A070910 -id:A070910 - cp's OEIS frontend

A197036 Decimal expansion of the Modified Bessel Function I of order 0 at 1.

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

Offset: 1

R. J. Mathar, Oct 08 2011

			1.26606587775200833559824462521471753760767031135496...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 51, page 504.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical functions, Chapter 9.6.
Eric Weisstein's World of Mathematics, Modified Bessel Function of the First Kind.

Cf. A002454, A242282.

Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), this sequence (I(0,1)), A334381 (I(0,sqrt(2))), A070910 (I(0,2)).

Maple
```
BesselI(0,1) ;evalf(%) ;
```

RealDigits[BesselJ[0, I], 10, 120][[1]] (* Amiram Eldar, Jun 15 2023 *)

besseli(0,1) \\ Charles R Greathouse IV, Feb 19 2014

I_0(1) = Sum_{k>=0} 1/(4^k*k!^2) = Sum_{k>=0} 1/A002454(k).
Equals (1/Pi)*Integral_{t=0..Pi} exp(cos(t)) dt.
Equals BesselJ(0,i). - Jianing Song, Sep 18 2021
From Amiram Eldar, Jul 09 2023: (Start)
Equals exp(-1) * Sum_{k>=0} binomial(2*k,k)/(2^k*k!).
Equals e * Sum_{k>=0} (-1/2)^k * binomial(2*k,k)/k!. (End)

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

A334380 Decimal expansion of Sum_{k>=0} (-1)^k/((2*k)!!)^2.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 25 2020

This constant is transcendental.

			1/(4^0*0!^2) - 1/(4^1*1!^2) + 1/(4^2*2!^2) - 1/(4^3*3!^2) + ... = 0.765197686557966551449717526...

Cf. A000165, A002454.

Bessel function values: this sequence (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2))), A070910 (I(0,2)).

RealDigits[BesselJ[0, 1], 10, 110] [[1]]

besselj(0, 1) \\ Michel Marcus, Apr 26 2020

Equals BesselJ(0,1).

Equals BesselI(0,i), where BesselI is the modified Bessel function of order 0. - Jianing Song, Sep 18 2021

A334383 Decimal expansion of Sum_{k>=0} (-1)^k/(2^k*(k!)^2).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 25 2020

			1/(2^0*0!^2) - 1/(2^1*1!^2) + 1/(2^2*2!^2) - 1/(2^3*3!^2) + ... = 0.5591341444189799174882684679...

Cf. A055546, A092605.

Bessel function values: A334380 (J(0,1)), this sequence (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2))), A070910 (I(0,2)).

RealDigits[BesselJ[0, Sqrt[2]], 10, 110] [[1]]

besselj(0, sqrt(2)) \\ Michel Marcus, Apr 26 2020

Equals BesselJ(0,sqrt(2)).

Equals BesselI(0,sqrt(2)*i), where BesselI is the modified Bessel function of order 0. - Jianing Song, Sep 18 2021

A253909 1 together with the positive squares.

Original entry on oeis.org

1, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500

Offset: 0

Omar E. Pol, Feb 12 2015

Also, right border of A246595 arranged as an irregular triangle.

a(n) are the Engel expansion of A070910. - Benedict W. J. Irwin, Dec 15 2016

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028310, A070910, A246595. Essentially the same as A000290 and A174902.

Join[{1}, Range[50]^2] (* Alonso del Arte, Feb 23 2015 *)
Range[0, 50]^2 /. 0 -> 1 (* Robert G. Wilson v, Dec 15 2016 *)

a(n)=max(n,1)^2 \\ Charles R Greathouse IV, Dec 16 2016

a(n) = A028310(n)^2.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>=4. - David Neil McGrath, May 23 2015
G.f.: (x^3-4*x^2+2*x-1)/(x-1)^3. - David Neil McGrath, May 25 2015
E.g.f.: 1 + exp(x)*x*(1 + x). - Stefano Spezia, Jan 30 2023

Keyword:mult added by Andrew Howroyd, Aug 06 2018

A334381 Decimal expansion of Sum_{k>=0} 1/(2^k*(k!)^2).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 25 2020

			1/(2^0*0!^2) + 1/(2^1*1!^2) + 1/(2^2*2!^2) + 1/(2^3*3!^2) + ... = 1.56608292975635053729238691...

Index entries for sequences related to Bessel functions or polynomials

Cf. A019774, A055546.

Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2))), A091681 (J(0,2)), A197036 (I(0,1)), this sequence (I(0,sqrt(2))), A070910 (I(0,2)).

RealDigits[BesselI[0, Sqrt[2]], 10, 110] [[1]]

suminf(k=0, 1/(2^k*(k!)^2)) \\ Michel Marcus, Apr 26 2020

besseli(0, sqrt(2)) \\ Michel Marcus, Apr 26 2020

Equals BesselI(0,sqrt(2)).

Equals BesselJ(0,sqrt(2)*i). - Jianing Song, Sep 18 2021

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)

A249590 E.g.f.: BesselI(0,2)*P(x) - Q(x), where P(x) = 1/Product_{n>=1} (1 - x^n/n^2) and Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k^2 - x^k).

Original entry on oeis.org

1, 1, 6, 63, 1162, 31263, 1207344, 61719326, 4103067834, 341454828363, 34946904263560, 4304483416099530, 629558493157805370, 107728435291299602135, 21346960361800584031800, 4847223770735591212039818, 1250978551922243595690043914, 364052135715732457875255719691

Offset: 0

Paul D. Hanna, Nov 01 2014

nonn

Here BesselI(0,2) = Sum_{n>=0} 1/n!^2 = 2.2795853023360672... (A070910).

