A073701 -id:A073701 - cp's OEIS frontend

A006040 a(n) = Sum_{i=0..n} (n!/(n-i)!)^2.

1, 2, 9, 82, 1313, 32826, 1181737, 57905114, 3705927297, 300180111058, 30018011105801, 3632179343801922, 523033825507476769, 88392716510763573962, 17324972436109660496553, 3898118798124673611724426, 997918412319916444601453057, 288398421160455852489819933474

Offset: 0

N. J. A. Sloane, Simon Plouffe

R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..250
D. Deford, Seating rearrangements on arbitrary graphs, 2013. (See Table 1)
D. Deford, Seating rearrangements on arbitrary graphs, involve, Vol. 7 (2014), No. 6, 787-805. (See Table 1)
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
Index entries for sequences related to Bessel functions or polynomials

Main diagonal of array A099597.

Cf. A073701.

a[0]:= 1:
for n from 1 to 30 do a[n]:= n^2*a[n-1] + 1 od:
seq(a[i],i=0..30); # Robert Israel, Dec 15 2014

a = 1; lst = {a}; Do[a = a * n^2 + 1; AppendTo[lst, a], {n, 1, 14}]; lst (* Zerinvary Lajos, Jul 08 2009 *)
Table[Sum[(n!/(n - k)!)^2, {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 15 2017 *)

a(n)=sum(k=0, n, (k!*binomial(n, k))^2 ); \\ Joerg Arndt, Dec 14 2014

def A006040_list(len):
    L = [1]
    for k in range(1,len): L.append(L[-1]*k^2+1)
    return L
A006040_list(18) # Peter Luschny, Dec 15 2014

a(n) = n^2*a(n-1) + 1.
The following formulas will need adjusting, since I have changed the offset. - N. J. A. Sloane, Dec 17 2013
a(n+1) = Nearest integer to BesselI(0, 2)*n!*n!, n >= 1.
a(n+1) = n!^2*Sum_{k = 0..n} 1/k!^2. BesselI(0, 2*sqrt(x))/(1-x) = Sum_{n>=0} a(n+1)*x^n/n!^2. - Vladeta Jovovic, Aug 30 2002
Recurrence: a(1) = 1, a(2) = 2, a(n+1) = (n^2 + 1)*a(n) - (n - 1)^2*a(n-1), n >= 2. The sequence defined by b(n) := (n-1)!^2 satisfies the same recurrence with the initial conditions b(1) = 1, b(2) = 1. It follows that a(n+1) = n!^2*(1 + 1/(1 - 1/(5 - 4/(10 - ... - (n - 1)^2/(n^2 + 1))))). Hence BesselI(0,2) := Sum_{k >= 0} 1/k!^2 = 1 + 1/(1 - 1/(5 - 4/(10 - ... - (n - 1)^2/(n^2 + 1 - ...)))). Cf. A073701. - Peter Bala, Jul 09 2008

Offset changed by N. J. A. Sloane, Dec 17 2013

A091681 Decimal expansion of BesselJ(0,2).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jan 28 2004

The Pierce Expansion of this number is the squares > 1: 4,9,16,25,... - Franklin T. Adams-Watters, May 22 2006

			0.223890779...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Eric Weisstein's World of Mathematics, Factorial Sums
Eric Weisstein's World of Mathematics, Pierce Expansion

Cf A000290, A068985, A073701.

Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), this sequence (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2))), A070910 (I(0,2)).

RealDigits[N[BesselJ[0, 2], 250]][[1]] (* G. C. Greubel, Dec 26 2016 *)

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

Equals Sum_{k>=0} (-1)^k/(k!)^2.
Continued fraction expansion: BesselJ(0,2) = 1/(4 + 4/(8 + 9/(15 + ... + (n - 1)^2/(n^2 + 1 + ...)))). See A073701 for a proof. - Peter Bala, Feb 01 2015
Equals BesselI(0,2*i), where BesselI is the modified Bessel function of order 0. - Jianing Song, Sep 18 2021

A337152 a(n) = 2^n * (n!)^2 * Sum_{k=0..n} 1 / ((-2)^k * (k!)^2).

