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A032357
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Convolution of Catalan numbers and powers of -1.
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6
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1, 0, 2, 3, 11, 31, 101, 328, 1102, 3760, 13036, 45750, 162262, 580638, 2093802, 7601043, 27756627, 101888163, 375750537, 1391512653, 5172607767, 19293659253, 72188904387, 270870709263, 1019033438061, 3842912963391, 14524440108761
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Absolute value of the alternating sum of Catalan Numbers. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 03 2006
Sums of two consecutive terms are A032357[n-1]+A032357[n]=1,2,5,14,42..=A000108[n] Catalan Numbers. The prime p divides a((p-3)/2) for p=11,19,29,31,41,59,61,71..=A045468[n] Primes congruent to {1, 4} mod 5. Prime p divides a(2p+1) for p=5,11,19,29,31,41,59,61,71..=A038872[n] Primes congruent to {0, 1, 4} mod 5. Also odd primes where 5 is a square mod p. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 03 2006
Hankel transform is F(2n+1). - Paul Barry (pbarry(AT)wit.ie), Jul 22 2008
Equals INVERTi transform of A000958 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]
Inverse binomial transform of A002212 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 17 2009]
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FORMULA
| G.f.: c(x)/(1+x), where c(x) = g.f. for Catalan numbers.
Sum((-1)^(n-k)*C(k), k=0..n), C(k)=A00108(k), Catalan numbers
a(n)=((-1)^(n+1)-binomial(2*(n+1), n+1)*sum((-5)^k*binomial(n+1, k)/binomial(2*k, k), k=0..n+1))/2
Also: a(n)=C[2n, n]/(n+1)-a(n-1)=A000108[n]-a(n-1) with a(0)=1. - Labos E. (labos(AT)ana.hu), Apr 26 2003
a(n) = Sum[ (-1)^(k+n)*CatalanNumber[k], {k,0,n}]. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 03 2006
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MATHEMATICA
| Table[Sum[(-1)^(k+n)*CatalanNumber[k], {k, 0, n}], {n, 0, 60}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 03 2006
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CROSSREFS
| Cf. A000108.
Cf. A000108, A014137, A014138, A033297, A045468, A038872, A064739.
A000958 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]
Sequence in context: A142957 A191058 A080155 * A144056 A062630 A159458
Adjacent sequences: A032354 A032355 A032356 * A032358 A032359 A032360
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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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EXTENSIONS
| More terms from Christian G. Bower (bowerc(AT)usa.net), Apr 15 1998.
More terms from Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 03 2006
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