A324963 G.f.: A(x) = Sum_{n>=0} x^n*((1+x)^n + sqrt(A(x)))^n / (1 + x*sqrt(A(x))*(1+x)^n)^(n+1).

1, 1, 2, 3, 8, 17, 61, 223, 1058, 5355, 29477, 171068, 1042319, 6646486, 44231512, 306592233, 2209107328, 16514226787, 127857487521, 1023541162850, 8459412252464, 72082183806141, 632429635304865, 5706629531494240, 52899935984638147, 503266172469569926, 4909051455794089278, 49053813732870894847, 501726197168709837165, 5248631560430224087649, 56117483960904047993635, 612815474656922971170469

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f.: A(x) = Sum_{n>=0} x^n*((1+x)^n + sqrt(A(x)))^n / (1 + x*sqrt(A(x))*(1+x)^n)^(n+1).

G.f.: A(x) = Sum_{n>=0} x^n*((1+x)^n - sqrt(A(x)))^n / (1 - x*sqrt(A(x))*(1+x)^n)^(n+1).

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 17*x^5 + 61*x^6 + 223*x^7 + 1058*x^8 + 5355*x^9 + 29477*x^10 + 171068*x^11 + 1042319*x^12 + ...
Let B = sqrt(A(x)) then
A(x) = 1/(1 + x*B) + x*((1+x) + B)/(1 + x*B*(1+x))^2 + x^2*((1+x)^2 + B)^2/(1 + x*B*(1+x)^2)^3 + x^3*((1+x)^3 + B)^3/(1 + x*B*(1+x)^3)^4 + x^4*((1+x)^4 + B)^4/(1 + x*B*(1+x)^4)^5 + x^5*((1+x)^5 + B)^5/(1 + x*B*(1+x)^5)^6 + ...
and
A(x) = 1/(1 - x*B) + x*((1+x) - B)/(1 - x*B*(1+x))^2 + x^2*((1+x)^2 - B)^2/(1 - x*B*(1+x)^2)^3 + x^3*((1+x)^3 - B)^3/(1 - x*B*(1+x)^3)^4 + x^4*((1+x)^4 - B)^4/(1 - x*B*(1+x)^4)^5 + x^5*((1+x)^5 - B)^5/(1 - x*B*(1+x)^5)^6 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n,
A = Vec( sum(m=0, #A, x^m*((1+x)^m + Ser(A)^(1/2))^m/(1 + x*Ser(A)^(1/2)*(1+x)^m)^(m+1)) )); A[n+1]}
for(n=0, 35, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n,
A = Vec( sum(m=0, #A, x^m*((1+x)^m - Ser(A)^(1/2))^m/(1 - x*Ser(A)^(1/2)*(1+x)^m)^(m+1)) )); A[n+1]}
for(n=0, 35, print1(a(n), ", "))

Crossrefs

Sequence in context: A345324 A290383 A099960 * A218090 A101182 A009207

Adjacent sequences: A324960 A324961 A324962 * A324964 A324965 A324966

Keyword

nonn

Author

Paul D. Hanna, Mar 23 2019

Status

approved