This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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. 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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