			E.g.f.: 1 + x + 6*x^2/2!^2 + 63*x^3/3!^2 + 1162*x^4/4!^2 + 31263*x^5/5!^2 +...
such that A(x) = BesselI(0,2)*P(x) - Q(x), where
P(x) = 1/Product_{n>=1} (1 - x^n/n^2) = Sum_{n>=0} A249588(n)*x^n/n!^2, and
Q(x) = Sum_{n>=1} 1/Product_{k=1..n} (k^2 - x^k).
More explicitly,
P(x) = 1/((1-x)*(1-x^2/4)*(1-x^3/9)*(1-x^4/16)*(1-x^5/25)*...);
Q(x) = 1/(1-x) + 1/((1-x)*(4-x^2)) + 1/((1-x)*(4-x^2)*(9-x^3)) + 1/((1-x)*(4-x^2)*(9-x^3)*(16-x^4)) + 1/((1-x)*(4-x^2)*(9-x^3)*(16-x^4)*(25-x^5)) +...
We can illustrate the initial terms a(n) in the following manner.
The coefficients q(n) in Q(x) = Sum_{n>=0} q(n)*x^n/n!^2 begin:
q(0) = 1.279585302336067267437204440811533...
q(1) = 1.279585302336067267437204440811533...
q(2) = 5.397926511680336337186022204057666...
q(3) = 48.69967981446729610442301759976513...
q(4) = 789.3250187996735809262470013346725...
q(5) = 19745.00072507184117617488656759887...
q(6) = 713288.6822890207712374724807435860...
q(7) = 34956701.28771539805703277298850790...
q(8) = 2239176303.370447012433955813571405...
q(9) = 181385849371.3820539848573249577420...
and the coefficients in P(x) = 1/Product_{n>=1} (1 - x^n/n^2) begin:
A249078 = [1, 1, 5, 49, 856, 22376, 842536, 42409480, 2782192064, ...];
from which we can generate this sequence like so:
a(0) = BesselI(0,2)*1 - q(0) = 1;
a(1) = BesselI(0,2)*1 - q(1) = 1;
a(2) = BesselI(0,2)*5 - q(2) = 6;
a(3) = BesselI(0,2)*49 - q(3) = 63;
a(4) = BesselI(0,2)*856 - q(4) = 1162;
a(5) = BesselI(0,2)*22376 - q(5) = 31263;
a(6) = BesselI(0,2)*842536 - q(6) = 1207344;
a(7) = BesselI(0,2)*42409480 - q(7) = 61719326;
a(8) = BesselI(0,2)*2782192064 - q(8) = 4103067834; ...

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A249607, A249592, A249588, A249078, A070910.

\p100 \\ set precision
{P=Vec(serlaplace(serlaplace(prod(k=1, 31, 1/(1-x^k/k^2 +O(x^31)))))); } \\ A249588
{Q=Vec(serlaplace(serlaplace(sum(n=1, 201, prod(k=1, n, 1./(k^2-x^k +O(x^31))))))); }
for(n=0, 30, print1(round(besseli(0,2)*P[n+1]-Q[n+1]), ", "))

A271574 Decimal expansion of Sum_{n>=0} 1/(n!)^3.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Apr 10 2016

			2.1297025489833064181345236105954134683192207470391693...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000

Cf. A001113, A070910.

evalf(Sum(1/n!^3, n=0..infinity), 120); # Vaclav Kotesovec, Apr 10 2016

RealDigits[HypergeometricPFQ[{}, {1, 1}, 1], 10, 120][[1]]
RealDigits[Total[1/(Range[0,200]!)^3],10,120][[1]] (* Harvey P. Dale, Mar 06 2024 *)

default(realprecision, 120); sumpos(n=0, 1/n!^3)

A070913 Continued fraction expansion for BesselI(0,2).

Original entry on oeis.org

2, 3, 1, 1, 2, 1, 3, 7, 4, 3, 1, 2, 2, 1, 2, 1, 1, 2, 7, 8, 1, 1, 21, 1, 16, 2, 1, 8, 1, 1, 8, 1, 35, 1, 2, 1, 1, 4, 1, 1, 1, 3, 132, 3, 1, 10, 2, 1, 1, 1, 1, 2, 2, 6, 100, 1, 1, 26, 1, 66, 1, 2, 16, 1, 4, 52, 2, 1, 1, 1, 16, 8, 1, 3, 172, 1, 3, 1, 3, 3, 1, 13, 1, 5, 2, 1, 4, 3, 1, 3, 4, 7, 1, 1, 23, 1

Offset: 0

Benoit Cloitre, May 20 2002

The continued fraction terms satisfy the Gauss-Kuzmin distribution. - A.H.M. Smeets, Aug 21 2018

A.H.M. Smeets, Table of n, a(n) for n = 0..20000

Cf. A070910 (decimal expansion). - A.H.M. Smeets, Aug 21 2018

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

BesselI(0, 2) = sum(k=>0, 1/k!^2) = 2.27958530233...

cp's OEIS Frontend

A197036 Decimal expansion of the Modified Bessel Function I of order 0 at 1.

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

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A334380 Decimal expansion of Sum_{k>=0} (-1)^k/((2*k)!!)^2.

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A334383 Decimal expansion of Sum_{k>=0} (-1)^k/(2^k*(k!)^2).

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A253909 1 together with the positive squares.

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A334381 Decimal expansion of Sum_{k>=0} 1/(2^k*(k!)^2).

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

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