A115369 -id:A115369 - cp's OEIS frontend

A001227 Number of odd divisors of n.

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 4, 2, 1, 4, 2, 4, 3, 2, 2, 4, 2, 2, 4, 2, 2, 6, 2, 2, 2, 3, 3, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 6, 1, 4, 4, 2, 2, 4, 4, 2, 3, 2, 2, 6, 2, 4, 4, 2, 2, 5, 2, 2, 4, 4, 2, 4, 2, 2, 6, 4, 2, 4, 2, 4, 2, 2, 3, 6, 3, 2, 4, 2, 2, 8

Offset: 1

N. J. A. Sloane

Also (1) number of ways to write n as difference of two triangular numbers (A000217), see A136107; (2) number of ways to arrange n identical objects in a trapezoid. - Tom Verhoeff
Also number of partitions of n into consecutive positive integers including the trivial partition of length 1 (e.g., 9 = 2+3+4 or 4+5 or 9 so a(9)=3). (Useful for cribbage players.) See A069283. - Henry Bottomley, Apr 13 2000
This has been described as Sylvester's theorem, but to reduce ambiguity I suggest calling it Sylvester's enumeration. - Gus Wiseman, Oct 04 2022
a(n) is also the number of factors in the factorization of the Chebyshev polynomial of the first kind T_n(x). - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003
Number of factors in the factorization of the polynomial x^n+1 over the integers. See also A000005. - T. D. Noe, Apr 16 2003
a(n) = 1 if and only if n is a power of 2 (see A000079). - Lekraj Beedassy, Apr 12 2005
Number of occurrences of n in A049777. - Philippe Deléham, Jun 19 2005
For n odd, n is prime if and only if a(n) = 2. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 10 2005
Also number of partitions of n such that if k is the largest part, then each of the parts 1,2,...,k-1 occurs exactly once. Example: a(9)=3 because we have [3,3,2,1],[2,2,2,2,1] and [1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 07 2006
Also the number of factors of the n-th Lucas polynomial. - T. D. Noe, Mar 09 2006
Lengths of rows of triangle A182469;
Denoted by Delta_0(n) in Glaisher 1907. - Michael Somos, May 17 2013
Also the number of partitions p of n into distinct parts such that max(p) - min(p) < length(p). - Clark Kimberling, Apr 18 2014
Row sums of triangle A247795. - Reinhard Zumkeller, Sep 28 2014
Row sums of triangle A237048. - Omar E. Pol, Oct 24 2014
A069288(n) <= a(n). - Reinhard Zumkeller, Apr 05 2015
A000203, A000593 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016
a(n) is equal to the number of ways to write 2*n-1 as (4*x + 2)*y + 4*x + 1 where x and y are nonnegative integers. Also a(n) is equal to the number of distinct values of k such that k/(2*n-1) + k divides (k/(2*n-1))^(k/(2*n-1)) + k, (k/(2*n-1))^k + k/(2*n-1) and k^(k/(2*n-1)) + k/(2*n-1). - Juri-Stepan Gerasimov, May 23 2016, Jul 15 2016
Also the number of odd divisors of n*2^m for m >= 0. - Juri-Stepan Gerasimov, Jul 15 2016
a(n) is odd if and only if n is a square or twice a square. - Juri-Stepan Gerasimov, Jul 17 2016
a(n) is also the number of subparts in the symmetric representation of sigma(n). For more information see A279387 and A237593. - Omar E. Pol, Nov 05 2016
a(n) is also the number of partitions of n into an odd number of equal parts. - Omar E. Pol, May 14 2017 [This follows from the g.f. Sum_{k >= 1} x^k/(1-x^(2*k)). - N. J. A. Sloane, Dec 03 2020]

			G.f. = q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + 2*q^10 + ...
From _Omar E. Pol_, Nov 30 2020: (Start)
For n = 9 there are three odd divisors of 9; they are [1, 3, 9]. On the other hand there are three partitions of 9 into consecutive parts: they are [9], [5, 4] and [4, 3, 2], so a(9) = 3.
Illustration of initial terms:
                              Diagram
   n   a(n)                         _
   1     1                        _|1|
   2     1                      _|1 _|
   3     2                    _|1  |1|
   4     1                  _|1   _| |
   5     2                _|1    |1 _|
   6     2              _|1     _| |1|
   7     2            _|1      |1  | |
   8     1          _|1       _|  _| |
   9     3        _|1        |1  |1 _|
  10     2      _|1         _|   | |1|
  11     2    _|1          |1   _| | |
  12     2   |1            |   |1  | |
...
a(n) is the number of horizontal line segments in the n-th level of the diagram. For more information see A286001. (End)

