A006040 -id:A006040 - cp's OEIS frontend

A375231 Interleaving A006040 and A228229.

1, 1, 3, 9, 19, 82, 229, 1313, 4581, 32826, 137431, 1181737, 5772103, 57905114, 323237769, 3705927297, 23273119369, 300180111058, 2094580743211, 30018011105801, 230403881753211, 3632179343801922, 30413312391423853, 523033825507476769, 4744476733062121069, 88392716510763573962

Offset: 0

Stefano Spezia, Aug 06 2024

nonn

Conjecture: a(n) is the permanent of the n X n matrix whose generic element is given by M_{i,j} = 1 if i = j or i + j = 1 (mod 2), with i,j in [n].

Cf. A006040, A078126 (determinant), A228229.

A006040[n_]:=Sum[(n!/(n - k)!)^2, {k, 0, n}]; A228229[n_]:= n!*(n + 1)!*Sum[ 1/(k!*(k + 1)!),{k,0,n}]; a[n_]:=If[OddQ[n],A006040[(n+1)/2],A228229[n/2]]; Array[a,26,0]

a(n) = A006040((n+1)/2) for odd n.

a(n) = A228229(n/2) for even n.

A070910 Decimal expansion of BesselI(0,2).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, May 20 2002

			2.2795853023360672674372044408115333532858411...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:5:5 at page 20.

Michael Penn, An exponential trigonometric integral., YouTube video, 2020.
Eric Weisstein's World of Mathematics, Factorial Sums.
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind.

Cf. A096789, A070913 (continued fraction), A006040.

Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2))), this sequence (I(0,2)).

RealDigits[ BesselI[0, 2], 10, 110] [[1]] (* Robert G. Wilson v, Jul 09 2004 *)
(* Or *) RealDigits[ Sum[ 1/(n!n!), {n, 0, Infinity}], 10, 110][[1]]

besseli(0,2) \\ Charles R Greathouse IV, Feb 19 2014

Equals Sum_{k>=0} 1/k!^2.
From Peter Bala, Aug 19 2013: (Start)
Continued fraction expansion: 1/(1 - 1/(2 - 1/(5 - 4/(10 - 9/(17 - ... - (n-1)^2/(n^2+1 - ...)))))). See A006040. Cf. A096789.
This continued fraction is the particular case k = 0 of the result BesselI(k,2) = Sum_{n = 0..oo} 1/(n!*(n+k)!) = 1/(k! - k!/((k+2) - (k+1)/((2*k+5) - 2*(k+2)/((3*k+10) - ... - n*(n+k)/(((n+1)*(n+k+1)+1) - ...))))). See the remarks in A099597 for a sketch of the proof. (End)
From Amiram Eldar, May 29 2021: (Start)
Equals (1/e^2) * Sum_{k>=0} binomial(2*k,k)/k! = e^2 * Sum_{k>=0} (-1)^k*binomial(2*k,k)/k!.
Equal (1/(2*Pi)) * Integral_{x=0..2*Pi} exp(2*sin(x)) dx. (End)
Equals BesselJ(0,2*i). - Jianing Song, Sep 18 2021

A073701 a(n) = n^2*a(n-1)+(-1)^n.

Original entry on oeis.org

1, 0, 1, 8, 129, 3224, 116065, 5687184, 363979777, 29482361936, 2948236193601, 356736579425720, 51370067437303681, 8681541396904322088, 1701582113793247129249, 382855975603480604081024, 98011129754491034644742145, 28325216499047909012330479904, 9177370145691522519995075488897

Offset: 0

Vladeta Jovovic, Aug 30 2002

nonn

The sequence b(n) := n!^2 satisfies the same recurrence below for a(n) with the initial conditions b(0) = 1, b(1) = 1. It follows that, for n >=3, a(n) = n!^2*(1/(4 + 4/(8 + 9/(15 +...+ (n-1)^2/(n^2-1))))). Hence BesselJ(0,2) := sum {k = 0..inf} (-1)^k/k!^2 = 1/(4 + 4/(8 + 9/(15 + ...+(n-1)^2/(n^2+1 + ...)))) = 0.22388 90779 ... . Cf. A006040. - Peter Bala, Jul 09 2008

Seiichi Manyama, Table of n, a(n) for n = 0..253

Cf. A000166, A006040.

