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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; row sums equal the factorials: F(n,1) = n!. 5

%I

%S 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 110,136,117,92,70,43,32,17,12,5,3,1,1,1,21,140,385,616,784,694,687,

%U 478,411,255,222,127,91,50,39,17,12,5,3,1,1,1,28,266,1106,2471,4032,4887,5189

%N 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; row sums equal the factorials: F(n,1) = n!.

%C Limit of reversed rows equals A126471. Largest term in rows equal A126472.

%H Paul D. Hanna, <a href="/A126470/b126470.txt">Rows n = 0..40, flattened.</a>

%F Contribution from _Paul D. Hanna_, Oct 04 2008: (Start)

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

%F 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 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 Number of terms in row n is: n*(n-1)/2 + 1.

%e Row functions B(n,q) begin:

%e F(0,q) = F(1,q) = 1;

%e F(1,q) = 1 + q;

%e F(2,q) = 1 + 3*q + q^2 + q^3;

%e F(3,q) = 1 + 6*q + 7*q^2 + 5*q^3 + 3*q^4 + q^5 + q^6.

%e Triangle begins:

%e 1;

%e 1;

%e 1, 1;

%e 1, 3, 1, 1;

%e 1, 6, 7, 5, 3, 1, 1;

%e 1, 10, 25, 25, 26, 11, 12, 5, 3, 1, 1;

%e 1, 15, 65, 110, 136, 117, 92, 70, 43, 32, 17, 12, 5, 3, 1, 1;

%e 1, 21, 140, 385, 616, 784, 694, 687, 478, 411, 255, 222, 127, 91, 50, 39, 17, 12, 5, 3, 1, 1; ...

%e Contribution from _Paul D. Hanna_, Oct 04 2008: (Start)

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

%e where A(x,q) = exp( Integral A(q*x,q) dx ),

%e A(q*x,q) = exp( q * Integral A(q^2*x,q) dx ),

%e A(q^2*x,q) = exp( q^2 * Integral A(q^3*x,q) dx ), ...

%e A(q^n*x,q) = exp( q^n * Integral A(q^(n+1)*x,q) dx ) for n>=0.

%e Here the operation 'Integral' does not introduce the constant of integration. (End)

%o (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 {T(n,k)=Vec(F(n,q)+O(q^(n*(n-1)/2+1)))[k+1]}

%o for(n=0,10,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print(""))

%o (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 for(n=0,10,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print(""))

%o (PARI) /* Faster to use: A(x,q) = 1 + Integral A(x,q)*A(qx,q) dx */

%o {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)} \\ From _Paul D. Hanna_, Oct 04 2008

%o for(n=0,10,for(k=0,n*(n-1)/2,print1(T(n,k),", "));print(""))

%Y Cf. A126471, A126472; Bell number variant: A126347.

%K nonn,tabl

%O 0,6

%A _Paul D. Hanna_, Dec 31 2006

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