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 487 Entry 47.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 306.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4).
Ronald. L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), see exercise 2.30 on p. 65.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 28.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
K. S. Brown's Mathpages, Partitions into Consecutive Integers.
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 18.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
A. Heiligenbrunner, Sum of adjacent numbers (in German).
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 11.
Gerzson Kéri, The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas, Annales Univ. Sci. Budapest (Hungary, 2022) Sect. Comp., Vol. 53, 93-108.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function tau_o(n).
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976.
N. J. A. Sloane, Transforms.
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article 99.1.6.
Eric Weisstein's World of Mathematics, Binomial Number and Odd Divisor Function.
Eric Weisstein's World of Mathematics, q-Polygamma Function.
Index entries for "core" sequences.
Index entries for sequences mentioned by Glaisher.

Cf. A000005, A000079, A000593, A010054 (char. func.), A038547 (positions of first appearances), A050999, A051000, A051001, A051002, A051731, A054844, A069283, A069288, A109814, A115369, A118235, A118236, A125911, A136655, A183063, A183064, A237593, A247795, A272887, A273401, A279387, A286001.
Cf. A000203, A001620, A002162, A053866, A060831.
If this sequence counts gapless sets by sum (by Sylvester's enumeration), these sets are ranked by A073485 and A356956. See also A055932, A066311, A073491, A107428, A137921, A333217, A356224, A356841, A356845.
Dirichlet inverse is A327276.

a001227 = sum . a247795_row
-- Reinhard Zumkeller, Sep 28 2014, May 01 2012, Jul 25 2011

[NumberOfDivisors(n)/Valuation(2*n, 2): n in [1..100]]; // Vincenzo Librandi, Jun 02 2019

for n from 1 by 1 to 100 do s := 0: for d from 1 by 2 to n do if n mod d = 0 then s := s+1: fi: od: print(s); od:
A001227 := proc(n) local a,d;
    a := 1 ;
    for d in ifactors(n)[2] do
        if op(1,d) > 2 then
            a := a*(op(2,d)+1) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jun 18 2015

f[n_] := Block[{d = Divisors[n]}, Count[ OddQ[d], True]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Aug 27 2004 *)
Table[Total[Mod[Divisors[n], 2]],{n,105}] (* Zak Seidov, Apr 16 2010 *)
f[n_] := Block[{d = DivisorSigma[0, n]}, If[ OddQ@ n, d, d - DivisorSigma[0, n/2]]]; Array[f, 105] (* Robert G. Wilson v *)
a[ n_] := Sum[  Mod[ d, 2], { d, Divisors[ n]}]; (* Michael Somos, May 17 2013 *)
a[ n_] := DivisorSum[ n, Mod[ #, 2] &]; (* Michael Somos, May 17 2013 *)
Count[Divisors[#],?OddQ]&/@Range[110] (* _Harvey P. Dale, Feb 15 2015 *)
(* using a262045 from A262045 to compute a(n) = number of subparts in the symmetric representation of sigma(n) *)
(* cl = current level, cs = current subparts count *)
a001227[n_] := Module[{cs=0, cl=0, i, wL, k}, wL=a262045[n]; k=Length[wL]; For[i=1, i<=k, i++, If[wL[[i]]>cl, cs++; cl++]; If[wL[[i]]Hartmut F. W. Hoft, Dec 16 2016 *)
a[n_] := DivisorSigma[0, n / 2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Jun 12 2022 *)

