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A097893
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Partial sums of the central trinomial coefficients (A002426).
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1
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1, 2, 5, 12, 31, 82, 223, 616, 1723, 4862, 13815, 39468, 113257, 326198, 942425, 2730032, 7926659, 23061590, 67214399, 196211252, 573590621, 1678941350, 4920076877, 14433305000, 42381641381, 124558477682, 366371703833
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n)=number of peaks at odd height in all Motzkin paths of length n+2. Example: a(2)=5 counts the peaks shown between parentheses in the 9 Motzkin paths of length 4: HHHH, HH(UD), H(UD)H, HUHD, (UD)HH, (UD)(UD), UHDH, UHHD and UUDD.
Binomial transform of 1,1,2,2,6,6,20,20,70,70...... (A000984 doubled). It would appear that the Hankel transform of this sequence is a signed version of A128055, with sign pattern given by s(n)=(2/3-sqrt(3)/3)cos(5*pi*n/6)-sin(5*pi*n/6)/3+(sqrt(3)/3+2/3)*cos(pi*n/6)-sin(pi*n/6)/3-cos(pi*n/2)/3+sin(pi*n/2)/3. - Paul Barry (pbarry(AT)wit.ie), Jan 03 2008
The subsequence of primes in this sequence of partial sums begins: 2, 5, 31, 223, 1723, no more through a(26). [From Jonathan Vos Post (jvospost3(AT)gmail.com), May 12 2010]
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FORMULA
| G.f. = 1/[(1-z)sqrt(1-2z-3z^2)].
a(n)=sum(0<=j<=i<=n, C(i, i-j)*C(j, i-j)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 23 2004
a(n):=sum{k=0..n, sum{j=0..n-k, C(k,j)C(n-k,j)C(2j,j)}}; - Paul Barry (pbarry(AT)wit.ie), Jan 03 2008
Logarithm g.f. atan(x*M(x)), M(x) - o.g.f. for Motzkin numbers (A001006). [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Aug 11 2010]
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MAPLE
| ser:=series(1/(1-z)/sqrt(1-2*z-3*z^2), z=0, 32): 1, seq(coeff(ser, z^n), n=1..31);
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PROG
| (PARI) a(n)=sum(i=0, n, sum(j=0, i, binomial(i, i-j)*binomial(j, i-j)))
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CROSSREFS
| Cf. A002426.
Sequence in context: A110035 A000635 A077556 * A093379 A076906 A071359
Adjacent sequences: A097890 A097891 A097892 * A097894 A097895 A097896
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2004
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