A132624 - OEIS

%I #22 Jan 08 2025 13:10:57

%S 1,1,3,14,87,669,6098,64050,759817,10028799,145575337,2302441248,

%T 39377544316,723627151168,14212023123570,296941929433826,

%U 6573946153123597,153673571064191583,3781352342496043197,97672909528404096334,2641852466110908004319

%N Column 0 of triangle A132623.

%C Triangle T=A132623 is generated by sums of matrix powers of itself such that: T(n,k) = Sum_{j=1..n-k-1} [T^j](n-1,k) with T(n+1,n) = n+1 and T(n,n)=0 for n>=0.

%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

%H Jun Yan, <a href="https://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv:2404.07958 [math.CO], 2024. See pp. 11, 31, 34.

%H Jun Yan, <a href="https://arxiv.org/abs/2501.01152">Lattice paths enumerations weighted by ascent lengths</a>, arXiv:2501.01152 [math.CO], 2025. See p. 16.

%F G.f.: x = Sum_{n>=1} a(n) * x^n*(1-x)^n / Product_{k=1..n-1} (1 + k*x).

%e G.f.: x = 1*x*(1-x) + 1*x^2*(1-x)^2/(1+x) + 3*x^3*(1-x)^3/((1+x)*(1+2*x)) + 14*x^4*(1-x)^4/((1+x)*(1+2*x)*(1+3*x)) + 87*x^5*(1-x)^5/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) +...

%t a[1] = 1; a[n_] := a[n] = SeriesCoefficient[x - Sum[a[k]*x^k*(1 - x)^k/ Product[1 + j*x + O[x]^n, {j, 0, k-1}], {k, 1, n-1}], {x, 0, n}];

%t Array[a, 21] (* _Jean-François Alcover_, Jul 26 2018, from PARI *)

%o (PARI) {a(n)=if(n<1, 0, polcoeff(x-sum(k=1, n-1, a(k)*x^k*(1-x)^k/prod(j=0, k-1, 1+j*x+x*O(x^n))), n))}

%o for(n=1,20,print1(a(n),", "))

%Y Cf. A132623, A208676, A208677, A208678.

%K nonn,changed

%O 1,3

%A _Paul D. Hanna_, Aug 25 2007