{a(n) = sumdiv(n, d, d%2)}; /* Michael Somos, Oct 06 2007 */

{a(n) = direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( 4, p) * X))[n]}; /* Michael Somos, Oct 06 2007 */

a(n)=numdiv(n>>valuation(n,2)) \\ Charles R Greathouse IV, Mar 16 2011

a(n)=sum(k=1,round(solve(x=1,n,x*(x+1)/2-n)),(k^2-k+2*n)%(2*k)==0) \\ Charles R Greathouse IV, May 31 2013

a(n)=sumdivmult(n,d,d%2) \\ Charles R Greathouse IV, Aug 29 2013

from functools import reduce
from operator import mul
from sympy import factorint
def A001227(n): return reduce(mul,(q+1 for p, q in factorint(n).items() if p > 2),1) # Chai Wah Wu, Mar 08 2021

def A001227(n): return len([1 for d in divisors(n) if is_odd(d)])
[A001227(n) for n in (1..80)]  # Peter Luschny, Feb 01 2012

Dirichlet g.f.: zeta(s)^2*(1-1/2^s).
Comment from N. J. A. Sloane, Dec 02 2020: (Start)
By counting the odd divisors f n in different ways, we get three different ways of writing the ordinary generating function.  It is:
A(x) =  x + x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + x^8 + 3*x^9 + 2*x^10 + ...
  = Sum_{k >= 1} x^(2*k-1)/(1-x^(2*k-1))
  = Sum_{k >= 1} x^k/(1-x^(2*k))
  = Sum_{k >= 1} x^(k*(k+1)/2)/(1-x^k)  [Ramanujan, 2nd notebook, p. 355.].
(This incorporates comments from Vladeta Jovovic, Oct 16 2002 and  Michael Somos, Oct 30 2005.) (End)
G.f.: x/(1-x) + Sum_{n>=1} x^(3*n)/(1-x^(2*n)), also L(x)-L(x^2) where L(x) = Sum_{n>=1} x^n/(1-x^n). - Joerg Arndt, Nov 06 2010
a(n) = A000005(n)/(A007814(n)+1) = A000005(n)/A001511(n).
Multiplicative with a(p^e) = 1 if p = 2; e+1 if p > 2. - David W. Wilson, Aug 01 2001
a(n) = A000005(A000265(n)). - Lekraj Beedassy, Jan 07 2005
Moebius transform is period 2 sequence [1, 0, ...] = A000035, which means a(n) is the Dirichlet convolution of A000035 and A057427.
a(n) = A113414(2*n). - N. J. A. Sloane, Jan 24 2006 (corrected Nov 10 2007)
a(n) = A001826(n) + A001842(n). - Reinhard Zumkeller, Apr 18 2006
Sequence = M*V = A115369 * A000005, where M = an infinite lower triangular matrix and V = A000005, d(n); as a vector: [1, 2, 2, 3, 2, 4, ...]. - Gary W. Adamson, Apr 15 2007
Equals A051731 * [1,0,1,0,1,...]; where A051731 is the inverse Mobius transform. - Gary W. Adamson, Nov 06 2007
a(n) = A000005(n) - A183063(n).
a(n) = d(n) if n is odd, or d(n) - d(n/2) if n is even, where d(n) is the number of divisors of n (A000005). (See the Weisstein page.) - Gary W. Adamson, Mar 15 2011
Dirichlet convolution of A000005 and A154955 (interpreted as a flat sequence). - R. J. Mathar, Jun 28 2011
a(A000079(n)) = 1; a(A057716(n)) > 1; a(A093641(n)) <= 2; a(A038550(n)) = 2; a(A105441(n)) > 2; a(A072502(n)) = 3. - Reinhard Zumkeller, May 01 2012
a(n) = 1 + A069283(n). - R. J. Mathar, Jun 18 2015
a(A002110(n)/2) = n, n >= 1. - Altug Alkan, Sep 29 2015
a(n*2^m) = a(n*2^i), a((2*j+1)^n) = n+1 for m >= 0, i >= 0 and j >= 0. a((2*x+1)^n) = a((2*y+1)^n) for positive x and y. - Juri-Stepan Gerasimov, Jul 17 2016
Conjectures: a(n) = A067742(n) + 2*A131576(n) = A082647(n) + A131576(n). - Omar E. Pol, Feb 15 2017
a(n) = A000005(2n) - A000005(n) = A099777(n)-A000005(n). - Danny Rorabaugh, Oct 03 2017
L.g.f.: -log(Product_{k>=1} (1 - x^(2*k-1))^(1/(2*k-1))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
G.f.: (psi_{q^2}(1/2) + log(1-q^2))/log(q), where psi_q(z) is the q-digamma function. - Michael Somos, Jun 01 2019
a(n) = A003056(n) - A238005(n). - Omar E. Pol, Sep 12 2021
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma + log(2)/2 - 1/2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000005(k) = log(2) (A002162). - Amiram Eldar, Mar 01 2023
a(n) = Sum_{i=1..n} (-1)^(i+1)*A135539(n,i). - Ridouane Oudra, Apr 13 2023

