A015083 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=2.

1, 1, 3, 17, 171, 3113, 106419, 7035649, 915028347, 236101213721, 121358941877763, 124515003203007345, 255256125633703622475, 1046039978882750301409545, 8571252355254982356001107795, 140448544236464264647066322058465, 4602498820363674769217316088142020635

Offset

0,3

Comments

Limit_{n->inf} a(n)/2^((n-1)(n-2)/2) = Product{k>=1} 1/(1-1/2^k) = 3.462746619455... (cf. A065446). - Paul D. Hanna, Jan 24 2005

It appears that the Hankel transform is 2^A002412(n). - Paul Barry, Aug 01 2008

Hankel transform of aerated sequence is A125791. - Paul Barry, Dec 15 2010

Links

Seiichi Manyama, Table of n, a(n) for n = 0..81

J. Fürlinger, J. Hofbauer, q-Catalan numbers, Journal of Combinatorial Theory, Series A, Volume 40, Issue 2, November 1985, Pages 248-264.

Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

Formula

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=2 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(2*x)) = 1/(1-x/(1-2*x/(1-2^2*x/(1-2^3*x/(1-...))))) (continued fraction). - Paul D. Hanna, Jan 24 2005

G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=0..n-1} 2^k*x*A(2^k*x). - Paul D. Hanna, May 17 2010

a(n) = the upper left term in M^(n-1), M = the infinite square production matrix:

1, 2, 0, 0, 0, ...

1, 2, 4, 0, 0, ...

1, 2, 4, 8, 0, ...

1, 2, 4, 8, 16, ...

...

Also, a(n+1) = sum of top row terms of M^(n-1). Example: top row of M^3 = (17, 34, 56, 64, 0, 0, 0, ...); where a(4) = 17 and a(5) = 171 = (17 + 34 + 56 + 64). - Gary W. Adamson, Jul 14 2011

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

Example

G.f. = 1 + x + 3*x^2 + 17*x^3 + 171*x^4 + 3113*x^5 + 106419*x^6 + 7035649*x^7 + ...
From Seiichi Manyama, Dec 05 2016: (Start)
a(1) = 1,
a(2) = 2^1 + 1 = 3,
a(3) = 2^3 + 2^2 + 2*2^1 + 1 = 17,
a(4) = 2^6 + 2^5 + 2*2^4 + 3*2^3 + 3*2^2 + 3*2^1 + 1 = 171. (End)

Mathematica

a[n_] := a[n] = Sum[2^i*a[i]*a[n - i - 1], {i, 0, n - 1}];
a[0] = 1; Array[a, 16, 0] (* Robert G. Wilson v, Dec 24 2016 *)
m = 17; ContinuedFractionK[If[i == 1, 1, -2^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

Prog

(PARI) a(n)=if(n==0, 1, sum(i=0, n-1, 2^i*a(i)*a(n-1-i))) \\  Paul D. Hanna
(PARI) {a(n) = my(A); if( n<1, n==0, A = vector(n, i, 1); for(k=0, n-1, A[k+1] = if( k<1, 1, A[k]*(1+2^k) + sum(i=1, k-1, 2^i * A[i] * A[k-i]))); A[n])}; /* Michael Somos, Jan 30 2005 */
(PARI) {a(n) = my(A); if( n<0, 0, A = O(x); for(k=1, n, A = 1 / (1 - x * subst(A, x, 2*x))); polcoeff(A, n))}; /* Michael Somos, Jan 30 2005 */
(Ruby)
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
  ary
end
def A015083(n)
  A(2, n)
end # Seiichi Manyama, Dec 24 2016

Crossrefs

Cf. A065446, A227543.

Cf. A015108 (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), A090192 (q=-1), A000108 (q=1), this sequence (q=2), A015084 (q=3), A015085 (q=4), A015086 (q=5), A015089 (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).

Column k=2 of A090182, A290759.

Sequence in context: A268758 A069856 A214346 * A263460 A053934 A159592

Adjacent sequences: A015080 A015081 A015082 * A015084 A015085 A015086

Keyword

nonn

Author

Olivier Gérard

Extensions

Offset changed to 0 by Seiichi Manyama, Dec 05 2016

Status

approved