A375448 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^2 for n >= 1.
1, 3, 6, 33, 357, 5283, 96534, 2067312, 50345955, 1367512761, 40875976152, 1331343423234, 46892513148468, 1775323414999818, 71885746640828286, 3100014000785085216, 141857882269044077865, 6866221878372182554395, 350521791594556907681202, 18824690900373744731703396
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 3*x + 6*x^2 + 33*x^3 + 357*x^4 + 5283*x^5 + 96534*x^6 + 2067312*x^7 + 50345955*x^8 + 1367512761*x^9 + 40875976152*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, 3, 6, 33, 357, 5283, 96534, ...]; n=2: [1, 6, 21, 102, 948, 13104, 230139, ...]; n=3: [1, 9, 45, 234, 1935, 24678, 414234, ...]; n=4: [1, 12, 78, 456, 3561, 41868, 667746, ...]; n=5: [1, 15, 120, 795, 6150, 67428, 1017540, ...]; n=6: [1, 18, 171, 1278, 10107, 105246, 1501578, ...]; n=7: [1, 21, 231, 1932, 15918, 160587, 2172807, ...]; ... in which the sum of the first n coefficients in A(x/n)^n equals n^2, as illustrated by 1 = 1; 4 = 1 + 6/2; 9 = 1 + 9/3 + 45/3^2; 16 = 1 + 12/4 + 78/4^2 + 456/4^3; 25 = 1 + 15/5 + 120/5^2 + 795/5^3 + 6150/5^4; 36 = 1 + 18/6 + 171/6^2 + 1278/6^3 + 10107/6^4 + 105246/6^5; 49 = 1 + 21/7 + 231/7^2 + 1932/7^3 + 15918/7^4 + 160587/7^5 + 2172807/7^6; ... RELATED SERIES. Let B(x) be the series reversion of x/A(x), B(x/A(x)) = x, then B(x) = x + 3*x^2 + 15*x^3 + 114*x^4 + 1230*x^5 + 17541*x^6 + 310401*x^7 + 6502368*x^8 + ... + A375449(n)*x^n + ... Further, let C(x) = x*B'(x)/(1 - B(x)) = x + 7*x^2 + 55*x^3 + 547*x^4 + 7081*x^5 + 116821*x^6 + 2351497*x^7 + 55390315*x^8 + ... 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, 7 = 1 + 6, 55 = 1 + 9 + 45, 547 = 1 + 12 + 78 + 456, 7081 = 1 + 15 + 120 + 795 + 6150, etc.
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^2 - sum(k=1, #V, V[k]) )*m^(m-2) ); H=A; 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^2 for n >= 1, where B(x/A(x)) = x and B(x) is the g.f. of A375449.
a(n) ~ c * n^(n+1), where c = 0.5698891168602..., conjecture: c = (exp(1)-1)/exp(3*exp(-1)). - Vaclav Kotesovec, Sep 13 2024