A115368 Decimal expansion of first zero of the Bessel function J_0(z).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jan 21 2006

"This [constant] arises from the study of a vibrating, homogeneous membrane that is uniformly stretched across the unit disk. [Its square] is the principal frequency of the sound one hears when a kettledrum is struck." - Quoted from the book by Steven R. Finch.

Siegel proves (the Main Theorem) that J_0(z) is transcendental if z is algebraic and nonzero, but since in our case J_0(z) = 0 is not transcendental it follows that z cannot be algebraic. - Charles R Greathouse IV, Oct 20 2020

			2.4048255576957727686...

Chi Keung Cheung et al., Getting Started with Mathematica, 2nd Ed. New York: J. Wiley (2005) p. 7.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 221.
C. Siegel, Über einige Anwendungen Diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1929/30, No. 1. Translated as "On some applications
of Diophantine approximations" by Clemens Fuchs.

Eric Weisstein's World of Mathematics, Bessel Function Zeros
Index entries for transcendental numbers

Cf. A115369, A115370, A115371, A115372, A115373.

Cf. A000275, A002190, A188489, A181167, A181168.

RealDigits[BesselJZero[0, 1], 10, 120][[1]] (* Alonso del Arte, May 06 2011 *)

solve(x=2,3,besselj(0,x)) \\ Charles R Greathouse IV, Feb 19 2014

besseljzero(0) \\ Charles R Greathouse IV, Aug 06 2022

A180874 Lassalle's sequence connected with Catalan numbers and Narayana polynomials.

Original entry on oeis.org

1, 1, 5, 56, 1092, 32670, 1387815, 79389310, 5882844968, 548129834616, 62720089624920, 8646340208462880, 1413380381699497200, 270316008395632253340, 59800308109377016336155, 15151722444639718679892150, 4359147487054262623576455600

Offset: 1

Jonathan Vos Post, Sep 22 2010

Defined by the recurrence formula in Theorem 1, page 2 of Lasalle.
From Tom Copeland, Jan 26 2016: (Start)
Let G(t) = Sum_{n>=0} t^(2n)/(n!(n+1)!) = exp(c.t) be the e.g.f. of the aerated Catalan numbers c_n of A126120.
R = x + H(D) = x + d/dD log[G(D)] = x + D - D^3/3! + 5 D^5/5! - 56 D^7/7! + ... = x + e^(r. D) generates a signed, aerated version of this entry's sequence a(n), (r.)^(2n+1) = r(2n+1) = (-1)^n a(n+1) for n>=0 and r(0) = a(0) = 0, and is, with D = d/dx, the raising operator for the Appell polynomials P(n,x) of A097610, where P(n,x) = (c. + x)^n = Sum{k=0 to n} binomial(n,k) c_k x^(n-k) with c_k = A126120(k), i.e., R P(n,x) = P(n+1,x).
d/dt log[G(t)] = e^(r.t) = e^(q.t) / e^(c.t) = Ev[c. e^(c.t)] / Ev[e^(c.t)] = e^(q.t) e^(d.t) = [Sum_{n>=0} 2n t^(2n-1)/(n!(n+1)!)] / [Sum_{n>=0} t^(2n)/(n!(n+1)!)] with Ev[..] denoting umbral evaluation, so q(n) = c(n+1) = A126120(n+1) and d(2n) = (-1)^n A238390(n) and vanishes otherwise. Then (r. + c.)^n = q(n) = Sum_{k=0..n} binomial(n,k) r(k) c(n-k) and (q. + d.)^n = r(n), relating A180874, A126120 (A000108), and A238390 through binomial convolutions.
The sequence can also be represented in terms of the Faber polynomials of A263916 as a(n) = |(2n-1)! F(2n,0,b(2),0,b(4),0,..)| = |h(2n)| where b(2n) = 1/(n!(n + 1)!) = A126120(2n)/(2n)! = A000108(n)/(2n)!, giving h(0) = 1, h(1) = 0, h(2) = 1, h(3) = 0, h(4) = -1, h(5) = 0, h(6) = 5, h(7) = 0, h(8) = -56, ..., implying, among other relations, that A000108(n/2)= A126120(n) = Bell(n,0,h(2),0,h(4),...), the Bell polynomials of A036040 which reduce to A257490 in this case.
(End)
From Colin Defant, Sep 06 2018: (Start)
a(n) is the number of pairs (rho,r), where rho is a matching on [2n] and r is an acyclic orientation of the crossing graph of rho in which the block containing 1 is the only source (see the Josuat-Verges paper or the Defant-Engen-Miller paper for definitions).
a(n) is the number of permutations of [2n-1] that have exactly 1 preimage under West's stack-sorting map.
a(n) is the number of valid hook configurations of permutations of [2n-1] that have n-1 hooks (see the paper by Defant, Engen, and Miller for definitions).
Say a binary tree is full if every vertex has either 0 or 2 children. If u is a left child in such a tree, then we can start at the sibling of u and travel down left edges until reaching a leaf v. Call v the leftmost nephew of u. A decreasing binary plane tree on [m] is a binary plane tree labeled with the elements of [m] in which every nonroot vertex has a label that is smaller than the label of its parent. a(n) is the number of full decreasing binary plane trees on [2n-1] in which every left child has a label that is larger than the label of its leftmost nephew.
(End)

