A326275 G.f. A(x) satisfies: Sum_{n>=0} A(x)^((n+1)^2) * x^n = Sum_{n>=0} (1 + A(x)^(n+1))^n * x^n.

1, 1, 1, 3, 15, 85, 535, 3623, 25951, 194520, 1514905, 12198161, 101200455, 862904340, 7548194457, 67646260677, 620496791884, 5821278151196, 55827806529613, 547097673373322, 5476831304418641, 55993353247707012, 584508694490025552, 6228747205436059856, 67743826781449323262, 751784959486903813202

Offset

0,4

Links

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

Formula

G.f. A(x) allows the following sums to be equal:

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

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

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

Example

G.f. A(x) = 1 + x + x^2 + 3*x^3 + 15*x^4 + 85*x^5 + 535*x^6 + 3623*x^7 + 25951*x^8 + 194520*x^9 + 1514905*x^10 + 12198161*x^11 + 101200455*x^12 + ...
such that the following sums are equal
B(x) = A(x) + A(x)^4*x + A(x)^9*x^2 + A(x)^16*x^3 + A(x)^25*x^4 + A(x)^36*x^5 + A(x)^49*x^6 + ... + A(x)^((n+1)^2)*x^n + ...
and
B(x) = 1 + (1 + A(x)^2)*x + (1 + A(x)^3)^2*x^2 + (1 + A(x)^4)^3*x^3 + (1 + A(x)^5)^4*x^4 + ... + (1 + A(x)^(n+1))^n*x^n + ...
also
B(x) = 1/(1 - x) + A(x)^2*x/(1 - x*A(x))^2 + A(x)^6*x/(1 - x*A(x)^2)^3 + A(x)^12*x/(1 - x*A(x)^3)^4 + ... + A(x)^(n*(n+1))*x^n/(1 - x*A(x)^n)^(n+1) + ...
where
B(x) = 1 + 2*x + 6*x^2 + 23*x^3 + 105*x^4 + 545*x^5 + 3118*x^6 + 19261*x^7 + 126615*x^8 + 876553*x^9 + 6342647*x^10 + 47701975*x^11 + 371337731*x^12 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, (Ser(A)^(m+1) + 1)^m*x^m - Ser(A)^((m+1)^2)*x^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A326287, A326560, A326561, A326562.

Sequence in context: A317133 A182016 A127085 * A093615 A191148 A001931

Adjacent sequences: A326272 A326273 A326274 * A326276 A326277 A326278

Keyword

nonn

Author

Paul D. Hanna, Jun 28 2019

Status

approved