A232433 E.g.f. satisfies: A(x,q) = exp( Integral A(x,q)*A(q*x,q) dx ).

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, 720, 1800, 2040, 1830, 1448, 992, 674, 408, 232, 128, 68, 32, 14, 6, 2, 1, 5040, 15120, 21000, 21840, 19824, 15834, 12144, 8758, 5904, 3860, 2442, 1482, 870, 492, 260, 142, 68, 32, 14, 6, 2, 1

Offset: 0

Author

Paul D. Hanna, Nov 23 2013

Keywords

• nonn
• tabf

Examples

			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!
+ (24 + 36*q + 22*q^2 + 14*q^3 + 6*q^4 + 2*q^5 + q^6)*x^4/4!
+ (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! +...
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:
[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];
[720, 1800, 2040, 1830, 1448, 992, 674, 408, 232, 128, 68, 32, 14, 6, 2, 1];
[5040, 15120, 21000, 21840, 19824, 15834, 12144, 8758, 5904, 3860, 2442, 1482, 870, 492, 260, 142, 68, 32, 14, 6, 2, 1];
[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]; ...
The limit of the reversed rows (A232434) begins:
[1, 2, 6, 14, 32, 68, 142, 276, 542, 1022, 1876, 3394, 6066, 10628, ...].
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..9919; rows 0..39 in flattened triangle.
• Miguel A. Mendez, Combinatorial differential operators in: Faà di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees, arXiv:1610.03602 [math.CO], 2016.

Crossrefs

Cf. A232434, A126470, A001286, A075183.

Programs

Mathematica

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}];
CoefficientList[#, q]& /@ (CoefficientList[A[x, q], x] Range[0, nmax-1]!) // Flatten (* Jean-François Alcover, Oct 27 2018 *)

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

Formula

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

Row sums equal the odd double factorials.

Limit of reversed rows yield A232434.