A319946 G.f. A(x) satisfies: [x^n] 1/(1 - (n+1)*x) / A(x)^2 = 0 for n >= 1.

1, 1, 3, 17, 149, 1779, 26745, 481947, 10079457, 239266403, 6343381625, 185548532427, 5931292679587, 205618230689389, 7681464906167799, 307596969286936725, 13142997272755615869, 596844383184584449467, 28705887156859657764753, 1457749861777513666057923, 77946590565516834838055775, 4377554664749646483714085785

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

a(n) ~ c * n^n, where c = 0.859140914229... - Vaclav Kotesovec, Oct 06 2020

Example

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 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in 1/(1 - (n+1)*x) / A(x)^2 begins:
n=1: [1, -1, -4, -24, -224, -2840, -44700, -832636, -17836768, ...];
n=2: [1, 0, -3, -26, -252, -3120, -48100, -884136, -18772404, ...];
n=3: [1, 1, 0, -20, -260, -3396, -52048, -944080, -19836372, ...];
n=4: [1, 2, 5, 0, -200, -3416, -55524, -1010032, -21044260, ...];
n=5: [1, 3, 12, 40, 0, -2616, -54940, -1062636, -22317312, ...];
n=6: [1, 4, 21, 106, 436, 0, -41860, -1039096, -23238708, ...];
n=7: [1, 5, 32, 204, 1228, 5980, 0, -787936, -22519684, ...];
n=8: [1, 6, 45, 340, 2520, 17544, 98492, 0, -17004132, ...];
n=9: [1, 7, 60, 520, 4480, 37704, 297476, 1889348, 0, ...];
n=10:[1, 8, 77, 750, 7300, 70384, 661980, 5831864, 41314508, 0, ...]; ...
in which the coefficient of x^(n-1) in row n forms a diagonal of zeros after an initial '1'.
RELATED SERIES.
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 + ...

Prog

(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]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A319945.

Sequence in context: A080253 A234289 A009813 * A213507 A305471 A368597

Adjacent sequences: A319943 A319944 A319945 * A319947 A319948 A319949

Keyword

nonn

Author

Paul D. Hanna, Oct 11 2018

Status

approved