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A105633
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Row sums of triangle A105632.
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17
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1, 2, 4, 9, 22, 57, 154, 429, 1223, 3550, 10455, 31160, 93802, 284789, 871008, 2681019, 8298933, 25817396, 80674902, 253106837, 796968056, 2517706037, 7977573203, 25347126630, 80738862085, 257778971504, 824798533933
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OFFSET
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0,2
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COMMENTS
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a(n) = the number of Dyck paths of semilength n+1 avoiding UUDU. a(n) = the number of Dyck paths of semilength n+1 avoiding UDUU = the number of binary trees without zigzag (i.e., with no node with a father, with a right son and with no left son). This sequence is the first column of the triangle A116424. E.g., a(2) = 4 because there exist four Dyck paths of semilength 3 that avoid UUDU: UDUDUD, UDUUDD, UUDDUD, UUUDDD, as well as four Dyck paths of semilength 3 that avoid UDUU: UDUDUD, UUDUDD, UUDDUD, UUUDDD. - I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006
The sequence beginning 1,1,2,4,9,... gives the diagonal sums of A130749, and has g.f. 1/(1-x-x^2/(1-x/(1-x-x^2/(1-x/(1-x-x^2/(1-... (continued fraction); and general term Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} binomial(n-k,j)*A090181(j,k). Its Hankel transform is A099443(n+1). - Paul Barry, Jun 30 2009
The number of plain lambda terms presented by de Bruijn indices, see Bendkowski et al. - Kellen Myers, Jun 15 2015
a(n) = the number of Dyck paths of semilength n+1 with no pairs of
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LINKS
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Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Permutations avoiding a simsun pattern, The Electronic Journal of Combinatorics (2020) Vol. 27, Issue 3, P3.45. (See a_n in Theorem 4.)
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FORMULA
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G.f.: A(x) = (1-x - sqrt((1-x)^2 - 4*x^2/(1-x)))/(2*x^2).
a(n) = 2*a(n-1) + Sum_{i=1..n-2} a(i)*(a(n-1-i) - a(n-2-i)). a(n) = Sum_{i=0..floor(n/2)} (-1)^i * binomial(n+1-i,i) * binomial(2*(n+1)-3*i, n-2*i) /(n+1-i). - I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006
G.f.: (1/(1-x)^2)c(x^2/(1-x)^3), where c(x) is the g.f. of A000108. - Paul Barry, May 22 2009
1/(1-x-x/(1-x^2/(1-x-x/(1-x^2/(1-x-x/(1-x^2/(1-... (continued fraction). - Paul Barry, Jun 30 2009
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} binomial(n-k,j)(0^(j+k)+(1/(j+0^j))*binomial(j,k)*binomial(j,k+1)). - Paul Barry, Jun 30 2009
G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x*(1-x)*A(x)). - Paul D. Hanna, Sep 12 2012
G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * (1-x)^k ). - Paul D. Hanna, Sep 12 2012
D-finite with recurrence: (n+2)*a(n) + (-4*n-3)*a(n-1) + (2*n+1)*a(n-2) + a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Nov 26 2012
The recurrence is true, since by holonomic transformation, it can be computed formally using GFUN, associated with the equation: x^3 + x^2 - 2x + (x^3 + 3 x^2 -3x +1) A(x) + (x^5 + 2x^3 -4 x^2 + x) A'(x) = 0. - Pierre Lescanne, Jun 30 2015
G.f.: (1 - 1/(G(0)-x))/x^2 where G(k) = 1 + x/(1 + x/(x^2 - 1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 16 2012
a(n) ~ 2^(n/3-1/6) * 3^(n+2) * (13+3*sqrt(33))^((n+1)/3) * sqrt(4*(2879 + 561*sqrt(33))^(1/3) + 8*(7822 + 1362*sqrt(33))^(1/3) - 91 - 21*sqrt(33)) / (((26+6*sqrt(33))^(2/3) - (26+6*sqrt(33))^(1/3) - 8)^(n+3/2) * (4*(26+6*sqrt(33))^(1/3) - (26+6*sqrt(33))^(2/3) + 8) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Mar 13 2014
a(n) = Sum_{i=0..n+1} (-1)^i*hypergeom([(i+1)/2, i/2+1, i-n-1], [1, 2], -4). - Peter Luschny, May 03 2018
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 4*x^2 + 9*x^3 + 22*x^4 + 57*x^5 + 154*x^6 + 429*x^7 + ...
with A(x)^2 = 1 + 4*x + 12*x^2 + 34*x^3 + 96*x^4 + 274*x^5 + 793*x^6 + ...
where A(x) = 1 + x*(2-x)*A(x) + x^2*(1-x)*A(x)^2.
The logarithm of the g.f. begins:
log(A(x)) = (1 + (1-x))*x + (1 + 2^2*(1-x) + (1-x)^2)*x^2/2 +
(1 + 3^2*(1-x) + 3^2*(1-x)^2 + (1-x)^3)*x^3/3 +
(1 + 4^2*(1-x) + 6^2*(1-x)^2 + 4^2*(1-x)^3 + (1-x)^4)*x^4/4 +
(1 + 5^2*(1-x) + 10^2*(1-x)^2 + 10^2*(1-x)^3 + 5^2*(1-x)^4 + (1-x)^5)*x^5/5 + ...
Explicitly,
log(A(x)) = 2*x + 4*x^2/2 + 11*x^3/3 + 32*x^4/4 + 97*x^5/5 + 301*x^6/6 + 947*x^7/7 + 3008*x^8/8 + 9623*x^9/9 + 30959*x^10/10 + ...
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MAPLE
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a := n -> add((-1)^i*hypergeom([(i+1)/2, i/2+1, i-n-1], [1, 2], -4), i=0..n+1):
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MATHEMATICA
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CoefficientList[Series[(1 - x - Sqrt[(1 - x)^2 - 4 x^2/(1 - x)])/(2 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 15 2014 *)
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PROG
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(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff(2/(1-X)/(1-X+sqrt((1-X)^2-4*X^2/(1-X))), n, x)}
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*sum(k=0, m, binomial(m, k)^2*(1-x)^(m-k) + x*O(x^n)))), n)} \\ Paul D. Hanna, Sep 12 2012
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More terms from I. Tasoulas (jtas(AT)unipi.gr), Feb 15 2006
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STATUS
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approved
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