%I A161881
%S A161881 1,1,2,8,46,333,2822,26884,280778,3162129,37962174,481796692,6424120440,
%T A161881 89561323131,1300606338522,19614272779492,306422062160964,
%U A161881 4948682216714809,82474329755007710,1416291364674413764
%N A161881 a(n) = Sum_{k=0..n-1} {[x^k] A(x)^(n-k)} * {[x^(n-k-1)} A(x)^(k+1)} for 
               n>0, with a(0)=1, where g.f. A(x) = Sum_{n>=0} a(n)*x^n.
%e A161881 G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 333*x^5 +...
%e A161881 Coefficients in powers of A(x) begin:
%e A161881 A^1: [1, 1, 2, 8, 46, 333, 2822, 26884, 280778, ...];
%e A161881 A^2: [1, 2, 5, 20, 112, 790, 6558, 61480, ...];
%e A161881 A^3: [1, 3, 9, 37, 204, 1407, 11450, 105627, ...];
%e A161881 A^4: [1, 4, 14, 60, 329, 2228, 17796, 161572, ...];
%e A161881 A^5: [1, 5, 20, 90, 495, 3306, 25960, 232050, ...];
%e A161881 A^6: [1, 6, 27, 128, 711, 4704, 36383, 320376, ...];
%e A161881 A^7: [1, 7, 35, 175, 987, 6496, 49595, 430550, ...];
%e A161881 A^8: [1, 8, 44, 232, 1334, 8768, 66228, 567376, ...];
%e A161881 A^9: [1, 9, 54, 300, 1764, 11619, 87030, 736596, ...];
%e A161881 ...
%e A161881 The recurrence involves products of the antidiagonals:
%e A161881 a(1) = 1*1 =1,
%e A161881 a(2) = 1*1 + 1*1 = 2,
%e A161881 a(3) = 1*2 + 2*2 + 2*1 = 8,
%e A161881 a(4) = 1*8 + 3*5 + 5*3 + 8*1 = 46,
%e A161881 a(5) = 1*46 + 4*20 + 9*9 + 20*4 + 46*1 = 333,
%e A161881 a(6) = 1*333 + 5*112 + 14*37 + 37*14 + 112*5 + 333*1 = 2822.
%o A161881 (PARI) {a(n)=local(A=1 + sum(j=1, n-1, a(j)*x^j)+x*O(x^n));if(n==0, 1, 
               sum(k=0, n-1, polcoeff(A^(n-k), k)*polcoeff(A^(k+1), n-k-1)))}
%Y A161881 Sequence in context: A141117 A145844 A005840 this_sequence A088791 A111552 
               A128085
%Y A161881 Adjacent sequences: A161878 A161879 A161880 this_sequence A161882 A161883 
               A161884
%K A161881 nonn
%O A161881 0,3
%A A161881 Paul D. Hanna (pauldhanna(AT)juno.com), Jun 21 2009