A265269 E.g.f.: Sum_{n>=0} (x^n + y^n)^n / n! - Sum_{n>=0} y^(n^2) / n! at y=2.

1, 8, 192, 16396, 5242880, 6442453824, 30786325577728, 576460752306003968, 42501298345826806983744, 12379400392853802758900285440, 14278816360970775978458864905355264, 65334214448820184984967924794323967844352, 1187470080331358621040493926581979953470445191168, 85819750288489776068067433520417314295130321163120541696, 24682568359818090632324537738360257574741037984503809538441871360

Offset

1,2

Links

Table of n, a(n) for n=1..15.

Formula

a(n) = Sum_{d|n} 2^(d^2-n) * binomial(d, n/d) * n!/d! for n>=1.

Example

E.g.f.: A(x) = x + 8*x^2/2! + 192*x^3/3! + 16396*x^4/4! + 5242880*x^5/5! + 6442453824*x^6/6! + 30786325577728*x^7/7! + 576460752306003968*x^8/8! + ...
such that
A(x) = [(x + y) + (x^2 + y^2)^2/2! + (x^3 + y^3)^3/3! + (x^4 + y^4)^4/4! + (x^5 + y^5)^5/5! + (x^6 + y^6)^6/6! + (x^7 + y^7)^7/7! + ...]
  - [y + y^4/2! + y^9/3! + y^16/4! + y^25/5! + y^36/6! + y^49/7! + ...]
evaluated at y=2.
Equivalently,
A(x) = x + 2*y^2*x^2/2! + 3*y^6*x^3/3! +
  (4*y^12 + 12)*x^4/4! +
  5*y^20*x^5/5! +
  (6*y^30 + 360*y^3)*x^6/6! +
  7*y^42*x^7/7! +
  (8*y^56 + 10080*y^8)*x^8/8! +
  (9*y^72 + 60480)*x^9/9! +
  (10*y^90 + 302400*y^15)*x^10/10! +
  11*y^110*x^11/11! +
  (12*y^132 + 9979200*y^24 + 79833600*y^4)*x^12/12! +
  13*y^156*x^13/13! +
  (14*y^182 + 363242880*y^35)*x^14/14! +
  (15*y^210 + 108972864000*y^10)*x^15/15! +
  (16*y^240 + 14529715200*y^48 + 871782912000)*x^16/16! + ...
evaluated at y=2.

Mathematica

a[n_] := DivisorSum[n, 2^(#^2-n) * Binomial[#, n/#] * n!/#! &]; Array[a, 15] (* Amiram Eldar, Aug 24 2023 *)

Prog

(PARI) {a(n, y=2) = my(A=1); A = sum(m=0, n, ((x^m + y^m +x*O(x^n))^m - y^(m^2))/m!); if(n==0, 0, n!*polcoeff(A, n))}
for(n=1, 20, print1(a(n), ", "))
(PARI) {a(n, y=2) = if(n<1, 0, sumdiv(n, d, y^(d^2-n) * binomial(d, n/d) * n!/d! ) )}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A259209, A265270.

Sequence in context: A003435 A071303 A128406 * A003956 A204820 A041269

Adjacent sequences: A265266 A265267 A265268 * A265270 A265271 A265272

Keyword

nonn

Author

Paul D. Hanna, Dec 26 2015

Status

approved