A355351 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

1, 1, 4, 16, 60, 231, 920, 3819, 16365, 71792, 320219, 1446517, 6602975, 30415725, 141231704, 660431602, 3107519738, 14701758926, 69891556656, 333700223891, 1599475107712, 7693580712200, 37125486197570, 179675330190428, 871910824853956, 4241603521253775

Offset

0,3

Comments

a(n) = Sum_{k=0..n} A355350(n,k) for n >= 0.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..600

Formula

G.f. A(x) satisfies:

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

(2) x*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.

Example

G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 60*x^4 + 231*x^5 + 920*x^6 + 3819*x^7 + 16365*x^8 + 71792*x^9 + 320219*x^10 + 1446517*x^11 + ...
where
x = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ...
also,
x*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/A(x)) * ...
where P(x) is the partition function and begins
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...

Prog

(PARI) {a(n) = my(A=[1, 1], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*n+9));
A[#A] = -polcoeff( sum(m=-t, t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A355350, A355352, A355353, A355354, A355355, A355356, A355357.

Sequence in context: A051043 A123620 A234008 * A203153 A126929 A338531

Adjacent sequences: A355348 A355349 A355350 * A355352 A355353 A355354

Keyword

nonn

Author

Paul D. Hanna, Jun 29 2022

Status

approved