G. C. Greubel, Table of n, a(n) for n = 1..250
Colin Defant, Descents in t-Sorted Permutations, arXiv:1904.02613 [math.CO], 2019.
Colin Defant, Michael Engen, and Jordan A. Miller, Stack-sorting, set partitions, and Lassalle's sequence, arXiv:1809.01340 [math.CO], 2018.
Colin Defant, Catalan Intervals and Uniquely Sorted Permutations, arXiv:1904.02627 [math.CO], 2019.
Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020. See p. 37.
Matthieu Josuat-Verges, Cumulants of the q-semicircular law, Tutte polynomials, and heaps, arXiv:1203.3157 [math.CO], 2012.
Michel Lassalle, Catalan numbers and a new integer sequence, arXiv:1009.4225 [math.CO], 2010-2012.
Michel Lassalle, Two integer sequences related to Catalan numbers, Journal of Combinatorial Theory, Series A, Volume 119, Issue 4, May 2012, Pages 923-935.
Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.

Cf. A000108, A001263, A188664, A115369.

Cf. A036040, A097610, A126120, A238390, A263916, A257490.

A000108 := proc(n) binomial(2*n,n)/(1+n) ; end proc:
A180874 := proc(n) option remember; if n = 1 then 1; else A000108(n)+add((-1)^j*binomial(2*n-1,2*j-1)*procname(j)*A000108(n-j),j=1..n-1) ;   %*(-1)^(n-1) ; end if; end proc: # R. J. Mathar, Apr 16 2011

nmax=20; a = ConstantArray[0,nmax]; a[[1]]=1; Do[a[[n]] = (-1)^(n-1)*(Binomial[2*n,n]/(n+1) + Sum[(-1)^j*Binomial[2n-1,2j-1]*a[[j]]* Binomial[2*(n-j),n-j]/(n-j+1),{j,1,n-1}]),{n,2,nmax}]; a (* Vaclav Kotesovec, Feb 28 2014 *)

a(n) = (-1)^(n-1) * (C(n)+Sum_{j=1..n-1} (-1)^j *binomial(2n-1,2j-1) * a(j) *C(n-j)), where C() = A000108(). - R. J. Mathar, Apr 17 2011, corrected by Vaclav Kotesovec, Feb 28 2014
E.g.f.: Sum_{k>=0} a(k)*x^(2*k+2)/(2*k+2)! = log(x/BesselJ(1,2*x)). - Sergei N. Gladkovskii, Dec 28 2011
a(n) ~ (n!)^2 / (sqrt(Pi) * n^(3/2) * r^n), where r = BesselJZero[1, 1]^2/16 = 0.917623165132743328576236110539381686855099186384686... - Vaclav Kotesovec, added Feb 28 2014, updated Mar 01 2014
Define E(m,n) by E(1,1) = 1, E(n,n) = 0 for n > 1, and E(m,n) = Sum_{j=1..m} Sum_{i=1..n-m-1} binomial(n-m-1,i-1) * F_j(i+j-1) * F_{m-j}(n-j-i) for 0 <= m < n, where F_m(n) = Sum_{j=m..n} E_j(n). Then a(n) = F_0(2n-1). - Colin Defant, Sep 06 2018

A103365 First column of triangle A103364, which equals the matrix inverse of the Narayana triangle (A001263).

