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A049027
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G.f.: (1-2*x*C)/(1-3*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.
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17
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| [a(n+1)] = [1,4,17,74,326,...] is the binomial transform of A059738. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 26 2009]
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REFERENCES
| 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
| W. 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.
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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 (pbarry(AT)wit.ie), Jun 22 2004
a(n) = (9*a(n-1)-Catalan(n-1))/2, n>1. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 08 2004
a(n+1)=Sum_{k, 0<=k<=n}A039598(n,k)*2^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 21 2007
G.f.: 2/ (3 -1/sqrt(1 -4*x)) . - Michael Somos Apr 08 2007
a(n)=Sum_{k, 0<=k<=n}A039599(n,k)*A001045(k), for n>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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). [From Milan R. Janjic (agnus(AT)blic.net), 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)
Conjecture: 2*n*a(n) +(12-17*n)*a(n-1) +18*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
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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 */
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CROSSREFS
| Row sums of triangle A035324. Cf. A000108, A001700.
Sequence in context: A184700 A125586 A086351 * A026751 A081568 A026378
Adjacent sequences: A049024 A049025 A049026 * A049028 A049029 A049030
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KEYWORD
| easy,nonn
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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