. Atanassova and Krassimir T. Atanassov, On the 43rd and 44th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5, No. 2, (1999), 86-88.
Li Jie, On the inferior and superior factorial part sequences, in Zhang Wenpeng (ed.), Research on Smarandache Problems in Number Theory (collected papers), 2004, pp. 47-48.
Florentin Smarandache, Only Problems, Not Solutions!.

Cf. A000142, A001113, A096789, A084558.

Table[k = 1; While[(k + 1)! <= n, k++]; k!, {n, 80}] (* Michael De Vlieger, Aug 30 2016 *)

a(n)=my(t=1,k=1);while(t<=n,t*=k++);t/k \\ Charles R Greathouse IV, Sep 19 2012

from sympy import factorial as f
def a(n):
    k=1
    while f(k + 1)<=n: k+=1
    return f(k)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 21 2017, after Mathematica code

n log log n / log n << a(n) <= n. - Charles R Greathouse IV, Sep 19 2012
From Amiram Eldar, Aug 02 2022: (Start)
Sum_{n>=1} 1/a(n)^m = Sum_{k>=1} k/k!^m (Li Jie, 2004).
In particular:
Sum_{n>=1} 1/a(n)^2 = e (A001113).
Sum_{n>=1} 1/a(n)^3 = BesselI(1,2) (A096789). (End)
a(n) = A000142(A084558(n)). - Ridouane Oudra, Aug 22 2025

A086880 a(n) = floor( sum(k=0, infinity, k^n/(k!)^2 ) ); related to generalized Bell numbers.

Original entry on oeis.org

2, 1, 2, 3, 7, 17, 45, 128, 391, 1287, 4524, 16889, 66657, 276982, 1207598, 5507362, 26203307, 129757596, 667358910, 3558097578, 19632277761, 111930731957, 658482495614, 3992062349412, 24911272290567, 159833355923362

Offset: 0

Paul D. Hanna, Sep 16 2003

nonn

Define B(n) = sum(k=0, infinity, k^n/(k!)^2), then there exists a complex linear relation: B(3) = B(2) + B(1); B(4) = 2*B(3); B(5) = 2*B(4) + B(2); B(6) = 5*B(4) + 3*B(2); B(7) = 7*B(5) + B(3); B(12) = B(11) + 11*B(10); ...

			a(5) = floor(1^5/(1!)^2 + 2^5/(2!)^2 + 3^5/(3!)^2 + 4^5/(4!)^2 +...)

Vaclav Kotesovec, Table of n, a(n) for n = 0..646

Cf. A000994, A000995, A006789.

Table[Floor[Sum[k^n/(k!)^2,{k,0,Infinity}]],{n,0,20}] (* Vaclav Kotesovec, Jul 31 2014 *)
Flatten[{2, 1, Table[Floor[HypergeometricPFQ[ConstantArray[2, n-2], ConstantArray[1, n-1], 1]], {n,2,20}]}] (* Vaclav Kotesovec, May 23 2015 *)

sum(k>=0, k^n/(k!)^2) = A000994(n)*BesselI(0, 2) + A000995(n)*BesselI(1, 2), using Bessel function values BesselI(0, 2)=2.2795853023..., BesselI(1, 2) = 1.5906368546... (A096789) and where A000994 and A000995 shift 2 places left under binomial transform: A000994={1, 0, 1, 1, 2, 5, 13, 36, 109, 359, 1266, 4731, ...} A000995={0, 1, 0, 1, 2, 4, 10, 29, 90, 295, 1030, 3838, ...}.

A103505 Denominator in expansion of (1-x)*log(1-x).

