A080893 Expansion of e.g.f. exp(x*C(x)) = exp((1-sqrt(1-4*x))/2), where C(x) is the g.f. of the Catalan numbers A000108.

1, 1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353, 28875761731, 1098127402131, 46150226651233, 2124008553358849, 106246577894593683, 5739439214861417731, 332993721039856822081, 20651350143685984386753

Offset

0,3

Comments

Essentially the same as A001517: a(n+1) = A001517(n).

For k >= 2, the difference a(n+k) - a(n) is divisible by k. It follows that for each k, the sequence formed by taking a(n) modulo k is periodic with period dividing k. For example, modulo 10 the sequence becomes [1, 1, 3, 9, 3, 1, 1, 3, 9, 3, ...], a purely periodic sequence of period 5. Cf. A047974. - Peter Bala, Feb 11 2025

Links

Table of n, a(n) for n=0..17.

W. Mlotkowski and A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.

Eric Weisstein's World of Mathematics, Bell Polynomial.

Formula

E.g.f.: exp((1-sqrt(1-4*x))/2).

D-finite with recurrence: a(n+2) = 2*(2*n + 1)*a(n+1) + a(n).

Recurrence: y(n+1) = Sum_{k = 0..n} binomial(n, k)*binomial(2k, k)*k!*y(n-k).

a(1 - n) = a(n). a(n + 1) = A001517(n). - Michael Somos, Apr 07 2012

G.f.: 1 + x/Q(0), where Q(k)= 1 - x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013

a(n) ~ 2^(2*n-3/2)*n^(n-1)/exp(n-1/2). - Vaclav Kotesovec, Jun 26 2013

a(n) = hypergeom([-n+1, n], [], -1). - Peter Luschny, Oct 17 2014

a(n) = Sum_{k=0..n} (-4)^(n-k) * Stirling1(n,k) * A009235(k) = (-4)^n * Sum_{k=0..n} (1/2)^k * Stirling1(n,k) * Bell_k(-1/2), where Bell_n(x) is n-th Bell polynomial. - Seiichi Manyama, Aug 02 2024

Mathematica

y[x_] := y[x] = 2(2x - 3)y[x - 1] + y[x - 2]; y[0] = 1; y[1] = 1; Table[y[n], {n, 0, 17}]
With[{nn=20}, CoefficientList[Series[Exp[(1-Sqrt[1-4x])/2], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Oct 30 2011 *)

Prog

(PARI) {a(n) = if( n<1, n = 1 - n); n! * polcoeff( exp( (1 - sqrt(1 - 4*x + x * O(x^n))) / 2), n)} /* Michael Somos, Apr 07 2012 */
(Sage)
A080893 = lambda n: hypergeometric([-n+1, n], [], -1)
[simplify(A080893(n)) for n in (0..19)] # Peter Luschny, Oct 17 2014

Crossrefs

Cf. A000108, A001517, A009235, A375173.

Sequence in context: A155805 A218261 A001517 * A028854 A222865 A108292

Adjacent sequences: A080890 A080891 A080892 * A080894 A080895 A080896

Keyword

easy,nice,nonn

Author

Emanuele Munarini, Mar 31 2003

Status

approved