A325585 G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1 + 5*x)^n - A(x))^(n+1), where A(0) = 0.

1, 9, 26, 224, 2476, 23224, 287376, 3757374, 52257501, 788478999, 12610776876, 212979510624, 3790898971876, 70777961166874, 1381742116784376, 28138911700128124, 596359893046340626, 13125567196055049999, 299467375895770156251, 7070998303069778171874, 172523963169549496984376, 4343634679080455046328124, 112703773700255237721093751, 3010180318995682243232265624, 82668711264219381166762578126

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} x^n * ((1 + 5*x)^n - A(x))^(n+1).

(2) 1 + x = Sum_{n>=0} x^n * (1 + 5*x)^(n*(n-1)) / (1 + x*(1 + 5*x)^n*A(x))^(n+1).

FORMULA FOR TERMS.

a(n) = (-1)^n (mod 5) for n >= 0.

Example

G.f.: A(x) = x + 9*x^2 + 26*x^3 + 224*x^4 + 2476*x^5 + 23224*x^6 + 287376*x^7 + 3757374*x^8 + 52257501*x^9 + 788478999*x^10 + 12610776876*x^11 + ...
such that
1 = (1 - A(x)) + x*((1+5*x) - A(x))^2 + x^2*((1+5*x)^2 - A(x))^3 + x^3*((1+5*x)^3 - A(x))^4 + x^4*((1+5*x)^4 - A(x))^5 + x^5*((1+5*x)^5 - A(x))^6 + x^6*((1+5*x)^6 - A(x))^7 + ...

Mathematica

a[n_] := Module[{A}, A={1}; For[i=1, i <= n, i++, AppendTo[A, 0]; A[[-1]] = Coefficient[Sum[x^m*((1 + 5*x + x*O[x]^Length[A])^m - x*(A.x^Range[0, Length[A]-1]))^(m+1), {m, 0, Length[A]}], x, Length[A]]]; A[[n+1]]];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, May 11 2019, from PARI *)

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, x^m*((1 + 5*x +x*O(x^#A))^m - x*Ser(A))^(m+1) ), #A); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A307940, A325582, A325583, A325584.

Sequence in context: A372668 A084813 A322764 * A056409 A056399 A256927

Adjacent sequences: A325582 A325583 A325584 * A325586 A325587 A325588

Keyword

nonn

Author

Paul D. Hanna, May 11 2019

Status

approved