Original entry on oeis.org

1, 1, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450

Offset: 0

Paul Barry, Feb 09 2005

Apart from initial terms, same as A002378.
See A002378 for many more comments and references.
Denominators for the sequence with o.g.f. (1-x)*log(1-x). Numerators are given by 1 - 0^n - 2(C(1,n) - C(0,n)). Also denominators for the sequence with o.g.f. (1+x)*log(1+x). This sequence has numerators (-1)^n - 0^n + 2(C(1,n) - C(0,n)).
Also the denominator of the least distance between two adjacent Farey fractions of order n. The numerator is 1. - Robert G. Wilson v, Apr 13 2014
For n>0, a(n) are the Engel expansion of A096789. - Benedict W. J. Irwin, Dec 15 2016
Number of permutations of length n>=0 avoiding the partially ordered pattern (POP) {1>2} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second one. - Sergey Kitaev, Dec 08 2020

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Cf. A000384.

[0^n+Binomial(1,n)-Binomial(0,n)+2*Binomial(n,2): n in [0..60]]; // Vincenzo Librandi, Dec 18 2016

CoefficientList[Series[(1-2x+2x^2+2x^3-x^4)/(1-x)^3,{x,0,50}],x] (* or *) Denominator/@CoefficientList[Normal[Series[(1-x)Log[1-x], {x,0,50}]], x]  (* Harvey P. Dale, Apr 20 2011 *)

G.f.: (1-2*x+2*x^2+2*x^3-x^4) / (1-x)^3;

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

A130820 Decimal expansion of number whose Engel expansion is given by the sequence: 1,1,2,2,3,3,4,4,...ceiling(n/2),...

Original entry on oeis.org

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

Offset: 1

Stephen Casey (hexomino(AT)gmail.com), Jul 17 2007

			2.8702221569733963308194588658111996012403192622809957012...

Engel, F. "Entwicklung der Zahlen nach Stammbruechen" Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg. pp. 190-191, 1913.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
Eric Weisstein's World of Mathematics, Engel Expansion.

Cf. A006784, A064648, A101689, A001405, A070910, A096789, A141827.

evalf(BesselI(0, 2) + BesselI(1, 2) - 1, 100); # Peter Bala, Jul 02 2016

First@ RealDigits@ N[Sum[1/Product[Ceiling[r/2], {r, n}], {n, 1000}], 100] (* Original program amended to generate output by Michael De Vlieger, Jul 03 2016 *)
RealDigits[3 - HypergeometricPFQ[{1, 1}, {3, 3, 3}, 1]/8, 10, 100][[1]] (* Vaclav Kotesovec, Jul 03 2016 *)

From Peter Bala, Jul 01 2016: (Start)
Constant c = 1/1 + 1/(1*1) + 1/(1*1*2) + 1/(1*1*2*2) + 1/(1*1*2*2*3) + 1/(1*1*2*2*3*3) + ... = Sum_{n >= 1} binomial(n,floor(n/2))/n!.
Alternative series representations:
c = 3 - Sum_{n >= 2} 1/(n*(n - 1)*n!^2);
c = 1 + Sum_{n >= 1} (n + 2)/(n!*(n + 1)!);
c = 5/3 + 1/3*Sum_{n >= 2} (n + 1)*(n + 2)/n!^2;
c = A070910 + A096789 - 1.
Continued fraction: c = 3 - 1/(8 - 4/(14 - 9/(32 - ... - (n-1)^2/(n^2 + n + 2 - ...)))). See comments in A141827. (End)

A222391 Decimal expansion of e^2/sqrt(Pi).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Mar 19 2013

			4.1688284832666922304213039077523102603866646811484996378300089546240432...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A072334, A087197, A222392.
Cf. A096789: Sum_{n >= 1} 1/(Gamma(n)*Gamma(n+1)).
Cf. A035009 (see fourth comment).

Digits:=100: evalf(exp(1)^2/sqrt(Pi)); # Wesley Ivan Hurt, Jan 09 2017

Mathematica
```
RealDigits[E^2/Sqrt[Pi], 10, 90][[1]]
```

(exp(1))^2/sqrt(Pi) \\ G. C. Greubel, Jan 09 2017

Equals A072334*A087197.

Equals decimal expansion of Sum_{n >= 1} 1/(Gamma(n/2)*Gamma((n+1)/2)).

A338697 a(n) = [x^n] Product_{k>=1} 1 / (1 - n^(k-1)*x^k).

