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A049027
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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.
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19
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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
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OFFSET
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0,3
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COMMENTS
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(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
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)
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REFERENCES
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L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.
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LINKS
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FORMULA
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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
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
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
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
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EXAMPLE
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G.f. = 1 + x + 4*x^2 + 17*x^3 + 74*x^4 + 326*x^5 + 1446*x^6 + 6441*x^7 + ...
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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