A323679 a(n) equals the coefficient of x^(n*(n+1)) in Sum_{m>=0} x^m * (1 + x^m)^m / (1 + x^(m+1))^(m+1) for n >= 0.

1, 3, 9, 15, 79, 657, 2789, 9679, 50187, 122379, 911783, 7942511, 71320919, 292307479, 1254424307, 5649367163, 25471489371, 151196109585, 513167692755, 4381764642655, 29243835423341, 209688880294187, 1635709191538759, 9050289283302795, 40510463137400579, 159199144373303265, 672166886239228581, 3264163837045382109, 22816461252418921287, 152055680315368539533, 570344377866479916661, 3765597736734681752239, 26013973013767849104195

Offset

0,2

Comments

a(n) = A323557(n*(n+1)) for n >= 0.

Appears to consist of all the odd terms in A323557.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..180

Formula

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

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

a(n) = [x^(n*(n+1))] Sum_{m>=0} x^m * Sum_{k=0..m} binomial(m,k) * (x^m - x^k)^(m-k).

a(n) = [x^(n*(n+1))] Sum_{m>=0} x^m * Sum_{k=0..m} binomial(m,k) * (-1)^k * (x^m + x^k)^(m-k).

a(n) = [x^(n*(n+1))] Sum_{m>=0} x^m * Sum_{k=0..m} binomial(m,k) * (-1)^k * Sum_{j=0..m-k} binomial(m-k,j) * x^((m-k)*(m-j)).

Example

Given the g.f. of A323557, G(x) = Sum_{n>=0} x^n * (1 + x^n)^n / (1 + x^(n+1))^(n+1), i.e.,
G(x) = 1/(1 + x) + x*(1 + x)/(1 + x^2)^2 + x^2*(1 + x^2)^2/(1 + x^3)^3 + x^3*(1 + x^3)^3/(1 + x^4)^4 + x^4*(1 + x^4)^4/(1 + x^5)^5 + x^5*(1 + x^5)^5/(1 + x^6)^6 + x^6*(1 + x^6)^6/(1 + x^7)^7 + x^7*(1 + x^7)^7/(1 + x^8)^8 + ...
and writing G(x) as a power series in x starting as
G(x) = 1 + 3*x^2 - 2*x^3 + 2*x^4 + 9*x^6 - 14*x^7 + 8*x^8 + 12*x^10 - 12*x^11 + 15*x^12 - 52*x^13 + 76*x^14 - 36*x^15 + 2*x^16 + 50*x^18 - 104*x^19 + 79*x^20 - 140*x^21 + 324*x^22 - 276*x^23 + 128*x^24 - 144*x^25 + 118*x^26 - 28*x^27 + 72*x^28 - 336*x^29 + 657*x^30 + ... + A323557(n)*x^n + ...
then the odd coefficients of x^n in G(x), occurring at n = k*(k+1) for k>=0, form this sequence.
RELATED SEQUENCE.
The terms A323557(n*(n+3)) begin
[1, 2, 12, 50, 72, 142, 5346, 38338, 240902, 1369462, ..., A323678(n), ...].

Mathematica

nmax = 20; s = CoefficientList[Series[Sum[x^m*(1 + x^m)^m/(1 + x^(m + 1))^(m + 1), {m, 0, nmax*(nmax + 1)}], {x, 0, nmax*(nmax + 1)}], x]; Table[s[[n*(n + 1) + 1]], {n, 0, nmax}] (* Vaclav Kotesovec, Jul 08 2019 *)

Prog

(PARI) {A323557(n) = my(A=sum(m=0, n, x^m * (1 + x^m +x*O(x^n))^m/(1 + x^(m+1) +x*O(x^n))^(m+1) )); polcoeff(A, n)}
a(n) = A323557(n*(n+1))
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A323557, A323678, A323677.

Cf. A326286.

Sequence in context: A110960 A192165 A050869 * A038547 A348199 A242438

Adjacent sequences: A323676 A323677 A323678 * A323680 A323681 A323682

Keyword

nonn

Author

Paul D. Hanna, Feb 03 2019

Status

approved