Original entry on oeis.org

1, 1, 9, 161, 5153, 257649, 18550729, 1817971441, 232700344449, 37697455800737, 7539491160147401, 1824556860755671041, 525472375897633259809, 177609663053400041815441, 69622987916932816391652873, 31330344562619767376243792849, 16041136416061320896636821938689

Offset: 0

Ilya Gutkovskiy, Jan 27 2021

nonn

Cf. A000354, A073701, A336804, A337153, A337154, A337155.

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

a(n) = 2^n * (n!)^2 * sum(k=0, n, 1 / ((-2)^k * (k!)^2)); \\ Michel Marcus, Jan 28 2021

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

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

A337153 a(n) = 3^n * (n!)^2 * Sum_{k=0..n} 1 / ((-3)^k * (k!)^2).

Original entry on oeis.org

1, 2, 25, 674, 32353, 2426474, 262059193, 38522701370, 7396358663041, 1797315155118962, 539194546535688601, 195727620392454962162, 84554332009540543653985, 42869046328837055632570394, 25206999241356188711951391673, 17014724487915427380567189379274, 13067308406719048228275601443282433

Offset: 0

Ilya Gutkovskiy, Jan 27 2021

nonn

Cf. A000180, A073701, A336805, A337152, A337154, A337155.

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

a(n) = 3^n * (n!)^2 * sum(k=0, n, 1 / ((-3)^k * (k!)^2)); \\ Michel Marcus, Jan 28 2021

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

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

A337154 a(n) = 4^n * (n!)^2 * Sum_{k=0..n} 1 / ((-4)^k * (k!)^2).

Original entry on oeis.org

1, 3, 49, 1763, 112833, 11283299, 1624795057, 318459831171, 81525716779777, 26414332236647747, 10565732894659098801, 5113814721015003819683, 2945557279304642200137409, 1991196720809938127292888483, 1561098229114991491797624570673, 1404988406203492342617862113605699

Offset: 0

Ilya Gutkovskiy, Jan 27 2021

nonn

Cf. A001907, A073701, A336807, A337152, A337153, A337155.

Table[4^n n!^2 Sum[1/((-4)^k k!^2), {k, 0, n}], {n, 0, 15}]
nmax = 15; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - 4 x), {x, 0, nmax}], x] Range[0, nmax]!^2

a(n) = 4^n * (n!)^2 * sum(k=0, n, 1 / ((-4)^k * (k!)^2)); \\ Michel Marcus, Jan 28 2021

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

a(0) = 1; a(n) = 4 * n^2 * a(n-1) + (-1)^n.

A337155 a(n) = 5^n * (n!)^2 * Sum_{k=0..n} 1 / ((-5)^k * (k!)^2).

Original entry on oeis.org

1, 4, 81, 3644, 291521, 36440124, 6559222321, 1607009468644, 514243029966081, 208268427136262804, 104134213568131402001, 63001199208719498210604, 45360863430278038711634881, 38329929598584942711331474444, 37563331006613243857104844955121, 42258747382439899339242950574511124

Offset: 0

Ilya Gutkovskiy, Jan 27 2021

nonn

Cf. A001908, A073701, A336808, A337152, A337153, A337154.

Table[5^n n!^2 Sum[1/((-5)^k k!^2), {k, 0, n}], {n, 0, 15}]
nmax = 15; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - 5 x), {x, 0, nmax}], x] Range[0, nmax]!^2

a(n) = 5^n * (n!)^2 * sum(k=0, n, 1 / ((-5)^k * (k!)^2)); \\ Michel Marcus, Jan 28 2021

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

a(0) = 1; a(n) = 5 * n^2 * a(n-1) + (-1)^n.

A354304 a(n) is the numerator of Sum_{k=0..n} (-1)^k / (k!)^2.

Original entry on oeis.org

1, 0, 1, 2, 43, 403, 23213, 118483, 51997111, 1842647621, 327581799289, 8918414485643, 4670006130663971, 361730891537680087, 130890931830249779173, 427294615628884602769, 6534075316966068976316143, 885163015595247156635327497, 41526561745210509140249210357

Offset: 0

Ilya Gutkovskiy, May 23 2022

			1, 0, 1/4, 2/9, 43/192, 403/1800, 23213/103680, 118483/529200, 51997111/232243200, 1842647621/8230118400, ...