[1] cat [ n eq 1 select 0 else n^2*Self(n-1)+(-1)^n:n in [1..15]]; // Marius A. Burtea, Feb 13 2020

Join[{a = 1}, Table[a = a*n^2 + (-1)^n, {n, 15}]] (* Jayanta Basu, Jul 08 2013 *)

a(n) = n!^2*Sum_{k=0..n} (-1)^k/k!^2.
BesselJ(0, 2*sqrt(x))/(1-x) = Sum_{n>=0} a(n)*x^n/n!^2.
a(n) = round(n!^2*BesselJ(0, 2)), n>0.
Recurrence: a(0) = 1, a(1) = 0, a(n) = (n^2-1)*a(n-1) + (n-1)^2*a(n-2), n >= 2. - Peter Bala, Jul 09 2008, corrected by Georg Fischer, Feb 13 2020
a(n) = Sum_{k=0..n} (-1)^(n-k)*(k!*binomial(n,k))^2. - Ridouane Oudra, Jul 11 2025

A228229 Recurrence a(n) = n(n + 1)a(n-1) + 1 with a(0) = 1.

Original entry on oeis.org

1, 3, 19, 229, 4581, 137431, 5772103, 323237769, 23273119369, 2094580743211, 230403881753211, 30413312391423853, 4744476733062121069, 863494765417306034559, 181333900737634267257391, 43520136177032224141773841, 11837477040152764966562484753

Offset: 0

Peter Bala, Aug 19 2013

Main subdiagonal (and main superdiagonal) of A099597. Cf. A006040 and A228230.

Seiichi Manyama, Table of n, a(n) for n = 0..252
E. W. Weisstein, Modified Bessel Function of the First Kind

Cf. A006040, A096789, A099597, A228230, A368837, A368838.

A228229 :=proc(n) option remember
    if n = 0 then 1
    else n*(n+1)*procname(n-1) + 1
    end if:
end proc:
seq(A228229(n), n = 0..20);

RecurrenceTable[{a[n] == n*(n + 1)*a[n-1] + 1, a[0] == 1},a,{n,0,20}] (* Vaclav Kotesovec, May 06 2015 *)

a(n) = n!*(n + 1)!*sum {k = 0..n} 1/(k!*(k + 1)!).
Generating function: 1/(1 - x)*1/sqrt(x)*BesselI(1, 2*sqrt(x)) = sum {n >= 0} a(n)*x^n/(n!*(n + 1)!).
Defining recurrence equation: a(n) = n*(n + 1)*a(n-1) + 1 with a(0) = 1.
Alternative recurrence equation: a(0) = 1, a(1) = 3, and for n >= 2, a(n) = (n*(n + 1) + 1)*a(n-1) - n*(n - 1)*a(n-2).
The sequence b(n) := n!*(n + 1)! satisfies the same recurrence with the initial conditions b(0) = 1, b(1) = 2. It follows that we have the finite continued fraction expansion a(n) = n!*(n + 1)!*(1/(1 - 1/(3 - 2/(7 - 6/(13 - … - n*(n - 1)/(n^2 + n + 1)))))). Taking the limit yields the continued fraction expansion for the modified Bessel function value BesselI(1,2) = sum {k = 0..inf} 1/(k!*(k + 1)!) = 1/(1 - 1/(3 - 2/(7 - 6/(13 - ...- n*(n - 1)/(n^2 + n + 1 - ...))))) = 1.590636... (see A096789).
a(n) ~ BesselI(1,2) * n!*(n+1)!. - Vaclav Kotesovec, May 06 2015

A099597 Array T(n,k) read by antidiagonals: expansion of exp(x+y)/(1-xy).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 9, 4, 1, 1, 5, 19, 19, 5, 1, 1, 6, 33, 82, 33, 6, 1, 1, 7, 51, 229, 229, 51, 7, 1, 1, 8, 73, 496, 1313, 496, 73, 8, 1, 1, 9, 99, 919, 4581, 4581, 919, 99, 9, 1, 1, 10, 129, 1534, 11905, 32826, 11905, 1534, 129, 10, 1, 1, 11, 163, 2377, 25733, 137431, 137431, 25733, 2377, 163, 11, 1

