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%I #20 May 21 2019 23:46:38
%S 1,1,2,8,28,128,616,3246,18277,109143,687012,4534864,31269292,
%T 224515730,1674200760,12937083460,103396000242,853266369103,
%U 7260005391139,63604089876714,573070009421688,5304309681566996,50386117123977631,490737093714045784,4896284341878168342,50004967889113301578,522348075672187803146,5576978585271447925911,60819137853681816353455,677034453647710270583612,7688697614189589138274128
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1+x)^n - A(x))^(n+1), where A(0) = 0.
%H Paul D. Hanna, <a href="/A307940/b307940.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} x^n * ((1+x)^n - A(x))^(n+1).
%F (2) 1 + x = Sum_{n>=0} x^n * (1+x)^(n*(n-1)) / (1 + x*(1+x)^n*A(x))^(n+1).
%F (3) 1 + x = Sum_{n>=0} x^n * (1-x)^(2*n) / ((1-x)^n - x*A(x/(1-x)))^n.
%F (4) 1/(1-x) = Sum_{n>=0} x^n * (1 - (1-x)^(n-1) * A(x/(1-x)))^n / (1-x)^(n^2).
%e G.f.: A(x) = x + x^2 + 2*x^3 + 8*x^4 + 28*x^5 + 128*x^6 + 616*x^7 + 3246*x^8 + 18277*x^9 + 109143*x^10 + 687012*x^11 + 4534864*x^12 + ...
%e such that
%e 1 = (1 - A(x)) + x*((1+x) - A(x))^2 + x^2*((1+x)^2 - A(x))^3 + x^3*((1+x)^3 - A(x))^4 + x^4*((1+x)^4 - A(x))^5 + x^5*((1+x)^5 - A(x))^6 + x^6*((1+x)^6 - A(x))^7 + ...
%e Expressed as a power series in x with A = A(x),
%e 1 = (1 - A) + x*(1 - 2*A + A^2) + x^2*(3 - 5*A + 3*A^2 - A^3) + x^3*(8 - 16*A + 12*A^2 - 4*A^3 + A^4) + x^4*(28 - 59*A + 49*A^2 - 22*A^3 + 5*A^4 - A^5) + x^5*(107 - 242*A + 225*A^2 - 112*A^3 + 35*A^4 - 6*A^5 + A^6) + x^6*(456 - 1096*A + 1101*A^2 - 619*A^3 + 215*A^4 - 51*A^5 + 7*A^6 - A^7) + x^7*(2119 - 5364*A + 5798*A^2 - 3556*A^3 + 1395*A^4 - 368*A^5 + 70*A^6 - 8*A^7 + A^8) + x^8*(10616 - 28215*A + 32433*A^2 - 21504*A^3 + 9246*A^4 - 2748*A^5 + 581*A^6 - 92*A^7 + 9*A^8 - A^9) + x^9*(56926 - 158050*A + 192030*A^2 - 136172*A^3 + 63420*A^4 - 20694*A^5 + 4914*A^6 - 864*A^7 + 117*A^8 - 10*A^9 + A^10) + ...
%e The coefficients in which form the triangle
%e [1, -1];
%e [1, -2, 1];
%e [3, -5, 3, -1];
%e [8, -16, 12, -4, 1];
%e [28, -59, 49, -22, 5, -1];
%e [107, -242, 225, -112, 35, -6, 1];
%e [456, -1096, 1101, -619, 215, -51, 7, -1];
%e [2119, -5364, 5798, -3556, 1395, -368, 70, -8, 1];
%e [10616, -28215, 32433, -21504, 9246, -2748, 581, -92, 9, -1];
%e [56926, -158050, 192030, -136172, 63420, -20694, 4914, -864, 117, -10, 1]; ...
%t m = 31; A[_] = 0; Do[A[x_] = Sum[x^n ((1+x)^n - A[x])^(n+1), {n, 1, k-1}] + O[x]^k // Normal, {k, m}]; CoefficientList[A[x], x] // Rest (* _Jean-François Alcover_, May 07 2019 *)
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); A[#A] = polcoeff( sum(m=0,#A, x^m*((1+x +x*O(x^#A))^m - x*Ser(A))^(m+1) ), #A););A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A307952, A307953, A307954, A307955, A325575.
%Y Cf. A325582, A325583, A325584, A325585.
%K nonn
%O 1,3
%A _Paul D. Hanna_, May 06 2019
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