Cf. A001044, A053557, A061354, A073701, A091681, A103816, A120265, A143382, A354302, A354305 (denominators).

Table[Sum[(-1)^k/(k!)^2, {k, 0, n}], {n, 0, 18}] // Numerator
nmax = 18; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator

Numerators of coefficients in expansion of BesselJ(0,2*sqrt(x)) / (1 - x).

A354305 a(n) is the denominator of Sum_{k=0..n} (-1)^k / (k!)^2.

Original entry on oeis.org

1, 1, 4, 9, 192, 1800, 103680, 529200, 232243200, 8230118400, 1463132160000, 39833773056000, 20858412072960000, 1615657835151360000, 584619573580922880000, 1908495817772544000000, 29184209113159670169600000, 3953548328298349068288000000, 185476873609942457647104000000

Offset: 0

Ilya Gutkovskiy, May 23 2022

			1, 0, 1/4, 2/9, 43/192, 403/1800, 23213/103680, 118483/529200, 51997111/232243200, 1842647621/8230118400, ...

Cf. A001044, A053556, A061355, A073701, A091681, A143383, A354303, A354304 (numerators).

Table[Sum[(-1)^k/(k!)^2, {k, 0, n}], {n, 0, 18}] // Denominator
nmax = 18; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Denominator
Accumulate[Table[(-1)^k/(k!)^2,{k,0,20}]]//Denominator (* Harvey P. Dale, Apr 25 2023 *)

Denominators of coefficients in expansion of BesselJ(0,2*sqrt(x)) / (1 - x).

A074702 a(n) = ((n+1)^2(n-1)a(n-1)+(-1)^(n+1))/n.

Original entry on oeis.org

1, 4, 43, 806, 23213, 947864, 51997111, 3685295242, 327581799289, 35673657942572, 4670006130663971, 723461783075360174, 130890931830249779173, 27346855400248614577216, 6534075316966068976316143, 1770326031190494313270654994, 539845302687736618823239734641

Offset: 1

Vladeta Jovovic, Sep 03 2002

Harvey P. Dale, Table of n, a(n) for n = 1..252

Cf. A073701.

nxt[{n_,a_}]:={n+1,(n*a*(n+2)^2+(-1)^(n+2))/(n+1)}; Transpose[ NestList[ nxt,{1,1},15]][[2]] (* Harvey P. Dale, Jun 26 2013 *)

a(n) = round((n+1)!^2*BesselJ(0, 2))/n.

More terms from Emeric Deutsch, Dec 18 2003

More terms from Harvey P. Dale, Jun 26 2013

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

Vladeta Jovovic, Sep 03 2002

Harvey P. Dale, Table of n, a(n) for n = 1..253

Cf. A006040, A073701.

nxt[{n_,a_}]:={n+1,(n+1)^2 a+1}; Transpose[NestList[nxt,{1,0},20]][[2]] (* Harvey P. Dale, Dec 11 2013 *)

a(n)=round((besseli(0,2)-2)*n!^2) \\ Charles R Greathouse IV, Feb 19 2014

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

cp's OEIS Frontend

A006040 a(n) = Sum_{i=0..n} (n!/(n-i)!)^2.

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A091681 Decimal expansion of BesselJ(0,2).

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A337152 a(n) = 2^n * (n!)^2 * Sum_{k=0..n} 1 / ((-2)^k * (k!)^2).

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A337153 a(n) = 3^n * (n!)^2 * Sum_{k=0..n} 1 / ((-3)^k * (k!)^2).

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A337154 a(n) = 4^n * (n!)^2 * Sum_{k=0..n} 1 / ((-4)^k * (k!)^2).

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A337155 a(n) = 5^n * (n!)^2 * Sum_{k=0..n} 1 / ((-5)^k * (k!)^2).

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A354304 a(n) is the numerator of Sum_{k=0..n} (-1)^k / (k!)^2.

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A354305 a(n) is the denominator of Sum_{k=0..n} (-1)^k / (k!)^2.

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A074702 a(n) = ((n+1)^2(n-1)a(n-1)+(-1)^(n+1))/n.