A144301 a(0) = a(1) = 1; thereafter a(n) = (2*n-3)*a(n-1) + a(n-2).

1, 1, 2, 7, 37, 266, 2431, 27007, 353522, 5329837, 90960751, 1733584106, 36496226977, 841146804577, 21065166341402, 569600638022431, 16539483668991901, 513293594376771362, 16955228098102446847, 593946277027962411007, 21992967478132711654106, 858319677924203716921141

Offset

0,3

Comments

A variant of A001515, which is the main entry.

a(n) = number of increasing ordered trees on the vertex set [0,n] (counted by the double factorials A001147) in which n is the label on the leaf that terminates the leftmost path from the root. - David Callan, Aug 24 2011

Links

G. C. Greubel, Table of n, a(n) for n = 0..400

E. Grosswald, Bessel Polynomials, Lecture Notes Math., Vol. 698, 1978.

Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From N. J. A. Sloane, Sep 15 2012

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

Formula

a(n) = A001515(n-1) for n>= 1.

E.g.f.: A(x) = exp(1-sqrt(1-2*x)) satisfies A'(x) = A(x)/(1-sqrt(1-2*x)).

Hence a(n+1) = Sum_{k=0..n} ( a(n-k)*binomial(n,k)*(2*k)!/(k!*2^k) ).

A''(x) = (A'(x)/(1-2*x))*(1 + 1/sqrt(1-2*x)).

A''(x) = 2*x*A''(x) + A'(x) + A(x), which is equivalent to the recurrence in the definition.

a(n) = Sum_{k=0..n-1} binomial(n+k-1,2*k)*(2*k)!/(k!*2^k) ). [See Grosswald, p. 6, Eq. (8).]

a(n) ~ exp(1)*(2n-1)!/(n!*2^n) as n -> oo. [See Grosswald, p. 124]

G.f.: 1+x/U(0) where U(k)= 1 - x - x*(2*k+1)/(1 - x - 2*x*(k+1)/U(k+1)) ; (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 06 2012

G.f.: 1+x*(1-x)/U(0) where U(k)= 1 - 3*x + x^2 - x*4*k - x^2*(2*k+1)*(2*k+2)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 06 2012

E.g.f.: E(0)/2, where E(k)= 1 + 1/(1 - 2*x/(2*x + (k+1)*(1+sqrt(1-2*x))/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 23 2013

G.f.: conjecture: 1 + x*(1-x)/(1-3*x+x^2)*Q(0), where Q(k) = 1 - 2*(k+1)*(2*k+1)*x^2/(2*(k+1)*(2*k+1)*x^2 - (1 - 3*x + x^2 - 4*x*k)*(1 - 7*x + x^2 - 4*x*k)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 19 2013

a(1 - n) = a(n) for all n in Z. (a(n+1) + a(n+2))^2 = a(n)*a(n+2) + a(n+1)*a(n+3) for all integer n. - Michael Somos, Nov 22 2013

G.f.: 1 + x/(1-x)*T(0), where T(k) = 1 - x*(k+1)/( x*(k+1) - (1-x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2013

Example

G.f. = 1 + x + 2*x^2 + 7*x^3 + 37*x^4 + 266*x^5 + 2431*x^6 + 27007*x^7 + ...

Mathematica

a[n_]:= HypergeometricPFQ[{n, 1 - n}, {}, -1/2]; (* Michael Somos, Nov 22 2013 *)
a[n_]:= With[{m= If[n<1, -n, n-1]}, Sum[(m+k)!/((m-k)! k! 2^k), {k, 0, m}]]; (* Michael Somos, Nov 22 2013 *)
RecurrenceTable[{a[0]==a[1]==1, a[n]==(2*n-3)*a[n-1] +a[n-2]}, a, {n, 25}] (* Vincenzo Librandi, Jul 23 2015 *)
nxt[{n_, a_, b_}]:={n+1, b, b(2n-1)+a}; NestList[nxt, {1, 1, 1}, 30][[All, 2]] (* Harvey P. Dale, Nov 29 2022 *)

Prog

(PARI) {a(n) = my(m = if( n<1, -n, n-1)); sum( k=0, m,  (m+k)! / (k! * (m-k)! * 2^k))}; /* Michael Somos, Nov 22 2013 */
(Magma) [1] cat [n le 1 select n+1 else (2*n-1)*Self(n) + Self(n-1): n in [0..20]]; // Vincenzo Librandi, Jul 23 2015
(SageMath)
def A144301(n): return int(n==0) + sum(binomial(n-1, k)*factorial(n+k-1)/(2^k*factorial(n-1)) for k in range(n))
[A144301(n) for n in range(31)] # G. C. Greubel, Sep 29 2023

Crossrefs

See A001515 for much more about this sequence.

See A144498 for first differences.

Sequence in context: A125515 A135920 A001515 * A036247 A083659 A107877

Adjacent sequences: A144298 A144299 A144300 * A144302 A144303 A144304

Keyword

nonn,easy

Author

David Applegate and N. J. A. Sloane, Dec 07 2008

Extensions

More terms from Vincenzo Librandi, Jul 23 2015

Status

approved