Offset: 0

Ralf Stephan, Oct 28 2004

Rows are polynomials in n whose coefficients are in A099599.
From Peter Bala, Aug 19 2013: (Start)
The k-th superdiagonal sequence of this square array occurs as the sequence of numerators in the convergents to a certain continued fraction representation of the constant BesselI(k,2), where BesselI(k,x) is a modified Bessel function of the first kind:
Let d_k(n) = T(n,n+k) = n! * (n+k)! * Sum_{i=0..n} 1/(i!*(i+k)!) denote the sequence of entries on the k-th superdiagonal. It satisfies the first-order recurrence equation d_k(n) = n*(n+k)*d_k(n-1) + 1 with d_k(0) = 1 and also the second-order recurrence d_k(n) = (n*(n+k)+1)*d_k(n-1) - (n-1)*(n-1+k)*d_k(n-2) with initial conditions d_k(0) = 1 and d_k(1) = k+2. This latter recurrence is also satisfied by the sequence n!*(n+k)!. From this observation we obtain the finite continued fraction expansion d_k(n) = n!*(n+k)!*(1/(k! - k!/((k+2) - (k+1)/((2*k+5) - 2*(k+2)/((3*k+10) - ... - n*(n+k)/(((n+1)*(n+k+1)+1) ))))).
Taking the limit as n -> infinity produces a continued fraction representation for the modified Bessel function value BesselI(k,2) = Sum_{i=0..inf} 1/(i!*(i+k)!) = 1/(k! - k!/((k+2) -(k+1)/((2*k+5) - 2*(k+2)/((3*k+10) - ... - n*(n+k)/(((n+1)*(n+k+1)+1) - ...))))). See A070910 for the case k = 0 and A096789 for the case k = 1. (End)

			1, 1,  1,   1,    1,     1,
1, 2,  3,   4,    5,     6,
1, 3,  9,  19,   33,    51,
1, 4, 19,  82,  229,   496,
1, 5, 33, 229, 1313,  4581,
1, 6, 51, 496, 4581, 32826,

E. W. Weisstein, Modified Bessel Function of the First Kind

Rows include A000012, A000027, A058331. Main diagonal is A006040. Antidiagonal sums are in A099598. Cf. A099599.

Cf. A088699. A228229 (main super and subdiagonal).

#A099597
T := proc(n,k) option remember;
if n = 0 then 1 elif k = 0 then 1
else n*k*thisproc(n-1,k-1) + 1
fi
end:
# Diplay entries by antidiagonals
seq(seq(T(n-k,k), k = 0..n), n = 0..10);
# Peter Bala, Aug 19 2013

T[, 0] = T[0, ] = 1;
T[n_, k_] := T[n, k] = n k T[n - 1, k - 1] + 1;
Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 02 2019 *)

T(n,k) = Sum_{i=0..min(n,k)} C(n,i)*C(k,i)*i!^2. The LDU factorization of this square array is P * D * transpose(P), where P is Pascal's triangle A007318 and D = diag(0!^2, 1!^2, 2!^2, ... ). Compare with A088699. - Peter Bala, Nov 06 2007
Recurrence equation: T(n,k) = n*k*T(n-1,k-1) + 1 with boundary conditions T(n,0) = T(0,n ) = 1.
Main subdiagonal and main superdiagonal [1, 3, 19, 229, ...] is A228229. - Peter Bala, Aug 19 2013
nth row/column o.g.f.: HypergeometricPFQ[{1,1,-n},{},x/(x-1)]/(1-x) (see comment in A099599). - Natalia L. Skirrow, Jul 18 2025

A217284 a(n) = Sum_{k=0..n} (n!/k!)^3.

Original entry on oeis.org

1, 2, 17, 460, 29441, 3680126, 794907217, 272653175432, 139598425821185, 101767252423643866, 101767252423643866001, 135452212975869985647332, 234061424022303335198589697, 514232948577000427431301564310, 1411055210895289172871491492466641, 4762311336771600958441283787074913376

Offset: 0

Vaclav Kotesovec, Sep 30 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..180

Cf. A000522, A006040, A271574, A336247.

Table[Sum[(n!/k!)^3, {k, 0, n}], {n, 0, 20}]

a(n) = sum(k=0, n, (n!/k!)^3); \\ Seiichi Manyama, May 02 2021

Recurrence: a(n) = (n+1)*(n^2-n+1)*a(n-1)-(n-1)^3*a(n-2).
a(n) ~ 2.12970254898330641813452361... * (n!)^3 = A271574 * (n!)^3.
a(n) = n^3 * a(n-1) + 1. - Seiichi Manyama, May 02 2021

A066998 a(0)=0; a(n) = n^2*a(n-1) + 1.

