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A063020 Reversion of y - y^2 - y^3 + y^4. 12
0, 1, 1, 3, 9, 32, 119, 466, 1881, 7788, 32868, 140907, 611871, 2685732, 11896906, 53115412, 238767737, 1079780412, 4909067468, 22424085244, 102865595140, 473678981820, 2188774576575, 10145798119530, 47165267330415, 219839845852692, 1027183096151244, 4810235214490986 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Seems to be the inverse of A007858. Can someone prove this?
a(n+1) counts paths from (0,0) to (n,n) which do not go above the line y=x, using steps (1,0) and (2k,1), where k ranges over the nonnegative integers. For example, the 9 paths from (0,0) to (3,3) are the 5 Catalan paths, as well as DNEN, DENN, EDNN and ENDN. Here E=(1,0), N=(0,1), D=(2,1). - Brian Drake, Sep 20 2007
LINKS
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
A. Mironov and A. Morozov, Algebra of quantum C-polynomials, arXiv:2009.11641 [hep-th], 2020.
Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.
FORMULA
a(n) = (1/n)*Sum_{k=0..n-1} binomial(n+k-1,n-1) * Sum_{j=0..k} binomial(j,n-3*k+2*j-1)*(-1)^(j-k)*binomial(k,j). - _Vladimir Kruchinin,_ Oct 11 2011
a(n) = (1/n)*Sum_{i=0..n-1} (-1)^(i)*binomial(n+i-1,i)*binomial(3*n-i-2,n-i-1), n > 0. - Vladimir Kruchinin, Feb 13 2014
Recurrence: 16*(n-1)*n*(2*n-1)*(17*n-27)*a(n) = (n-1)*(1819*n^3 - 6527*n^2 + 7350*n - 2520)*a(n-1) + 8*(2*n-3)*(4*n-9)*(4*n-7)*(17*n-10)*a(n-2). - Vaclav Kotesovec, Feb 13 2014
a(n) ~ sqrt(11-3/sqrt(17))/16 * (107+51*sqrt(17))^n / (sqrt(Pi) * n^(3/2) * 2^(6*n)). - Vaclav Kotesovec, Feb 13 2014
The g.f. A(x) satisfies x*A'(x)/A(x) = 1 + x + 5*x^2 + 19*x^3 + 85*x^4 + ..., the g.f. of A348410. - Peter Bala, Feb 22 2022
MAPLE
A:= series(RootOf(_Z-_Z^2-_Z^3+_Z^4-x), x, 21): seq(coeff(A, x, i), i=0..20); # Brian Drake, Sep 20 2007
MATHEMATICA
CoefficientList[InverseSeries[Series[y - y^2 - y^3 + y^4, {y, 0, 30}], x], x]
PROG
(Maxima)
a(n):=sum((sum(binomial(j, n-3*k+2*j-1)*(-1)^(j-k)*binomial(k, j), j, 0, k))*binomial(n+k-1, n-1), k, 0, n-1)/n; /* Vladimir Kruchinin, Oct 11 2011 */
(PARI) x='x+O('x^66); concat([0], Vec(serreverse(x-x^2-x^3+x^4))) \\ Joerg Arndt, May 28 2013
(Maxima)
a(n):=sum((-1)^(i)*binomial(n+i-1, i)*binomial(3*n-i-2, n-i-1), i, 0, n-1)/n; /* Vladimir Kruchinin, Feb 13 2014 */
(SageMath)
def b(n):
h = binomial(3*n + 1, n) * hypergeometric([-n, n + 1], [-3*n - 1], -1) / (n + 1)
return simplify(h)
print([0] + [b(n) for n in range(27)]) # Peter Luschny, Sep 21 2023
CROSSREFS
Sequence in context: A122452 A192206 A091841 * A104184 A339230 A193621
KEYWORD
nonn,easy
AUTHOR
Olivier Gérard, Jul 05 2001
STATUS
approved

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Last modified May 29 07:06 EDT 2024. Contains 372926 sequences. (Running on oeis4.)