Original entry on oeis.org

1, -1, 2, -7, 39, -321, 3681, -56197, 1102571, -27036487, 810263398, -29139230033, 1238451463261, -61408179368043, 3513348386222286, -229724924077987509, 17023649385410772579, -1419220037471837658603, 132236541042728184852942, -13690229149108218523467549

Offset: 1

Paul D. Hanna, Feb 02 2005

sign

			From _Paul D. Hanna_, Jan 31 2009: (Start)
G.f.: A(x) = 1 - x + 2*x^2/3 - 7*x^3/18 + 39*x^4/180 - 321*x^5/2700 +...
G.f.: A(x) = 1/B(x) where:
B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 + x^5/2700 +...+ x^n/[n!*(n+1)!/2^n] +... (End)

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A001263, A103364, A103366, A103367.
Cf. A155926, A155927. [Paul D. Hanna, Jan 31 2009]
Cf. A238390, A180874, A115369.

Table[(-1)^((n-1)/2) * (CoefficientList[Series[x/BesselJ[1,2*x],{x,0,40}],x])[[n]] * ((n+1)/2)! * ((n-1)/2)!,{n,1,41,2}] (* Vaclav Kotesovec, Mar 01 2014 *)

a(n)=if(n<1,0,(matrix(n,n,m,j,binomial(m-1,j-1)*binomial(m,j-1)/j)^-1)[n,1])

{a(n)=local(B=sum(k=0,n,x^k/(k!*(k+1)!/2^k))+x*O(x^n));polcoeff(1/B,n)*n!*(n+1)!/2^n} \\ Paul D. Hanna, Jan 31 2009

From Paul D. Hanna, Jan 31 2009: (Start)
G.f.: A(x) = 1/B(x) where A(x) = Sum_{n>=0} (-1)^n*a(n)*x^n/[n!*(n+1)!/2^n] and B(x) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n].
G.f. satisfies: A(x) = 1/F(x*A(x)) and F(x) = 1/A(x*F(x)) where F(x) = Sum_{n>=0} A155926(n)*x^n/[n!*(n+1)!/2^n].
G.f. satisfies: A(x) = 1/G(x/A(x)) and G(x) = 1/A(x/G(x)) where G(x) = Sum_{n>=0} A155927(n)*x^n/[n!*(n+1)!/2^n]. (End)
a(n) ~ (-1)^(n+1) * c * n! * (n-1)! * d^n, where d = 4/BesselJZero[1, 1]^2 = 0.2724429913055159309179376055957891881897555639652..., and c = 9.11336321311226744479181866135367355200240221549667284076... = BesselJZero[1, 1]^2 / (4*BesselJ[2, BesselJZero[1, 1]]). - Vaclav Kotesovec, Mar 01 2014, updated Apr 01 2018

A103364 Matrix inverse of the Narayana triangle A001263.

Original entry on oeis.org

1, -1, 1, 2, -3, 1, -7, 12, -6, 1, 39, -70, 40, -10, 1, -321, 585, -350, 100, -15, 1, 3681, -6741, 4095, -1225, 210, -21, 1, -56197, 103068, -62916, 19110, -3430, 392, -28, 1, 1102571, -2023092, 1236816, -377496, 68796, -8232, 672, -36, 1, -27036487, 49615695, -30346380, 9276120, -1698732, 206388

Offset: 1

Paul D. Hanna, Feb 02 2005

The first column is A103365. The second column is A103366. Row sums are all zeros (for n > 1). Absolute row sums form A103367.