Original entry on oeis.org

1, 1, 3, 13, 101, 931, 12391, 178809, 3331721, 66288142, 1589753211, 40104031166, 1183380156013, 36187564837217, 1262524447510383, 45533370885563716, 1834219414937219601, 76016894083755947753, 3479900167920331954531, 162982921698852088968886, 8341707623665223127224821

Offset: 0

Ilya Gutkovskiy, Apr 24 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..385

Cf. A008284, A075900, A124577, A300579, A338673, A338674, A338675, A338676, A338677, A338678, A338679, A344095.

Table[SeriesCoefficient[Product[1/(1 - n^(k - 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Join[{1}, Table[Sum[Length[IntegerPartitions[n, {k}]] n^(n - k), {k, 0, n}], {n, 1, 20}]]
Join[{1}, Table[SeriesCoefficient[x + (n-1)/(n*QPochhammer[1/n, n*x]), {x, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, May 09 2021 *)

a(n) = Sum_{k=0..n} p(n,k) * n^(n-k), where p(n,k) is the number of partitions of n into k parts.

a(n) ~ c * n^(n-1), where c = BesselI(1,2) = A096789 = 1.590636854637329... - Vaclav Kotesovec, May 09 2021

A247844 Decimal expansion of the value of the continued fraction [1; 1, 2, 3, 4, 5, ...].

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 25 2014

Equals 1+A052119.

			1.697774657964007982006790592551752599486658262998...

MathOverflow, Is any particular algebraic number known to have unbounded continued fraction coefficients?
Eric Weisstein's MathWorld, Continued Fraction Constant
Eric Weisstein's MathWorld, Continued Fraction

Cf. A001040, A001053, A052119, A060997, A070910, A096789.

FromContinuedFraction[Join[{1}, Range[50]]] // RealDigits[#, 10, 105]& // First
(* or *) 1+BesselI[1, 2]/BesselI[0, 2] // RealDigits[#, 10, 105]& // First

1+besseli(1,2)/besseli(0,2) \\ Charles R Greathouse IV, Oct 23 2023

1 + I_1(2) / I_0(2), where I_n(x) gives the modified Bessel function of the first kind.

A261879 Decimal expansion of BesselI(3,2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Nov 19 2015

			0.212739959239852655272354393375932037291752272915691833255184450497...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Eric Weisstein's MathWorld, Modified Bessel Function of the First Kind

Cf. A070910 (BesselI(0,2)), A096789 (BesselI(1,2)), A229020 (BesselI(2,2)).

Mathematica
```
RealDigits[BesselI[3, 2], 10, 105][[1]]
```

besseli(3,2) \\ Altug Alkan, Nov 19 2015

Sum_{k>=1} 1/((k - 2)!*(k + 1)!).

Also S(2) - S(1), using Peter Bala's notation in A229020.

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

Dumitru Damian, Oct 25 2021

			0.5767248077568733872...

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)).

RealDigits[BesselJ[1, 2], 10, 100][[1]] (* Amiram Eldar, Oct 25 2021 *)

besselj(1, 2) \\ Michel Marcus, Oct 25 2021

Sage
```
bessel_J(1, 2).n(digits=100)
```

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

cp's OEIS Frontend

A340103 a(n) = [x^n] Product_{k>=1} (1 + n^(k-1)*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A048764 Largest factorial <= n.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A086880 a(n) = floor( sum(k=0, infinity, k^n/(k!)^2 ) ); related to generalized Bell numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A103505 Denominator in expansion of (1-x)*log(1-x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A130820 Decimal expansion of number whose Engel expansion is given by the sequence: 1,1,2,2,3,3,4,4,...ceiling(n/2),...

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A222391 Decimal expansion of e^2/sqrt(Pi).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A338697 a(n) = [x^n] Product_{k>=1} 1 / (1 - n^(k-1)*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A247844 Decimal expansion of the value of the continued fraction [1; 1, 2, 3, 4, 5, ...].

Views

Author