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A049027 G.f.: (1-2*x*c(x))/(1-3*x*c(x)) where c(x) = (1 - sqrt(1-4*x))/(2*x) is the g.f. for Catalan numbers A000108. 19
1, 1, 4, 17, 74, 326, 1446, 6441, 28770, 128750, 576944, 2587850, 11615932, 52167688, 234383146, 1053386937, 4735393794, 21291593238, 95747347176, 430624242942, 1936925461644, 8712882517188, 39195738193836, 176335080590442, 793336332850164, 3569368545752076 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Row sums of triangle A035324.
a(n+1) = {1, 4, 17, 74, 326, ...} is the binomial transform of A059738. - Philippe Deléham, Nov 26 2009
(1, 4, 17, 74, 326, ...) is the invert transform of the odd-indexed central binomial coefficients, A001700. - David Callan, Oct 14 2012
The sequence starting with index 1 is the INVERT transform of A001700: (1, 3, 10, 35, 126, ...) and the second INVERT transform of the Catalan numbers starting with index 1: (1, 2, 5, 14, 42, ...). - Gary W. Adamson, Jun 23 2015
From Peter Bala, Jan 27 2020: (Start)
This sequence is the main diagonal of the lower triangular array formed by taking the first column (k = 0) of the array equal to (1,1,3,9,27,...) - powers of 3 with 1 prepended - and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1. See my link in A001517.
1
1 1
3 4 4
9 13 17 17
27 40 57 74 74
81 121 178 252 326 326
...
(End)
REFERENCES
L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.
LINKS
José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, On One-Parameter Catalan Arrays, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1 and arXiv version, arXiv:1505.05568 [math.CO], 2015.
Paul Barry and Arnauld Mesinga Mwafise, Classical and Semi-Classical Orthogonal Polynomials Defined by Riordan Arrays, and Their Moment Sequences, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.5.
Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, Catalan-Spitzer permutations, arXiv:2310.06288 [math.CO], 2023. See p. 20.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
FORMULA
G.f.: x*c(x)/(1-3*x*c(x)), c(x)= g.f. of Catalan numbers A000108.
a(n+1) = Sum_{k=0..n} 2^k*comb(2n+1, n-k)*2*(k+1)/(n+k+2) - Paul Barry, Jun 22 2004
a(n) = (9*a(n-1) - Catalan(n-1))/2, n > 1. - Vladeta Jovovic, Aug 08 2004
a(n+1) = Sum_{k=0..n} A039598(n,k)*2^k. - Philippe Deléham, Mar 21 2007
G.f.: 2 / (3 - 1 / sqrt(1 - 4*x)). - Michael Somos, Apr 08 2007
a(n) = Sum_{k=0..n} A039599(n,k)*A001045(k), for n >= 1. - Philippe Deléham, Jun 10 2007
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i <= j), and A[i,j]=0, otherwise. Then, for n >= 1, a(n+1) = (-1)^n*charpoly(A,-3). - Milan Janjic, Jul 08 2010
From Gary W. Adamson, Jul 25 2011: (Start)
a(n) = upper left term in M^(n-1), M = an infinite square production matrix as follows:
4, 1, 0, 0, 0, ...
1, 1, 1, 0, 0, ...
1, 1, 1, 1, 0, ...
1, 1, 1, 1, 1, ...
... (End)
D-finite with recurrence: 2*n*a(n) + (12-17*n)*a(n-1) + 18*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 14 2011
a(n) ~ 3^(2*n-1)/2^(n+1). - Vaclav Kotesovec, Oct 08 2012
0 = a(n)*(1296*a(n+1) - 1098*a(n+2) + 180*a(n+3)) + a(n+1)*(-126*a(n+1) + 253*a(n+2) - 58*a(n+3)) + a(n+2)*(-10*a(n+2) + 4*a(n+3)) if n > 0. - Michael Somos, Jan 23 2014
O.g.f.: A(x) = 1/(1 - (1/2)*Sum_{n >= 1} binomial(2*n,n)*x^n). - Peter Bala, Sep 01 2016
a(n) = 3^(2*n-1)/2^(n+1) + 2^n * (2*n-1)!! * hypergeom([1,n+1], [n+2], 8/9)/(9*(n+1)!) + 0^n * 2/3. - Vladimir Reshetnikov, Oct 08 2016
EXAMPLE
G.f. = 1 + x + 4*x^2 + 17*x^3 + 74*x^4 + 326*x^5 + 1446*x^6 + 6441*x^7 + ...
MAPLE
a:= proc(n) option remember; `if`(n<3, 1+3*n*(n-1)/2,
(17/2-6/n)*a(n-1)-(18-27/n)*a(n-2))
end:
seq(a(n), n=0..28); # Alois P. Heinz, Jan 28 2020
MATHEMATICA
Table[SeriesCoefficient[2/(3-1/Sqrt[1-4*x]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2012 *)
FunctionExpand@Table[3^(2n-1)/2^(n+1) + 2^n (2n-1)!! Hypergeometric2F1[1, n + 1/2, n + 2, 8/9]/(9 (n + 1)!) + 2 KroneckerDelta[n]/3, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 08 2016 *)
PROG
(PARI) {a(n) = if( n<1, n==0, polcoeff( serreverse( x * (1 + 2*x) / (1 + 3*x)^2 + x * O(x^n) ), n))}; /* Michael Somos, Apr 08 2007 */
(PARI) {a(n) = if( n<0, 0, polcoeff( 2 / (3 - 1 / sqrt(1 - 4*x + x * O(x^n))), n))}; /* Michael Somos, Apr 08 2007 */
(Magma) [1] cat [n eq 1 select 1 else (9*Self(n-1)-Catalan(n-1))/2: n in [1..30]]; // Vincenzo Librandi, Jun 25 2015
(Sage) (2/(3-1/sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 02 2019
CROSSREFS
Sequence in context: A184700 A125586 A086351 * A026751 A227504 A363496
KEYWORD
easy,nonn
AUTHOR
STATUS
approved

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Last modified February 26 13:52 EST 2024. Contains 370352 sequences. (Running on oeis4.)