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%I #7 Jul 10 2013 00:57:41
%S 1,1,2,6,18,59,199,693,2465,8937,32880,122513,461331,1753037,6713758,
%T 25888515,100427611,391657635,1534674930,6039078032,23855475724,
%U 94561195899,376019415794,1499554893338,5996061250461,24034238674758,96554979145357,388711331661818,1567919554600690
%N Antidiagonal sums of triangle A227372.
%C The g.f. of triangle A227372 satisfies: G(x,q) = 1 + x*G(q*x,q)*G(x,q)^2.
%F G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2*B(x), where B(x) = 1 + x^2*B(x)^2*C(x), C(x) = 1 + x^3*C(x)^2*D(x), D(x) = 1 + x^4*D(x)^2*E(x), etc.
%e G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 18*x^4 + 59*x^5 + 199*x^6 + 693*x^7 +...
%e and equals a series involving row polynomials of triangle A227372:
%e A(x) = 1 + x*(1) + x^2*(2 + x) + x^3*(5 + 4*x + 2*x^2 + x^3)
%e + x^4*(14 + 15*x + 10*x^2 + 9*x^3 + 4*x^4 + 2*x^5 + x^6)
%e + x^5*(42 + 56*x + 45*x^2 + 43*x^3 + 34*x^4 + 23*x^5 + 14*x^6 + 9*x^7 + 4*x^8 + 2*x^9 + x^10) +...
%e RELATED SERIES.
%e G.f. A(x) = 1 + x*A(x)^2*B(x), where
%e B(x) = 1 + x^2 + 2*x^4 + x^5 + 5*x^6 + 4*x^7 + 16*x^8 + 16*x^9 + 52*x^10 +...
%e and B(x) = 1 + x^2*B(x)^2*C(x), where
%e C(x) = 1 + x^3 + 2*x^6 + x^7 + 5*x^9 + 4*x^10 + 2*x^11 + 15*x^12 +...
%e and C(x) = 1 + x^3*C(x)^2*D(x), where
%e D(x) = 1 + x^4 + 2*x^8 + x^9 + 5*x^12 + 4*x^13 + 2*x^14 + x^15 + 14*x^16 +...
%e and D(x) = 1 + x^4*D(x)^2*E(x), where
%e E(x) = 1 + x^5 + 2*x^10 + x^11 + 5*x^15 + 4*x^16 + 2*x^17 + x^18 + 14*x^20 +...
%e etc.
%o (PARI) /* From g.f. of A227372: G(x,q) = 1 + x*G(q*x,q)*G(x,q)^2: */
%o {a(n)=local(G=1);for(i=1,n,G=1+x*subst(G,x,q*x)*G^2 +x*O(x^n));polcoeff(sum(m=0,n,q^m*polcoeff(G,m,x))+q*O(q^n),n,q)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A227372, A227377.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 10 2013
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