login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A338178 G.f.: A(x) satisfies 1 = Sum_{n>=0} Product_{k=n..2*n-1} ( (1+x)^k - A(x) ). 3
1, 1, 2, 23, 383, 8456, 228657, 7268077, 264627570, 10842464809, 493454895161, 24695915073150, 1348165575801780, 79740451015249487, 5080946666286668723, 347046897108675507548, 25300042941361404342404, 1960975025091063811051791, 161045636717926154800016147, 13970534533583992498224467231 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare the g.f. A(x) to functions B(x) and C(x) defined by

A338181: 1 = Sum_{n>=0} Product_{k=2*n..3*n-1} ((1+x)^k - B(x)) ;

A338183: 1 = Sum_{n>=0} Product_{k=3*n..4*n-1} ((1+x)^k - C(x)).

LINKS

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

FORMULA

G.f. A(x) satisfies:

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

(2) 1 = Sum_{n>=0} (1+x)^(n*(3*n-1)/2) / Product_{k=n..2*n} (1 + (1+x)^k*A(x)).

a(n) ~ c * d^n * n! / sqrt(n), where d = 4.6634423082484267335372948079179... and c = 0.10798828318774... - Vaclav Kotesovec, Oct 24 2020

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 23*x^3 + 383*x^4 + 8456*x^5 + 228657*x^6 + 7268077*x^7 + 264627570*x^8 + 10842464809*x^9 + 493454895161*x^10 + ...

Let A = A(x), then g.f. A(x) satisfies

1 = 1 + ((1+x) - A) + ((1+x)^2 - A)*((1+x)^3 - A) + ((1+x)^3 - A)*((1+x)^4 - A)*((1+x)^5 - A) + ((1+x)^4 - A)*((1+x)^5 - A)*((1+x)^6 - A)*((1+x)^7 - A) + ((1+x)^5 - A)*((1+x)^6 - A)*((1+x)^7 - A)*((1+x)^8 - A)*((1+x)^9 - A) + ... + Product_{k=n..2*n-1} ((1+x)^k - A(x)) + ...

also

1 = 1/(1 + A) + (1+x)/((1 + (1+x)*A)*(1 + (1+x)^2*A)) + (1+x)^5/((1 + (1+x)^2*A)*(1 + (1+x)^3*A)*(1 + (1+x)^4*A)) + (1+x)^12/((1 + (1+x)^3*A)*(1 + (1+x)^4*A)*(1 + (1+x)^5*A)*(1 + (1+x)^6*A)) + (1+x)^22/((1 + (1+x)^4*A)*(1 + (1+x)^5*A)*(1 + (1+x)^6*A)*(1 + (1+x)^7*A)*(1 + (1+x)^8*A)) + ... + (1+x)^(n*(3*n-1)/2)/( Product_{k=n..2*n} (1 + (1+x)^(k*A(x)) ) + ...

PROG

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

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

CROSSREFS

Cf. A338181, A338183.

Sequence in context: A239109 A266923 A060941 * A219890 A119774 A074649

Adjacent sequences: A338175 A338176 A338177 * A338179 A338180 A338181

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 14 2020

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 8 14:17 EST 2022. Contains 358693 sequences. (Running on oeis4.)