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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A207397 G.f.: Sum_{n>=0} Product_{k=1..n} (q^k - 1) where q = (1+x)/(1+x^2). 4
1, 1, 1, 2, 11, 74, 557, 4799, 47004, 516717, 6302993, 84502346, 1235198136, 19552296646, 333212892221, 6083009119262, 118433569748072, 2449663066933397, 53643715882853914, 1239875630317731463, 30163779836127304106, 770476745704778418686 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Motivated by Peter Bala's identity described in A158690:

Sum_{n>=0} Product_{k=1..n} (q^k - 1) =

Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1),

here q = (1+x)/(1+x^2). See cross-references for other examples.

At present Bala's identity is conjectural and needs formal proof.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..240

FORMULA

G.f.: Sum_{n>=0} q^(-n^2) * Product_{k=1..n} (q^(2*k-1) - 1) where q = (1+x)/(1+x^2). [Based on Peter Bala's conjecture in A158690]

a(n) ~ c * 12^n * n! / Pi^(2*n), where c = 6*sqrt(2) / (Pi^2 * exp(Pi^2/8)) = 0.250367043877216848533826021231826... . - Vaclav Kotesovec, May 06 2014, updated Aug 22 2017

EXAMPLE

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 11*x^4 + 74*x^5 + 557*x^6 + 4799*x^7 +...

Let q = (1+x)/(1+x^2), then

A(x) = 1 + (q-1) + (q-1)*(q^2-1) + (q-1)*(q^2-1)*(q^3-1) + (q-1)*(q^2-1)*(q^3-1)*(q^4-1) + (q-1)*(q^2-1)*(q^3-1)*(q^4-1)*(q^5-1) +...

which also is proposed to equal:

A(x) = 1 + (q-1)/q + (q-1)*(q^3-1)/q^4 + (q-1)*(q^3-1)*(q^5-1)/q^9 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)/q^16 + (q-1)*(q^3-1)*(q^5-1)*(q^7-1)*(q^9-1)/q^25 +...

PROG

(PARI) {a(n)=local(A=1+x, q=(1+x)/(1+x^2 +x*O(x^n))); A=sum(m=0, n, prod(k=1, m, (q^k-1))); polcoeff(A, n)}

(PARI) {a(n)=local(A=1+x, q=(1+x)/(1+x^2 +x*O(x^n))); A=sum(m=0, n, q^(-m^2)*prod(k=1, m, (q^(2*k-1)-1))); polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A158690, A158691, A179525, A207386, A207433.

Sequence in context: A199417 A114179 A231556 * A319743 A166992 A058789

Adjacent sequences:  A207394 A207395 A207396 * A207398 A207399 A207400

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 17 2012

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 November 15 11:18 EST 2019. Contains 329144 sequences. (Running on oeis4.)