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Contribution from _From _Paul D. Hanna_, Apr 23 2010: (Start)

Contribution from _From _Paul D. Hanna_, May 04 2010: (Start)

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m = 25; A[_] = 0;

Do[A[x_] = 1 + Sum[x^k A[x]^(k^2) + O[x]^j, {k, 1, j}], {j, m}];

CoefficientList[A[x], x] (* Jean-François Alcover, Nov 05 2019 *)

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Paul D. Hanna, <a href="/A107595/b107595.txt">Table of n, a(n) for n = 0..200</a>

1, 1, 2, 7, 31, 158, 884, 5292, 33385, 219797, 1500449, 10573815, 76688602, 571232869, 4363912280, 34161879247, 273906591562, 2248935278231, 18909284838057, 162842178607893, 1436660527685476, 12988076148036405, 120345643023918566, 1143054910071718088, 11129160383826078389

G.f. A(x) = (1/x)*series-reversionSeries_Reversion(x/G107594F(x)) and thus A(x) = G107594F(x*A(x)) where G107594F(x) is the g.f. of A107594. G.f. A(x) = x/series-reversion(x*G107596(x)) and thus A(x) = G107596(x/A(x)) where G107596(x) is the g.f. of A107596.

G.f. A(x) = x/Series_Reversion(x*G(x)) and thus A(x) = G(x/A(x)) where G(x) is the g.f. of A107596.

A = 1/(1 - A*x/(1 - (A^3-A)*x/(1 - A^5*x/(1 - (A^7-A^3)*x/(1 - A^9*x/(1- (A^11-A^5)*x/(1 - A^13*x/(1 - (A^15-A^7)*x/(1 - ...)))))))))

A = Sum_{n>=0} x^n*A^n * Product_{k=1..n} (1 - x*A^(4k-3)) / (1 - x*A^(4k-1))

G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 158*x^5 + 884*x^6 + 5292*x^7 +...

Let A = g.f. A(x) then

= 1 + (x *(1 + x^2 + 2*x^3 2 + 7*x^4 3 + 31*x^5 4 + 158*x^5 + 884*x^6 +...)

+ (x^2 *(1 + 4*x^3 + 14*x^4 2 + 56*x^5 3 + 257*x^6 4 + 1312*x^5 +...)

+ (x^3 *(1 + 9*x^4 + 54*x^5 2 + 291*x^6 3 + 1557*x^7 4 + 8568*x^5 +..)

+ (x^4 *(1 + 16*x^5 + 152*x^6 2 + 1152*x^7 3 + 7836*x^4 +...) +...

+ x^5*(1 + 25*x + 350*x^2 + 3675*x^3 + 32625*x^4 +...)

+ x^6*(1 + 36*x + 702*x^2 + 9912*x^3 + 114201*x^4 +...) +...

for(n=0, 30, print1(a(n), ", "))

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Cf. A176719, A107590, A107594, A107596.

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