cp's OEIS Frontend

A375457 Expansion of the g.f. A(x) with the property that the sum of the first n coefficients in A(x/n)^n equals n for n >= 1.

%I A375457 #32 Sep 13 2024 07:00:09
%S A375457 1,1,2,11,105,1375,22390,430954,9512029,235992263,6488607220,
%T A375457 195627162152,6414053158664,227170447034030,8643069830739980,
%U A375457 351580969750713450,15228097928340597681,699791999466718937425,34010355409897760336176,1743142054929355666550574,93975675621720312817066020
%N A375457 Expansion of the g.f. A(x) with the property that the sum of the first n coefficients in A(x/n)^n equals n for n >= 1.
%H A375457 Paul D. Hanna, <a href="/A375457/b375457.txt">Table of n, a(n) for n = 0..300</a>
%F A375457 G.f. A(x) satisfies [x^n] x*B'(x/n) / (1 - n*B(x/n)) = n for n >= 1, where B(x/A(x)) = x and B(x) is the g.f. of A375452.
%F A375457 a(n) ~ c * n^n, where c = 1.189395759976..., conjecture: c = (exp(1)-1)/exp(exp(-1)). - _Vaclav Kotesovec_, Sep 13 2024
%e A375457 G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 105*x^4 + 1375*x^5 + 22390*x^6 + 430954*x^7 + 9512029*x^8 + 235992263*x^9 + 6488607220*x^10 + ...
%e A375457 The defining property of g.f. A(x) is described below.
%e A375457 The table of coefficients in A(x)^n begins:
%e A375457   n=1: [1, 1,  2,  11,  105,  1375,  22390, ...];
%e A375457   n=2: [1, 2,  5,  26,  236,  3004,  48071, ...];
%e A375457   n=3: [1, 3,  9,  46,  399,  4932,  77498, ...];
%e A375457   n=4: [1, 4, 14,  72,  601,  7212, 111194, ...];
%e A375457   n=5: [1, 5, 20, 105,  850,  9906, 149760, ...];
%e A375457   n=6: [1, 6, 27, 146, 1155, 13086, 193886, ...];
%e A375457   n=7: [1, 7, 35, 196, 1526, 16835, 244363, ...];
%e A375457   ...
%e A375457 in which the sum of the first n coefficients in A(x/n)^n equals n, as illustrated by
%e A375457   1 = 1;
%e A375457   2 = 1 + 2/2;
%e A375457   3 = 1 + 3/3 + 9/3^2;
%e A375457   4 = 1 + 4/4 + 14/4^2 + 72/4^3;
%e A375457   5 = 1 + 5/5 + 20/5^2 + 105/5^3 + 850/5^4;
%e A375457   6 = 1 + 6/6 + 27/6^2 + 146/6^3 + 1155/6^4 + 13086/6^5;
%e A375457   7 = 1 + 7/7 + 35/7^2 + 196/7^3 + 1526/7^4 + 16835/7^5 + 244363/7^6;
%e A375457   ...
%e A375457 RELATED SERIES.
%e A375457 Let B(x) be the series reversion of x/A(x), B(x/A(x)) = x, then
%e A375457 B(x) = x + x^2 + 3*x^3 + 18*x^4 + 170*x^5 + 2181*x^6 + 34909*x^7 + 663152*x^8 + 14493060*x^9 + ... + A375452(n)*x^n + ...
%e A375457 Further, let C(x) = x*B'(x)/(1 - B(x)) = x + 3*x^2 + 13*x^3 + 91*x^4 + 981*x^5 + 14421*x^6 + 262963*x^7 + 5630843*x^8 + 137203969*x^9 + ...
%e A375457 then the coefficient of x^n in C(x) equals the sum of the initial n terms of A(x)^n for n >= 1; 1 = 1, 3 = 1 + 2, 13 = 1 + 3 + 9, 91 = 1 + 4 + 14 + 72, 981 = 1 + 5 + 20 + 105 + 850, etc.
%e A375457 The logarithmic derivative of g.f. A(x) begins
%e A375457 A(x)'/A(x) = 1 + 3*x + 28*x^2 + 375*x^3 + 6306*x^4 + 125286*x^5 + 2845200*x^6 + 72355095*x^7 + 2031897160*x^8 + 62371350558*x^9 + 2076430998588*x^10 + ...
%e A375457 Notice that the coefficient of x^n in A(x)'/A(x) appears to be divisible by (n+2) for n > 0.
%o A375457 (PARI) {a(n) = my(A=[1],m,V); for(i=0,n, A = concat(A,0); m=#A; V=Vec( subst(Ser(A)^m, x, x/m) );
%o A375457 A[m] = (m - sum(k=1,#V,V[k]) )*m^(m-2) ); A[n+1]}
%o A375457 for(n=0,20,print1(a(n),", "))
%Y A375457 Cf. A375452, A375448, A088358.
%K A375457 nonn
%O A375457 0,3
%A A375457 _Paul D. Hanna_, Sep 08 2024