|
%I #10 Jun 29 2013 19:43:06
%S 1,1,3,16,119,1116,12522,162863,2404103,39673456,723567188,
%T 14452217803,313777135454,7358812996185,185417876158777,
%U 4995923835850536,143354000575456167,4364618600823015848,140542706037271723068
%N a(n) = Sum_{k=0..n} C(n,k)* [x^(n-k)] A(x)^k for n>0, with a(0)=1.
%F G.f. A(x) satisfies [from _Paul D. Hanna_, Jun 29 2013]:
%F exp( Integral (A(x) - 1)/x dx ) = (1/x)*Series_Reversion(x/(1 + x*A(x)).
%e A(x) = 1 + x + 3*x^2 + 16*x^3 + 119*x^4 + 1116*x^5 + 12522*x^6 +...
%e From the table of n-th self-convolutions:
%e A^0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
%e A^1: [1, 1, 3, 16, 119, 1116, 12522, 162863, 2404103, 39673456, ...];
%e A^2: [1, 2, 7, 38, 279, 2566, 28246, 361274, 5258937, 85798608, ...];
%e A^3: [1, 3, 12, 67, 489, 4425, 47844, 601923, 8639097, ...];
%e A^4: [1, 4, 18, 104, 759, 6780, 72106, 892660, 12631271, ...];
%e A^5: [1, 5, 25, 150, 1100, 9731, 101955, 1242665, 17336065, ...];
%e A^6: [1, 6, 33, 206, 1524, 13392, 138463, 1662636, 22870059, ...];
%e illustrate a(n) = Sum_{k=0..n} C(n,k)*[x^(n-k)] A(x)^k by:
%e a(1) = 1*(0) + 1*(1) = 1;
%e a(2) = 1*(0) + 2*(1) + 1*(1) = 3;
%e a(3) = 1*(0) + 3*(3) + 3*(2) + 1*(1) = 16;
%e a(4) = 1*(0) + 4*(16) + 6*(7) + 4*(3) + 1*(1) = 119;
%e a(5) = 1*(0) + 5*(119) + 10*(38) + 10*(12) + 5*(4) + 1*(1) = 1116.
%e ALTERNATE FORMULA.
%e Define B(x) = (1/x)*Series_Reversion(x/(1 + x*A(x)),
%e then B(x) satisfies:
%e . B'(x)/B(x) = (A(x) - 1)/x;
%e . B(x) = 1 + x*B(x) * A(x*B(x));
%e . B( x/(1 + x*A(x)) ) = 1 + x*A(x).
%e Explicitly, B(x) begins:
%e B(x) = 1 + x + 2*x^2 + 7*x^3 + 37*x^4 + 265*x^5 + 2394*x^6 + 26033*x^7 +...
%e Note that
%e log(B(x)) = x + 3*x^2/2 + 16*x^3/3 + 119*x^4/4 + 1116*x^5/5 + 12522*x^6/6 +...
%o (PARI) {a(n)=local(A=1+sum(k=1,n-1,a(k)*x^k));if(n==0,1,sum(k=0,n,binomial(n,k)*polcoeff(A^k,n-k)))}
%Y Cf. A125223, A222658.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 24 2006
|
