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A227373
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Antidiagonal sums of triangle A227372.
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2
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1, 1, 2, 6, 18, 59, 199, 693, 2465, 8937, 32880, 122513, 461331, 1753037, 6713758, 25888515, 100427611, 391657635, 1534674930, 6039078032, 23855475724, 94561195899, 376019415794, 1499554893338, 5996061250461, 24034238674758, 96554979145357, 388711331661818, 1567919554600690
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OFFSET
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0,3
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COMMENTS
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The g.f. of triangle A227372 satisfies: G(x,q) = 1 + x*G(q*x,q)*G(x,q)^2.
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LINKS
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FORMULA
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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.
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 18*x^4 + 59*x^5 + 199*x^6 + 693*x^7 +...
and equals a series involving row polynomials of triangle A227372:
A(x) = 1 + x*(1) + x^2*(2 + x) + x^3*(5 + 4*x + 2*x^2 + x^3)
+ x^4*(14 + 15*x + 10*x^2 + 9*x^3 + 4*x^4 + 2*x^5 + x^6)
+ 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) +...
RELATED SERIES.
G.f. A(x) = 1 + x*A(x)^2*B(x), where
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 +...
and B(x) = 1 + x^2*B(x)^2*C(x), where
C(x) = 1 + x^3 + 2*x^6 + x^7 + 5*x^9 + 4*x^10 + 2*x^11 + 15*x^12 +...
and C(x) = 1 + x^3*C(x)^2*D(x), where
D(x) = 1 + x^4 + 2*x^8 + x^9 + 5*x^12 + 4*x^13 + 2*x^14 + x^15 + 14*x^16 +...
and D(x) = 1 + x^4*D(x)^2*E(x), where
E(x) = 1 + x^5 + 2*x^10 + x^11 + 5*x^15 + 4*x^16 + 2*x^17 + x^18 + 14*x^20 +...
etc.
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PROG
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(PARI) /* From g.f. of A227372: G(x, q) = 1 + x*G(q*x, q)*G(x, q)^2: */
{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)}
for(n=0, 40, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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