OFFSET
0,3
COMMENTS
a(n) has the Lucas property, namely a(n) is congruent to a(n_0)a(n_1)...a(n_k) modulo p for any prime p where n_0,n_1,... are the base p digits of n. (Carlitz via McIntosh)
Let P_n be the Segre product of the Boolean lattice with itself. (-1)^n*a(n) = mu(P_n). See Yifei Li and Sheila Sundaram link below. - Geoffrey Critzer, Jan 20 2026
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..261
Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (8) (with t=0 and m=2) on p. 249.
Leonid Bedratyuk and Nataliia Luno, Connection problems for the generalized hypergeometric Appell polynomials, Carpathian Math. Publ. (2020) Vol. 12, No. 1, 10-18.
Leonard Carlitz, The coefficients of the reciprocal of J_0(x), Archiv. Math. 6 (1955), 121-127.
Leonard Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc. 80(5) (1974), 881-884.
Leonard Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc. 80(5) (1974), 881-884. [Annotated scanned copy]
Leonard Carlitz, N. J. A. Sloane, and Colin L. Mallows, Correspondence, 1975.
Theo Douvropoulos and Matthieu Josuat-Vergès, Homology character of the parabolic coset poset, arXiv:2509.11905 [math.CO], 2025. See p. 18.
Jan Geuenich, Tilting modules for the Auslander algebra of K[x]/(x^n), arXiv:1803.10707 [math.RT], 2018.
Yifei Li and Sheila Sundaram, Segre powers of posets preserve EL-shellability, arXiv:2512.10297 [math.CO], 2025.
Gunnar Þór Magnússon, The inner product on exterior powers of a complex vector space, arXiv preprint arXiv:1401.4048 [math.AG], 2014.
Richard J. McIntosh, A generalization of a congruential property of Lucas, Amer. Math. Monthly 99(3) (1992), 231-238; see page 232. MR1216210 (95b:11008)
John Riordan, Inverse relations and combinatorial identities, Amer. Math. Monthly, 71 (1964), 485-498.
Jonathan D. H. Smith, Commutative Moufang loops and Bessel functions, Invent. Math. 67(1) (1982), 173-187.
FORMULA
a(n) = Sum_{r=0..n-1} (-1)^(r+n+1) binomial(n, r)^2 a(r), if n > 0.
Sum_{n>=0} a(n) * x^n / n!^2 = 1 / J_0(sqrt(4*x)). - Michael Somos, May 17 2004
From Peter Bala, Aug 08 2011: (Start)
Conjectural formula: 1 = Sum_{n>=0} a(n)*x^n*Sum_{k>=0} binomial(n+k,k)^2*(-x)^k.
Apart from the initial term, first column of A192721. (End)
E.g.f.: 1/J_0(sqrt(4*x)) = 1 + x/Q(0), where Q(k) = (k+1)^2 - x + (k+1)^2*x/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2013
a(n) ~ c * (n!)^2 / r^n, where r = (1/4)*BesselJZero[0,1]^2 = 1.4457964907366961302939989396139517587678604516... and c = 1/(sqrt(r) * BesselJ(1, 2*sqrt(r))) = 1.60197469692804662664846689139151227422675123376219... - Vaclav Kotesovec, Mar 02 2014, updated Apr 01 2018
EXAMPLE
From Peter Bala, Aug 08 2011: (Start)
a(3) = 19: The 19 pairs of permutations in the group S_3 x S_3 with no common rises correspond to the zero entries in the table below.
======================================
Number of common rises in S_3 x S_3
======================================
| 123 132 213 231 312 321
======================================
123| 2 1 1 1 1 0
132| 1 1 0 1 0 0
213| 1 0 1 0 1 0
231| 1 1 0 1 0 0
312| 1 0 1 0 1 0
321| 0 0 0 0 0 0
(End)
G.f. = 1 + x + 3*x^2 + 19*x^3 + 211*x^4 + 3651*x^5 + 90921*x^6 + ...
MAPLE
A000275 := proc(n) sum(z^k/k!^2, k = 0..infinity);
series(%^x, z=0, n+1): n!^2*coeff(%, z, n); add(abs(coeff(%, x, k)), k=0..n) end:
seq(A000275(n), n=0..17); # Peter Luschny, May 27 2017
MATHEMATICA
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(r+n+1)*Binomial[n, r]^2 a[r], {r, 0, n-1}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Aug 05 2013 *)
CoefficientList[Series[1/BesselJ[0, Sqrt[4*x]], {x, 0, 20}], x]* Range[0, 20]!^2 (* Vaclav Kotesovec, Mar 02 2014 *)
a[ n_] := If[ n < 0, 0, (n! 2^n)^2 SeriesCoefficient[ 1 / BesselJ[ 0, x], {x, 0, 2 n}]]; (* Michael Somos, Aug 20 2015 *)
PROG
(PARI) {a(n) = if( n<0, 0, n!^2 * 4^n * polcoeff( 1 / besselj(0, x + x * O(x^(2*n))), 2*n))}; /* Michael Somos, May 17 2004 */
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Christian G. Bower, Apr 25 2000
STATUS
approved