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A126470 Triangle, read by rows, where row n lists coefficients of q in F(n,q) that satisfies: F(n,q) = Sum_{k=0..n-1} C(n-1,k)*F(k,q)*F(n-k-1,q)*q^k for n>0, with F(0,q) = 1.

%I A126470 #33 Jul 26 2018 03:35:29
%S A126470 1,1,1,1,1,3,1,1,1,6,7,5,3,1,1,1,10,25,25,26,11,12,5,3,1,1,1,15,65,
%T A126470 110,136,117,92,70,43,32,17,12,5,3,1,1,1,21,140,385,616,784,694,687,
%U A126470 478,411,255,222,127,91,50,39,17,12,5,3,1,1,1,28,266,1106,2471,4032,4887,5189
%N A126470 Triangle, read by rows, where row n lists coefficients of q in F(n,q) that satisfies: F(n,q) = Sum_{k=0..n-1} C(n-1,k)*F(k,q)*F(n-k-1,q)*q^k for n>0, with F(0,q) = 1.
%C A126470 Row sums equal the factorials: F(n,1) = n!.
%C A126470 Limit of reversed rows equals A126471. Largest term in rows equal A126472.
%H A126470 Paul D. Hanna, <a href="/A126470/b126470.txt">Rows n = 0..40, flattened.</a>
%H A126470 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 A126470 From _Paul D. Hanna_, Oct 04 2008: (Start)
%F A126470 E.g.f. satisfies: A(x,q) = exp( Integral A(q*x,q) dx ); further,
%F A126470 A(q^n*x,q) = exp( q^n * Integral A(q^(n+1)*x,q) dx ) for n>=0, where A(x,q) = Sum_{n>=0} x^n*[Sum_{k=0..n(n-1)/2} T(n,k)*q^k]/n!. (End)
%F A126470 E.g.f. satisfies: d/dx A(x,q) = A(x,q) * A(q*x,q) with A(0,q)=1; i.e., the logarithmic derivative of A(x,q) with respect to x equals A(q*x,q). - _Paul D. Hanna_, Oct 04 2008
%e A126470 Number of terms in row n is: n*(n-1)/2 + 1.
%e A126470 Row functions B(n,q) begin:
%e A126470 F(0,q) = F(1,q) = 1;
%e A126470 F(2,q) = 1 + q;
%e A126470 F(3,q) = 1 + 3*q + q^2 + q^3;
%e A126470 F(4,q) = 1 + 6*q + 7*q^2 + 5*q^3 + 3*q^4 + q^5 + q^6.
%e A126470 Triangle begins:
%e A126470 1;
%e A126470 1;
%e A126470 1, 1;
%e A126470 1, 3, 1, 1;
%e A126470 1, 6, 7, 5, 3, 1, 1;
%e A126470 1, 10, 25, 25, 26, 11, 12, 5, 3, 1, 1;
%e A126470 1, 15, 65, 110, 136, 117, 92, 70, 43, 32, 17, 12, 5, 3, 1, 1;
%e A126470 1, 21, 140, 385, 616, 784, 694, 687, 478, 411, 255, 222, 127, 91, 50, 39, 17, 12, 5, 3, 1, 1;
%e A126470 1, 28, 266, 1106, 2471, 4032, 4887, 5189, 4832, 4240, 3426, 2658, 2143, 1534, 1143, 790, 575, 351, 262, 151, 99, 58, 39, 17, 12, 5, 3, 1, 1;
%e A126470 1, 36, 462, 2730, 8589, 17892, 28519, 35613, 40639, 39200, 37934, 31508, 28076, 21570, 18288, 13451, 11009, 7747, 6120, 4089, 3106, 2056, 1530, 943, 683, 396, 289, 160, 108, 58, 39, 17, 12, 5, 3, 1, 1;
%e A126470 1, 45, 750, 6000, 25977, 70497, 141499, 220500, 291877, 336945, 357638, 347396, 323795, 288162, 247473, 207630, 170336, 139565, 109967, 87581, 66534, 51411, 37845, 28948, 20626, 15284, 10727, 7810, 5169, 3731, 2446, 1700, 1063, 733, 426, 299, 170, 108, 58, 39, 17, 12, 5, 3, 1, 1;
%e A126470 ...
%e A126470 E.g.f.: A(x,q) = 1 + x + x^2*(1+q)/2! + x^3*(1+3*q+q^2+q^3)/3! +...
%e A126470 where A(x,q) = exp( Integral A(q*x,q) dx ),
%e A126470 A(q*x,q) = exp( q * Integral A(q^2*x,q) dx ),
%e A126470 A(q^2*x,q) = exp( q^2 * Integral A(q^3*x,q) dx ), ...
%e A126470 A(q^n*x,q) = exp( q^n * Integral A(q^(n+1)*x,q) dx ) for n>=0.
%e A126470 Here the Integral is always in the limits 0..x.
%t A126470 F[0, _] = 1; F[n_, q_] := F[n, q] = Sum[Binomial[n-1, k] F[k, q] F[n-k-1, q] q^k, {k, 0, n-1}];
%t A126470 row[n_] := CoefficientList[F[n, q], q];
%t A126470 Table[row[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Jul 26 2018 *)
%o A126470 (PARI) F(n,q)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*F(k,q)*F(n-k-1,q)*q^k))
%o A126470 {T(n,k)=Vec(F(n,q)+O(q^(n*(n-1)/2+1)))[k+1]}
%o A126470 for(n=0,10,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print(""))
%o A126470 (PARI) T(n,k)=local(A=vector(n+2, j, 1+j*x)); for(i=0, n+1, for(j=0, n, m=n+1-j; A[m]=exp(q^(m-1)*intformal(A[m+1]+x*O(x^n))))); polcoeff(n!*polcoeff(A[1], n, x),k,q) \\ From _Paul D. Hanna_, Oct 04 2008
%o A126470 for(n=0,10,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print(""))
%o A126470 (PARI) /* Faster to use: A(x,q) = 1 + Integral A(x,q)*A(qx,q) dx */
%o A126470 {T(n,k)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+intformal(A*subst(A,x,q*x))); polcoeff(n!*polcoeff(A,n,x),k,q)}
%o A126470 for(n=0,10,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print()) \\ _Paul D. Hanna_, Oct 04 2008
%Y A126470 Cf. A126471, A126472; Bell number variant: A126347.
%K A126470 nonn,tabl
%O A126470 0,6
%A A126470 _Paul D. Hanna_, Dec 31 2006