A222391 -id:A222391 - cp's OEIS frontend

A035009 STIRLING transform of [1,1,2,4,8,16,32,...].

1, 1, 3, 11, 47, 227, 1215, 7107, 44959, 305091, 2206399, 16913987, 136823263, 1163490499, 10366252031, 96491364675, 935976996127, 9440144423875, 98800604237119, 1071092025420867, 12008090971866207, 139014305916844739, 1659578039401022079, 20405708646650507075

Offset: 0

N. J. A. Sloane

Numerators of sequence that shifts left one place under 1/2 order binomial transform. (Denominators are 2^(n-1) for n > 0.) - Franklin T. Adams-Watters, Jul 31 2005
Row sums of triangle A137597 starting (1, 3, 11, 47, 227, ...). - Gary W. Adamson, Jan 29 2008
From Gary W. Adamson, Jul 22 2011: (Start)
a(n)/2^(n-1) = upper left term in M^n, M = an infinite square production matrix in which a column of (1/2, 1/2, 1/2, ...) is appended to the right of Pascal's triangle, as follows:
  1,  1/2,  0,   0,   0,   0,  ...
  1,   1,  1/2,  0,   0,   0,  ...
  1,   2,   1,  1/2,  0,   0,  ...
  1,   3,   3,   1,  1/2,  0,  ...
  1,   4,   6,   4,   1,  1/2, ..., etc.
(End)
From Bruno Berselli, Mar 20 2013: (Start)
Note that, for t=A222391:
a(1)*t = Sum_{n >= 1} 1  /(Gamma(n/2)*Gamma((n+1)/2)),
a(2)*t = Sum_{n >= 1} n  /(Gamma(n/2)*Gamma((n+1)/2)),
a(3)*t = Sum_{n >= 1} n^2/(Gamma(n/2)*Gamma((n+1)/2)),
a(4)*t = Sum_{n >= 1} n^3/(Gamma(n/2)*Gamma((n+1)/2)),
a(5)*t = Sum_{n >= 1} n^4/(Gamma(n/2)*Gamma((n+1)/2)),
a(6)*t = Sum_{n >= 1} n^5/(Gamma(n/2)*Gamma((n+1)/2)), etc.
(End)
Except for the initial term, the main diagonal of A129340. - Peter Bala, Apr 14 2017

			Given the production matrix M, upper left term of M^5 = a(5)/2^4 = 227/16.

Seiichi Manyama, Table of n, a(n) for n = 0..558
Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.
Mikhail Khovanov, Victor Ostrik and Yakov Kononov, Two-dimensional topological theories, rational functions and their tensor envelopes, arXiv:2011.14758 [math.QA], 2020.
Toufik Mansour and Mark Shattuck, A recurrence related to the Bell Numbers, Integers 11 (2011), #A67.

Cf. A000110, A011782, A137597, A222391, A129340.

A035009 := proc(n) local a,b,i;
a := [seq(2,i=1..n-1)]; b := [seq(1,i=1..n-1)];
exp(-x)*hypergeom(a,b,x); round(evalf(subs(x=2,%), 10+2*n)) end:
seq(A035009(n),n=0..19);  # Peter Luschny, Mar 30 2011
# second Maple program:
b:= proc(n, m) option remember;
     `if`(n=0, ceil(2^(m-1)), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 03 2021

1/(2*E^2)*Sum[(i + j)^n/(i!*j!), {i, 0, Infinity}, {j, 0, Infinity}] (* Starting from the 2nd term *) (* Vladimir Reshetnikov, Dec 31 2008 *)
Join[{1}, Table[BellB[n, 2]/2, {n, 1, 25}]] (* Vaclav Kotesovec, Jun 26 2022 *)

x='x+O('x^99); Vec(serlaplace((1 + exp(2*exp(x)-2))/2)) \\ Joerg Arndt, Apr 01 2011

a(n) = (1/2)*A001861(n), n > 0.
E.g.f.: (1 + exp(2*exp(x)-2))/2. - Emeric Deutsch, Feb 09 2002
a(n+1) = 1 + 2*Sum_{j=1..n} binomial(n, j)*a(j). - Jon Perry, Apr 25 2005
Define f_1(x), f_2(x), ... such that f_1(x)=e^x and for n=2,3,... f_{n+1}(x) = (d/dx)(x*f_n(x)). Then a(n) = e^(-2)*f_n(2). - Milan Janjic, May 30 2008
G.f.: 1 + x/(Q(0) - 2*x) where Q(k) =  1 - x*(k+1)/( 1 - 2*x/Q(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 22 2013
G.f.: 1/Q(0), where Q(k)= 1 - x - 2*x/(1 - x*(2*k+1)/(1 - x - 2*x/(1 - x*(2*k+2)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, May 13 2013
G.f.: 1 + Sum_{k>=1} 2^(k-1)*x^k/Product_{j=1..k} (1 - j*x). - Ilya Gutkovskiy, Jun 19 2018

A222392 Decimal expansion of Sum_{n>=1} 1/Gamma(n/2).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Mar 19 2013

			5.57316966431003975325790404977558240053838531356466534303248120641829030...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:5:12 at page 415.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A222391.

Cf. A001113, (Sum_{n>=1} 1/Gamma(n)).

RealDigits[(1 + Erf[1]) E + 1/Sqrt[Pi], 10, 90][[1]]

Equals (1+erf(1))*e+1/sqrt(Pi), where erf is the error function (see A099286).

A217249 Decimal expansion of Pi^2/sqrt(e).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Mar 20 2013

			5.9862176684954389749324507474236892501651010658993869833349491850614...

Cf. A002388, A092605, A222391.

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

Equals A002388*A092605.

cp's OEIS Frontend

A035009 STIRLING transform of [1,1,2,4,8,16,32,...].

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A222392 Decimal expansion of Sum_{n>=1} 1/Gamma(n/2).

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Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

A217249 Decimal expansion of Pi^2/sqrt(e).

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula