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.
1, 1, 2, 11, 105, 1375, 22390, 430954, 9512029, 235992263, 6488607220, 195627162152, 6414053158664, 227170447034030, 8643069830739980, 351580969750713450, 15228097928340597681, 699791999466718937425, 34010355409897760336176, 1743142054929355666550574, 93975675621720312817066020
Offset: 0
Keywords
Examples
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 + ... The defining property of g.f. A(x) is described below. The table of coefficients in A(x)^n begins: n=1: [1, 1, 2, 11, 105, 1375, 22390, ...]; n=2: [1, 2, 5, 26, 236, 3004, 48071, ...]; n=3: [1, 3, 9, 46, 399, 4932, 77498, ...]; n=4: [1, 4, 14, 72, 601, 7212, 111194, ...]; n=5: [1, 5, 20, 105, 850, 9906, 149760, ...]; n=6: [1, 6, 27, 146, 1155, 13086, 193886, ...]; n=7: [1, 7, 35, 196, 1526, 16835, 244363, ...]; ... in which the sum of the first n coefficients in A(x/n)^n equals n, as illustrated by 1 = 1; 2 = 1 + 2/2; 3 = 1 + 3/3 + 9/3^2; 4 = 1 + 4/4 + 14/4^2 + 72/4^3; 5 = 1 + 5/5 + 20/5^2 + 105/5^3 + 850/5^4; 6 = 1 + 6/6 + 27/6^2 + 146/6^3 + 1155/6^4 + 13086/6^5; 7 = 1 + 7/7 + 35/7^2 + 196/7^3 + 1526/7^4 + 16835/7^5 + 244363/7^6; ... RELATED SERIES. Let B(x) be the series reversion of x/A(x), B(x/A(x)) = x, then 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 + ... 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 + ... 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. The logarithmic derivative of g.f. A(x) begins 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 + ... Notice that the coefficient of x^n in A(x)'/A(x) appears to be divisible by (n+2) for n > 0.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
Programs
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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) ); A[m] = (m - sum(k=1,#V,V[k]) )*m^(m-2) ); A[n+1]} for(n=0,20,print1(a(n),", "))
Formula
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.
a(n) ~ c * n^n, where c = 1.189395759976..., conjecture: c = (exp(1)-1)/exp(exp(-1)). - Vaclav Kotesovec, Sep 13 2024