A341373 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: [Sum_{n>=0} x^n/(1 - x^(n+1))]^2 = Sum_{n>=0} a(n)*x^n/(1 - x^(n+1))^2.

1, 2, 5, 6, 15, 6, 30, 16, 34, 14, 69, 12, 95, 20, 51, 46, 152, 8, 179, 34, 90, 34, 253, 26, 210, 46, 174, 52, 371, -30, 402, 128, 179, 36, 254, 40, 527, 78, 225, 58, 647, -56, 673, 140, 178, 38, 813, 46, 600, 32, 334, 104, 963, -24, 467, 180, 381, 26, 1169, -10, 1119, 120, 318, 236, 649

Offset

0,2

Links

Table of n, a(n) for n=0..64.

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

(1) [ Sum_{n>=0} x^n/(1 - x^(n+1)) ]^2 = Sum_{n>=0} a(n) * x^n / (1 - x^(n+1))^2.

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

Example

G.f.: A(x) = 1 + 2*x + 5*x^2 + 6*x^3 + 15*x^4 + 6*x^5 + 30*x^6 + 16*x^7 + 34*x^8 + 14*x^9 + 69*x^10 + 12*x^11 + 95*x^12 + 20*x^13 + 51*x^14 + 46*x^15 + ...
such that
D(x)^2 = 1/(1-x)^2 + 2*x/(1-x^2)^2 + 5*x^2/(1-x^3)^2 + 6*x^3/(1-x^4)^2 + 15*x^4/(1-x^5)^2 + 6*x^5/(1-x^6)^2 + 30*x^6/(1-x^7)^2 + ... + a(n)*x^n/(1-x^(n+1))^2 + ...
and
D(x)^2 = A(x) + 2*x*A(x^2) + 3*x^2*A(x^3) + 4*x^3*A(x^4) + 5*x^4*A(x^5) + 6*x^5*A(x^6) + 7*x^6*A(x^7) + ... + (n+1)*x^n*A(x^(n+1)) + ...
where
D(x)^2 = 1 + 4*x + 8*x^2 + 14*x^3 + 20*x^4 + 28*x^5 + 37*x^6 + 44*x^7 + 58*x^8 + 64*x^9 + 80*x^10 + 86*x^11 + 108*x^12 + ... + A055507(n+1)*x^n + ...
D(x) = 1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + 4*x^7 + 3*x^8 + 4*x^9 + 2*x^10 + 6*x^11 + 2*x^12 + ... + A000005(n+1)*x^n + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=0, #A, x^n/(1 - x^(n+1) +x*O(x^#A)) )^2 - sum(n=0, #A-1, A[n+1]*x^n/(1 - x^(n+1) + x*O(x^#A))^2 ), #A-1) ); A[n+1]}
for(n=0, 100, print1(a(n), ", "))

Crossrefs

Cf. A341374, A341375, A055507, A000005.

Sequence in context: A221472 A076624 A205385 * A363132 A287203 A291211

Adjacent sequences: A341370 A341371 A341372 * A341374 A341375 A341376

Keyword

sign

Author

Paul D. Hanna, Feb 11 2021

Status

approved