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A265942 G.f. A(x) satisfies: x = Sum_{n>=1} x^n * A(-n*x). 1

%I #25 Oct 01 2023 18:33:41

%S 1,1,1,2,10,118,3074,170402,19685482,4679048902,2269261320050,

%T 2232902808762146,4440206282955006346,17793903288234980266774,

%U 143417291429469295762696226,2321454058916677584825960932258,75384639189021086218383176211366826,4907135862070128733441621809087271706086,639945541949598669937210207492354528460950418,167124499527470730841323059848701978687632400092450

%N G.f. A(x) satisfies: x = Sum_{n>=1} x^n * A(-n*x).

%C a(n) ~ c * 2^(n*(n-1)/2), where c = 0.055207780612423352604210024412903589... - _Paul D. Hanna_, Oct 01 2023

%H Paul D. Hanna, <a href="/A265942/b265942.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = Sum_{k=0..n-1} (-1)^(n-k+1) * (n-k+1)^k * a(k), for n>0 with a(0)=1.

%F 0 = Sum_{k=0..n} (-1)^k * (n-k)^k * a(k), for n>1.

%F a(n) = 2 (mod 4) for n>2 [conjecture].

%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 10*x^4 + 118*x^5 + 3074*x^6 + 170402*x^7 + 19685482*x^8 + 4679048902*x^9 + 2269261320050*x^10 + ...

%e where

%e x = x*A(-x) + x^2*A(-2*x) + x^3*A(-3*x) + x^4*A(-4*x) + x^5*A(-5*x) + x^6*A(-6*x) + x^7*A(-7*x) + x^8*A(-8*x) + x^9*A(-9*x) + ...

%e The array of coefficients in A(-n*x) begins:

%e n=1: [1, -1, 1, -2, 10, -118, 3074, -170402, 19685482]

%e n=2: [1, -2, 4, -16, 160, -3776, 196736, -21811456, ...];

%e n=3: [1, -3, 9, -54, 810, -28674, 2240946, -372669174, ...];

%e n=4: [1, -4, 16, -128, 2560, -120832, 12591104, -2791866368, ...];

%e n=5: [1, -5, 25, -250, 6250, -368750, 48031250, -13312656250, ...];

%e n=6: [1, -6, 36, -432, 12960, -917568, 143420544, -47701654272, ...];

%e n=7: [1, -7, 49, -686, 24010, -1983226, 361653026, -140333374286, ...];

%e n=8: [1, -8, 64, -1024, 40960, -3866624, 805830656, -357358895104, ...]; ...

%e in which the antidiagonal sums yield [1,0,0,0,0,0,0,0,...].

%t nmax = 20; sol = {a[0] -> 1}; Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[x - Sum[x^k A[-k x], {k, 1, n}] + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];

%t sol /. Rule -> Set;

%t a /@ Range[0, nmax-1] (* _Jean-François Alcover_, Nov 03 2019 *)

%o (PARI) {a(n)=my(A=[1]);for(i=1,n,A=concat(A,0);A[#A]=(-1)^(#A)*Vec(sum(m=1,#A,subst(Ser(A),x,-m*x)*x^m))[#A]);A[n+1]}

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

%o (PARI) {a(n) = if(n==0,1, sum(k=0,n-1, (-1)^(n-k+1)*a(k)*(n-k+1)^k ) )}

%o for(n=0, 20, print1(a(n), ", "))

%o (PARI) /* Quick print of terms 0..30 */

%o {A=[1]; for(i=1, 30, A=concat(A, 0);

%o A[#A]=(-1)^(#A)*Vec(sum(n=1, #A, subst(Ser(A), x, -n*x)*x^n))[#A] ); A}

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jan 06 2016

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