A306089 G.f. A(x) satisfies: Sum_{n>=0} (-1)^n * Product_{k=1..n} x^(n+1-k) + (-A(x))^k = 1.

1, 1, 1, 1, 1, 6, 21, 68, 186, 495, 1335, 3744, 10870, 32120, 95565, 284830, 850580, 2548436, 7669604, 23192434, 70443076, 214768128, 656857897, 2014416494, 6192794179, 19081689920, 58923909932, 182331403224, 565289067360, 1755737915942, 5462257817753, 17019938788706, 53109742992332, 165952503650622, 519222849063545, 1626498619326355, 5100995860701418

Offset

1,6

Links

Paul D. Hanna, Table of n, a(n) for n = 1..100

Formula

G.f. A(x) satisfies: A(-A(-x)) = x.

Example

G.f.: A(x) = x + x^2 + x^3 + x^4 + x^5 + 6*x^6 + 21*x^7 + 68*x^8 + 186*x^9 + 495*x^10 + 1335*x^11 + 3744*x^12 + 10870*x^13 + 32120*x^14 + 95565*x^15 + ...
such that
1  =  1  -  (x - A(x))  +  (x + A(x)^2)*(x^2 - A(x))  -  (x - A(x)^3)*(x^2 + A(x)^2)*(x^3 - A(x))  +  (x + A(x)^4)*(x^2 - A(x)^3)*(x^3 + A(x)^2)*(x^4 - A(x))  -  (x - A(x)^5)*(x^2 + A(x)^4)*(x^3 - A(x)^3)*(x^4 + A(x)^2)*(x^5 - A(x)) + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = -Vec( sum(m=0, #A, (-1)^m * prod(k=1, m, x^(m+1-k) + (-x)^k*Ser(A)^k ) ) )[#A+1]); A[n]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A306088, A306090.

Sequence in context: A134931 A119103 A180795 * A107653 A123653 A375297

Adjacent sequences: A306086 A306087 A306088 * A306090 A306091 A306092

Keyword

nonn

Author

Paul D. Hanna, Jun 21 2018

Status

approved