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A063019 Reversion of y - y^2 + y^3 - y^4. 5
0, 1, 1, 1, 1, 2, 7, 22, 57, 132, 308, 793, 2223, 6328, 17578, 47804, 130169, 360924, 1019084, 2900484, 8252860, 23445510, 66717135, 190750110, 548178735, 1580970612, 4568275692, 13217653582, 38306172442, 111248832992 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Apparently: For n>0 number of Dyck (n-1)-paths with each ascent length equal to 0 or 1 modulo 4. - David Scambler, May 09 2012

LINKS

R. J. Mathar, Table of n, a(n) for n = 0..104

John Engbers, David Galvin, Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016. See p. 8.

Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.

Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Index entries for reversions of series

FORMULA

Conjecture: 32*n*(n-1)*(n-2)*a(n) -8*(n-1)*(n-2)*(16*n-9)*a(n-1) +2*(n-2)*(71*n^2+46*n-549)*a(n-2) +(97*n^3-2250*n^2+10859*n-14850)*a(n-3) -12*(4n-15)*(4*n-14)*(4*n-17)*a(n-4)=0. - R. J. Mathar, Oct 01 2012

Conjecture confirmed for n >= 5 using the fact that the G.f. satisfies

(24*x + 96)*g(x) + (-1104*x^2 - 1302*x + 456)*g'(x)

     + (-2688*x^3 - 1086*x^2 + 1086*x - 312)*g''(x)

     + (-768*x^4 + 97*x^3 + 142*x^2 - 128*x + 32)*g'''(x) = 6*x+24. It

is not true for n=4. - Robert Israel, Jan 08 2019

Recurrence: 16*(n-2)*(n-1)*n*(5*n-14)*a(n) = 4*(n-2)*(n-1)*(110*n^2 - 473*n + 468)*a(n-1) - (n-2)*(1015*n^3 - 6902*n^2 + 15391*n - 11232)*a(n-2) + 8*(2*n-5)*(4*n-13)*(4*n-11)*(5*n-9)*a(n-3). - Vaclav Kotesovec, Feb 12 2014

Conjecture: a(n+1) = 1/(n + 1)*sum(k = 0, floor(n/4), binomial(n + 1, n - 4*k)*binomial(n + k, n) ) (compare to the formula from Peter Bala in A215340). - Joerg Arndt, Apr 01 2019

MAPLE

F:= RootOf(y-y^2+y^3-y^4=x, y):

S:= series(F, x, 40):

seq(coeff(S, x, n), n=0..39); # Robert Israel, Jan 08 2019

MATHEMATICA

CoefficientList[InverseSeries[Series[y - y^2 + y^3 - y^4, {y, 0, 30}], x], x]

g[d_] := g[d] = If[OddQ[d], 3, 1]; f[x_, y_, d_] := f[x, y, d] = If[x < 0 || y < x, 0, If[x == 0 && y == 0, 1, f[x - 1, y, 0] + f[x, y - If[d == 0, 1, g[d]], If[d == 0, 1, g[d] + d]]]]; Join[{0}, Table[f[n - 1, n - 1, 0], {n, 30}]] (* David Scambler, May 09 2012 *)

PROG

(Maxima)

a(n):=if n<2 then n else (-1)^(n+1)*sum((sum(binomial(j, n-3*k+2*j-1)*(-1)^(2*j-k)*binomial(k, j), j, 0, k))*binomial(n+k-1, n-1), k, 1, n-1)/n; /* Vladimir Kruchinin, May 10 2011 */

(PARI)

x='x+O('x^66); Vec(serreverse(x-x^2+x^3-x^4)) /* Joerg Arndt, May 12 2011 */

(Sage) # Function Reversion defined in A063022.

Reversion(x - x^2 + x^3 - x^4, 30) # Peter Luschny, Jan 08 2019

CROSSREFS

Cf. A063022, A063023.

Sequence in context: A099131 A212384 A306347 * A183156 A018039 A198888

Adjacent sequences:  A063016 A063017 A063018 * A063020 A063021 A063022

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard, Jul 05 2001

STATUS

approved

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Last modified December 8 09:32 EST 2019. Contains 329862 sequences. (Running on oeis4.)