A232433 - OEIS

%I #14 Oct 27 2018 06:19:32

%S 1,1,2,1,6,6,2,1,24,36,22,14,6,2,1,120,240,210,160,104,56,32,14,6,2,1,

%T 720,1800,2040,1830,1448,992,674,408,232,128,68,32,14,6,2,1,5040,

%U 15120,21000,21840,19824,15834,12144,8758,5904,3860,2442,1482,870,492,260,142,68,32,14,6,2,1

%N E.g.f. satisfies: A(x,q) = exp( Integral A(x,q)*A(q*x,q) dx ).

%H Paul D. Hanna, <a href="/A232433/b232433.txt">Table of n, a(n) for n = 0..9919; rows 0..39 in flattened triangle.</a>

%H Miguel A. Mendez, <a href="https://arxiv.org/abs/1610.03602">Combinatorial differential operators in: Faà di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees</a>, arXiv:1610.03602 [math.CO], 2016.

%F E.g.f. satisfies: d/dx A(x,q) = A(x,q)^2 * A(q*x,q).

%F Row sums equal the odd double factorials.

%F Limit of reversed rows yield A232434.

%e E.g.f.: A(x,q) = 1 + (1)*x + (2 + q)*x^2/2! + (6 + 6*q + 2*q^2 + q^3)*x^3/3!

%e + (24 + 36*q + 22*q^2 + 14*q^3 + 6*q^4 + 2*q^5 + q^6)*x^4/4!

%e + (120 + 240*q + 210*q^2 + 160*q^3 + 104*q^4 + 56*q^5 + 32*q^6 + 14*q^7 + 6*q^8 + 2*q^9 + q^10)*x^5/5! +...

%e The triangle of coefficients T(n,k) of x^n*q^k, for n>=0, k=0..n*(n-1)/2, in e.g.f. A(x,q) begins:

%e [1];

%e [1];

%e [2, 1];

%e [6, 6, 2, 1];

%e [24, 36, 22, 14, 6, 2, 1];

%e [120, 240, 210, 160, 104, 56, 32, 14, 6, 2, 1];

%e [720, 1800, 2040, 1830, 1448, 992, 674, 408, 232, 128, 68, 32, 14, 6, 2, 1];

%e [5040, 15120, 21000, 21840, 19824, 15834, 12144, 8758, 5904, 3860, 2442, 1482, 870, 492, 260, 142, 68, 32, 14, 6, 2, 1];

%e [40320, 141120, 231840, 275520, 280056, 251496, 212112, 170424, 129716, 95248, 67632, 46616, 31280, 20576, 13142, 8232, 5004, 2954, 1706, 966, 524, 276, 142, 68, 32, 14, 6, 2, 1]; ...

%e The limit of the reversed rows (A232434) begins:

%e [1, 2, 6, 14, 32, 68, 142, 276, 542, 1022, 1876, 3394, 6066, 10628, ...].

%t nmax = 8; A[_, _] = 0; Do[A[x_, q_] = Exp[Integrate[A[x, q] A[q x, q], x]] + O[x]^n // Normal // Simplify, {n, nmax}];

%t CoefficientList[#, q]& /@ (CoefficientList[A[x, q], x] Range[0, nmax-1]!) // Flatten (* _Jean-François Alcover_, Oct 27 2018 *)

%o (PARI) {T(n,k)=local(A=1+x);for(i=1,n,A=exp(intformal(A*subst(A,x,x*y +x*O(x^n)),x)));n!*polcoeff(polcoeff(A,n,x),k,y)}

%o for(n=0,12,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print(""))

%Y Cf. A232434, A126470, A001286, A075183.

%K nonn,tabf

%O 0,3

%A _Paul D. Hanna_, Nov 23 2013