login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A326274 E.g.f.: Sum_{n>=0} ((1+x)^n - 1)^n * 4^n / n!. 4
1, 4, 64, 1920, 86464, 5304320, 418131456, 40727959552, 4765747597312, 655794545577984, 104360850604687360, 18948720298674028544, 3882059495694122090496, 889053986706845142876160, 225799026538694916941283328, 63163063632830911303738982400, 19344290761718462120859544846336, 6452149866509553556278434299117568, 2332867461867950308492384248149311488, 910538103145382496893587688740637114368, 382208425560563535419125500691963382333440 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

More generally, the following sums are equal:

(1) Sum_{n>=0} (q^n + p)^n * r^n/n!,

(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

here, q = (1+x) and p = -1, r = 4.

In general, let F(x) be a formal power series in x such that F(0)=1, then

Sum_{n>=0} m^n * F(q^n*r)^p * log( F(q^n*r) )^n / n! =

Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);

here, F(x) = exp(x), q = 1+x, p = -1, r = 4, m = 1.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

E.g.f. may be expressed by the following sums.

(1) Sum_{n>=0} ((1+x)^n - 1)^n * 4^n / n!.

(2) Sum_{n>=0} (1+x)^(n^2) * exp(-4*(1+x)^n) * 4^n / n!.

EXAMPLE

E.g.f: A(x) = 1 + 4*x + 64*x^2/2! + 1920*x^3/3! + 86464*x^4/4! + 5304320*x^5/5! + 418131456*x^6/6! + 40727959552*x^7/7! + 4765747597312*x^8/8! + 655794545577984*x^9/9! + 104360850604687360*x^10/10! +...

such that

A(x) = 1 + 4*((1+x) - 1) + 4^2*((1+x)^2 - 1)^2/2! + 4^3*((1+x)^3 - 1)^3/3! + 4^4*((1+x)^4 - 1)^4/4! + 4^5*((1+x)^5 - 1)^5/5! + 4^6*((1+x)^6 - 1)^6/6! + 4^7*((1+x)^7 - 1)^7/7! + ...

also

A(x) = 1 + 4*(1+x)*exp(-4*(1+x)) + 4^2*(1+x)^4*exp(-4*(1+x)^2)/2! + 4^3*(1+x)^9*exp(-4*(1+x)^3)/3! + 4^4*(1+x)^16*exp(-4*(1+x)^4)/4! + 4^5*(1+x)^25*exp(-4*(1+x)^5)/5! + 4^6*(1+x)^36*exp(-4*(1+x)^6)/6! + 4^7*(1+x)^49*exp(-4*(1+x)^7)/7! + ...

PROG

(PARI) {a(n)=n!*polcoeff(sum(m=0, n, 4^m*((1+x+x*O(x^n))^m-1)^m/m!), n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A192935, A326272, A326273.

Cf. A326094.

Sequence in context: A098698 A301583 A181398 * A081559 A298433 A261042

Adjacent sequences:  A326271 A326272 A326273 * A326275 A326276 A326277

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 22 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 13 03:41 EST 2019. Contains 329968 sequences. (Running on oeis4.)