A319946 - OEIS

%I #8 Oct 06 2020 03:09:03

%S 1,1,3,17,149,1779,26745,481947,10079457,239266403,6343381625,

%T 185548532427,5931292679587,205618230689389,7681464906167799,

%U 307596969286936725,13142997272755615869,596844383184584449467,28705887156859657764753,1457749861777513666057923,77946590565516834838055775,4377554664749646483714085785

%N G.f. A(x) satisfies: [x^n] 1/(1 - (n+1)*x) / A(x)^2 = 0 for n >= 1.

%H Paul D. Hanna, <a href="/A319946/b319946.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) ~ c * n^n, where c = 0.859140914229... - _Vaclav Kotesovec_, Oct 06 2020

%e G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 149*x^4 + 1779*x^5 + 26745*x^6 + 481947*x^7 + 10079457*x^8 + 239266403*x^9 + 6343381625*x^10 + ...

%e ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k in 1/(1 - (n+1)*x) / A(x)^2 begins:

%e n=1: [1, -1, -4, -24, -224, -2840, -44700, -832636, -17836768, ...];

%e n=2: [1, 0, -3, -26, -252, -3120, -48100, -884136, -18772404, ...];

%e n=3: [1, 1, 0, -20, -260, -3396, -52048, -944080, -19836372, ...];

%e n=4: [1, 2, 5, 0, -200, -3416, -55524, -1010032, -21044260, ...];

%e n=5: [1, 3, 12, 40, 0, -2616, -54940, -1062636, -22317312, ...];

%e n=6: [1, 4, 21, 106, 436, 0, -41860, -1039096, -23238708, ...];

%e n=7: [1, 5, 32, 204, 1228, 5980, 0, -787936, -22519684, ...];

%e n=8: [1, 6, 45, 340, 2520, 17544, 98492, 0, -17004132, ...];

%e n=9: [1, 7, 60, 520, 4480, 37704, 297476, 1889348, 0, ...];

%e n=10:[1, 8, 77, 750, 7300, 70384, 661980, 5831864, 41314508, 0, ...]; ...

%e in which the coefficient of x^(n-1) in row n forms a diagonal of zeros after an initial '1'.

%e RELATED SERIES.

%e A(x)^2 = 1 + 2*x + 7*x^2 + 40*x^3 + 341*x^4 + 3958*x^5 + 58231*x^6 + 1033124*x^7 + 21365965*x^8 + 503022874*x^9 + ... + A319945(n)*x^n + ...

%o (PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( 1/Ser(A)/(1-m*x +x^2*O(x^m)))[m] ); Vec(Ser(A)^(1/2))[n+1]}

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

%Y Cf. A319945.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 11 2018