Original entry on oeis.org

0, 1, 5, 46, 737, 18426, 663337, 32503514, 2080224897, 168498216658, 16849821665801, 2038828421561922, 293591292704916769, 49616928467130933962, 9724917979557663056553, 2188106545400474187724426

Offset: 0

Benoit Cloitre, Jan 27 2002

nonn

if s(n) is a sequence defined as s(0)=x, s(n) = n^2*s(n-1) + k, then s(n) = n!^2*x + a(n)*k. - Gary Detlefs, Feb 20 2010

Harry J. Smith, Table of n, a(n) for n = 0..100
Alexandre Silva, Carlos Pereira dos Santos, João Pedro Neto, and Richard J. Nowakowski, Disjunctive sums of quasi-nimbers, Theor. Comp. Sci. (2022).

This is the same recurrence relation as A006040 except A006040 has a(0) = 1.

RecurrenceTable[{a[0]==0,a[n]==n^2 a[n-1]+1},a,{n,20}] (* Harvey P. Dale, Jan 24 2019 *)

a(n) = { my(a=0); for (n=1, n, a=n^2*a + 1); a } \\ Harry J. Smith, Apr 24 2010

a(n) = (n!)^2*Sum_{i=1..n} 1/(i!)^2.
a(n) = floor((1-BesselI(0, 2))*(n!)^2). - Benoit Cloitre, Nov 02 2002
Sum_{n>=0} a(n) * x^n / (n!)^2 = (BesselI(0,2*sqrt(x)) - 1) / (1 - x). - Ilya Gutkovskiy, Jan 23 2021

Better description from James D. Klein, Feb 25 2002

A336247 a(n) = (n!)^n * Sum_{k=0..n} 1 / (k!)^n.

Original entry on oeis.org

1, 2, 9, 460, 684545, 50547203126, 280807908057046657, 165858480204085842350156792, 13997217669604247492958380810030809089, 218434494471443385260764665498960241287478619115850, 792268399795067334328715213043856435592857850955707257780000000001

Offset: 0

Ilya Gutkovskiy, Jul 14 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..30

Cf. A000522, A006040, A036740, A060943, A217284, A336248.

Main diagonal of A343863.

Table[(n!)^n Sum[1/(k!)^n, {k, 0, n}], {n, 0, 10}]

a(n) = (n!)^n * sum(k=0, n, 1/(k!)^n); \\ Michel Marcus, Jul 14 2020

A336804 a(n) = (n!)^2 * Sum_{k=0..n} 2^(n-k) / (k!)^2.

Original entry on oeis.org

1, 3, 25, 451, 14433, 721651, 51958873, 5091969555, 651772103041, 105587080692643, 21117416138528601, 5110414705523921443, 1471799435190889375585, 497468209094520608947731, 195007537965052078707510553, 87753392084273435418379748851, 44929736747147998934210431411713

Offset: 0

Ilya Gutkovskiy, Jan 27 2021

nonn

Cf. A006040, A010844, A228513, A336805, A336807, A336808.

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

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

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

A336805 a(n) = (n!)^2 * Sum_{k=0..n} 3^(n-k) / (k!)^2.

Original entry on oeis.org

1, 4, 49, 1324, 63553, 4766476, 514779409, 75672573124, 14529134039809, 3530579571673588, 1059173871502076401, 384480115355253733564, 166095409833469612899649, 84210372785569093740122044, 49515699197914627119191761873, 33423096958592373305454439264276, 25668938464198942698589009354963969

Offset: 0

Ilya Gutkovskiy, Jan 27 2021

nonn

Cf. A006040, A010845, A336804, A336807, A336808.

Table[n!^2 Sum[3^(n - k)/k!^2, {k, 0, n}], {n, 0, 16}]
nmax = 16; CoefficientList[Series[BesselI[0, 2 Sqrt[x]]/(1 - 3 x), {x, 0, nmax}], x] Range[0, nmax]!^2

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

a(0) = 1; a(n) = 3 * n^2 * a(n-1) + 1.

cp's OEIS Frontend

A375231 Interleaving A006040 and A228229.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A070910 Decimal expansion of BesselI(0,2).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A073701 a(n) = n^2*a(n-1)+(-1)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A228229 Recurrence a(n) = n*(n + 1)*a(n-1) + 1 with a(0) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A099597 Array T(n,k) read by antidiagonals: expansion of exp(x+y)/(1-xy).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A217284 a(n) = Sum_{k=0..n} (n!/k!)^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A066998 a(0)=0; a(n) = n^2*a(n-1) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A336247 a(n) = (n!)^n * Sum_{k=0..n} 1 / (k!)^n.

Views

A228229 Recurrence a(n) = n(n + 1)a(n-1) + 1 with a(0) = 1.