Let E(y) = Sum_{n >= 0} y^n/(n!*(n+1)!) = (1/sqrt(y))*BesselI(1,2*sqrt(y)). Then this triangle is the generalized Riordan array (1/E(y), y) with respect to the sequence n!*(n+1)! as defined in Wang and Wang. - Peter Bala, Aug 07 2013

			Rows begin:
        1;
       -1,        1;
        2,       -3,       1;
       -7,       12,      -6,       1;
       39,      -70,      40,     -10,     1;
     -321,      585,    -350,     100,   -15,     1;
     3681,    -6741,    4095,   -1225,   210,   -21,   1;
   -56197,   103068,  -62916,   19110, -3430,   392, -28,   1;
  1102571, -2023092, 1236816, -377496, 68796, -8232, 672, -36, 1;
  ...
From _Peter Bala_, Aug 09 2013: (Start)
The real zeros of the row polynomials R(n,x) appear to converge to zeros of E(alpha*x) as n increases, where alpha = -3.67049 26605 ... ( = -(A115369/2)^2).
Polynomial | Real zeros to 5 decimal places
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
R(5,x)     | 1, 3.57754, 3.81904
R(10,x)    | 1, 3.35230, 7.07532,  9.14395
R(15,x)    | 1, 3.35231, 7.04943, 12.09668, 15.96334
R(20,x)    | 1, 3.35231, 7.04943, 12.09107, 18.47845, 24.35255
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Function   |
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
E(alpha*x) | 1, 3.35231, 7.04943, 12.09107, 18.47720, 26.20778, ...
Note: The n-th zero of E(alpha*x) may be calculated in Maple 17 using the instruction evalf( BesselJZeros(1,n)/ BesselJZeros(1,1))^2 ). (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Cf. A000108, A001263, A055133, A086646, A103365, A103366, A103367, A104033.

T[n_, 1]:= Last[Table[(-1)^(n - 1)*(CoefficientList[Series[x/BesselJ[1, 2*x], {x, 0, 1000}], x])[[k]]*(n)!*(n - 1)!, {k, 1, 2*n - 1, 2}]]
T[n_, n_] := 1; T[2, 1] := -1; T[3, 1] := 2; T[n_, k_] := T[n, k] = T[n - 1, k - 1]*n*(n - 1)/(k*(k - 1)); Table[T[n, k], {n, 1, 50}, {k, 1, n}] // Flatten (* G. C. Greubel, Jan 04 2016 *)
T[n_, n_] := 1; T[n_, 1]/;n>1 := T[n, 1] = -Sum[T[n, j], {j, 2, n}]; T[n_, k_]/;1Oliver Seipel, Jan 01 2025 *)

PARI
```
T(n,k)=if(n
				
```

From Peter Bala, Aug 07 2013: (Start)
Let E(y) = Sum_{n >= 0} y^n/(n!*(n+1)!) = (1/sqrt(y))* BesselI(1,2*sqrt(y)). Generating function: E(x*y)/E(y) = 1 + (-1 + x)*y/(1!*2!) + (2 - 3*x + x^2)*y^2/(2!*3!) + (-7 + 12*x - 6*x^2 + x^3)*y^3/(3!*4!) + .... The n-th power of this array has a generating function E(x*y)/E(y)^n. In particular, the matrix inverse A001263 has a generating function E(y)*E(x*y).
Recurrence equation for the row polynomials: R(n,x) = x^n - Sum_{k = 0..n-1} 1/(n-k+1)*binomial(n,k)*binomial(n+1,k+1) *R(k,x) with initial value R(0,x) = 1.
Let alpha denote the root of E(x) = 0 that is smallest in absolute magnitude. Numerically, alpha = -3.67049 26605 ... = -(A115369/2)^2. It appears that for arbitrary complex x we have lim_{n->oo} R(n,x)/R(n,0) = E(alpha*x). Cf. A055133, A086646 and A104033.
A stronger result than pointwise convergence may hold: the convergence may be uniform on compact subsets of the complex plane. This would explain the observation that the real zeros of the polynomials R(n,x) seem to converge to the zeros of E(alpha*x) as n increases. Some numerical examples are given below. (End)
From Werner Schulte, Jan 04 2017, corrected May 05 2025: (Start)
T(n,k) = T(n-1,k-1)*n*(n-1)/(k*(k-1)) for 1 < k <= n;
T(n,k) = T(n+1-k,1)*A001263(n,k) for 1 <= k <= n;
Sum_{k=1..n} T(n,k)*A000108(k) = 1 for n > 0. (End)

A238390 E.g.f.: x / BesselJ(1, 2*x) (even powers only).

