A132624 Column 0 of triangle A132623.

1, 1, 3, 14, 87, 669, 6098, 64050, 759817, 10028799, 145575337, 2302441248, 39377544316, 723627151168, 14212023123570, 296941929433826, 6573946153123597, 153673571064191583, 3781352342496043197, 97672909528404096334, 2641852466110908004319

Offset

1,3

Comments

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.

Links

Table of n, a(n) for n=1..21.

J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See pp. 11, 31, 34.

Formula

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

Example

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)) +...

Mathematica

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}];
Array[a, 21] (* Jean-François Alcover, Jul 26 2018, from PARI *)

Prog

(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))}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A132623, A208676, A208677, A208678.

Sequence in context: A091102 A323771 A325140 * A121587 A332256 A335849

Adjacent sequences: A132621 A132622 A132623 * A132625 A132626 A132627

Keyword

nonn,changed

Author

Paul D. Hanna, Aug 25 2007

Status

approved