A242282 -id:A242282 - 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)

A064618 Stirling transform of (n!)^2.

Original entry on oeis.org

1, 1, 5, 49, 821, 21121, 775205, 38516689, 2490976661, 203419086241, 20474978755205, 2490729330118129, 360263844701062901, 61114158974786823361, 12017074366801186956005, 2711409826920884006692369, 695820350706240448128979541, 201526362605605903609254528481

Offset: 0

Karol A. Penson, Sep 26 2001

From Thomas Wieder, Oct 21 2004: (Start)
"Also the number of hierarchies with labeled elements and labeled levels where the levels are permuted. Let l_x denote level x, e.g. l_2 is level 2. Let 1 denote an element and 2 a second element and so on. Then l_1:123 means elements 1,2 and 3 are on level 1.
"Let | indicate separation between levels. Then l_1:1|l_2:346|l_3:5 denotes a hierarchy of n=6 unlabeled elements with element 1 on level 1, elements 3,4 and 6 on level 2 and element 5 on level 3.
"E.g. for n=3 one has a(3) = 49 possible hierarchies:
"l_1:123,
"l_1:12|l_2:3, l_1:13|l_2:2, l_1:23|l_2:1,
"l_2:12|l_1:3, l_2:13|l_1:2, l_2:23|l_1:1,
"l_1:1|l_2:23, l_1:2|l_2:13, l_1:3|l_2:12,
"l_2:1|l_1:23, l_2:2|l_1:13, l_2:3|l_1:12,
"l_1:1|l_2:2|l_3:3 and further five permutations of the elements with levels fixed,
"l_3:1|l_1:2|l_2:3 and further five permutations of the elements with levels fixed,. etc., up to
"l_3:1|l_2:2|l_1:3 and further five permutations of the elements with levels fixed. this gives 1 + 6 +6 + 6*6 = 49 = a(3) possible hierarchies.
"See A001339 for the number of hierarchies with unlabeled elements and labeled levels."
(End)
Conjecture: for fixed k = 1,2,..., the sequence a(n) (mod k) is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010. For example, modulo 10 the sequence becomes (1, 1, 5, 9, 1, 1, 5, 9, ...), with an apparent period 1, 1, 5, 9 of length 4 = phi(10) beginning at a(0). - Peter Bala, Jan 15 2018

Alois P. Heinz, Table of n, a(n) for n = 0..253

Cf. A001044.
Cf. A001339.
Cf. A242282, A242283, A019774, A320502.

a:= n-> add(Stirling2(n, k)*(k!^2), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 21 2012

Table[Sum[(k!)^2*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 10 2014 *)

/* By Vladeta Jovovic's formula: */
{a(n) = my(X=x+x*O(x^n)); n!*polcoeff( sum(m=0,n, m!*(exp(X)-1)^m), n)} /* Paul D. Hanna, Feb 15 2012 */

a(n) = Sum_{k=0..n} Stirling2(n, k)*(k!)^2.
E.g.f: hypergeom([1, 1], [], exp(x)-1). - Vladeta Jovovic, Sep 14 2003
O.g.f.: Sum_{n>=0} n!^2 * Product_{k=1..n} x/(1 - k*x). - Paul D. Hanna, Nov 25 2012
a(n) ~ exp(1/2) * (n!)^2. - Vaclav Kotesovec, May 10 2014

A242283 a(n) = Sum_{k=0..n} (k!)^6 * StirlingS2(n,k)^3.

Original entry on oeis.org

1, 1, 65, 48385, 201202625, 3177816192001, 149444281172914625, 17688550295661103160065, 4659004670032668841494537665, 2485460204094055083075883434816001, 2493268982658347340546535733064008565185, 4428569787044987118931586341533071670315481345

Offset: 0

Vaclav Kotesovec, May 10 2014

Generally, for p>=1 is Sum_{k=0..n} (k!)^(2*p) * StirlingS2(n,k)^p asymptotic to c * (n!)^(2*p), where c = 1 + Sum_{n>=1} 1/(Product_{k=1..n} (2*k)^p).

Cf. A064618 (p=1), A242282 (p=2).

a:= n-> add(k!^6*Stirling2(n,k)^3, k=0..n):
seq(a(n), n=0..15);  # Alois P. Heinz, Oct 23 2023

Table[Sum[(k!)^6 * StirlingS2[n,k]^3,{k,0,n}],{n,0,20}]

a(n) ~ c * (n!)^6, where c = 1.1269621849236767... = 1 + Sum_{n>=1} 1/(Product_{k=1..n} (2*k)^3) = HypergeometricPFQ[{}, {1, 1}, 1/8].

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

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A064618 Stirling transform of (n!)^2.

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A242283 a(n) = Sum_{k=0..n} (k!)^6 * StirlingS2(n,k)^3.

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