Original entry on oeis.org

1, 1, 4, 35, 546, 13482, 485892, 24108513, 1576676530, 131451399794, 13609184032808, 1712978776719938, 257612765775847132, 45620136452519144700, 9396239458048330569840, 2227147531572856811691105, 601916577165056911293330930, 183994483721828524163677628370

Offset: 0

Vaclav Kotesovec, Mar 01 2014

nonn

Aerated, the e.g.f. is e^(a.t) =  1/AC(i*t) = 1/[I_1(2i*t)/(it)] = 1/Sum_{n>=0} (-1)^n t^(2n) / [n!(n+1)!] = a_0 + a_2 t^2/2! + a_4 t^4/4! + ... = 1 + t^2/2! + 4 t^4/4! + 35 t^6/6! + ..., where AC(t) is the e.g.f. for the aerated Catalan numbers c_n of A126120 and I_n(t) are the modified Bessel functions of the first kind (i = sqrt(-1)). The signed, aerated sequence b_n = (i)^n a_n has the e.g.f. e^(b.t) = 1/AC(t) and, therefore, (i*a. + c.)^n = Sum_{k=0..n} binomial(n,k) i^k a_k c_(n-k) vanishes except for n=0 for which it's unity. - Tom Copeland, Jan 23 2016
With q(n) = A126120(n+1) and q(0) = 0, d(2n) =  (-1)^n A238390(n) and zero for odd arguments, and r(2n+1) = (-1)^n A180874(n+1) and zero for even arguments, then r(n) = (q. + d.)^n = Sum_{k=0..n} binomial(n,k) q(k) d(n-k), relating these sequences (and A000108) through binomial convolutions. Then also, (r. + c. + d.)^n = r(n). See A180874 for proofs and for relations to A097610. For quick reference, q = (0, 1, 0, 2, 0, 5, 0, 14, ..), d = (1, 0, -1, 0, 4, 0, -35, 0, ..), and r = (0, 1, 0, -1, 0, 5, 0, -56, ..). - Tom Copeland, Jan 28 2016
Aerated and signed, this sequence contains the moments m(n) of the Appell polynomial sequence UMT(n,h1,h2) that is the umbral compositional inverse of the Appell sequence of Motzkin polynomials MT(n,h1,h2) of A097610 with exp[x UMT(.,h1,h2)] = e^(x*h1) / AC(x*y) where y = sqrt(h2) and AC is defined above. UMT(n,h1,h2) = (m.y + h1)^n with (m.)^(2n) = m(2n) = (-1)^n A238390(n) and zero otherwise. Consequently, the associated lower triangular matrices A007318(n,k)*m(n-k) and A007318(n,k)*A126120(n-k) form an inverse pair (cf. also A133314), and MT(n,UMT(.,h1,h2),h2) = h1^n = UMT(n,MT(.,h1,h2),h2). - Tom Copeland, Jan 30 2016

G. C. Greubel, Table of n, a(n) for n = 0..98

Cf. A103365, A180874, A115369, A000275, A002190.

Cf. A000108, A007318, A097610, A126120, A133314.

S:= series(x/BesselJ(1,2*x),x,102):
seq((2*j)!*coeff(S,x,2*j),j=0..50); # Robert Israel, Jan 31 2016

Table[(CoefficientList[Series[x/BesselJ[1, 2*x], {x, 0, 40}], x] * Range[0, 40]!)[[n]], {n, 1, 41, 2}]

a(n) ~ c * (n!)^2 / (sqrt(n) * r^n), where r = BesselJZero[1, 1]^2/16 = 0.91762316513274332857623611, and c = 1/(Sqrt[Pi]*BesselJ[2, BesselJZero[1, 1]]) = 1.4008104828035425937394082168... - Vaclav Kotesovec, Mar 01 2014, updated Apr 01 2018

cp's OEIS Frontend

A129561 A054523 * A115369.

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A001227 Number of odd divisors of n.

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A115368 Decimal expansion of first zero of the Bessel function J_0(z).

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A180874 Lassalle's sequence connected with Catalan numbers and Narayana polynomials.

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A103365 First column of triangle A103364, which equals the matrix inverse of the Narayana triangle (A001263).

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A103364 Matrix inverse of the Narayana triangle A001263.

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A238390 E.g.f.: x / BesselJ(1, 2*x) (even powers only).