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 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 Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019. 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 * A365153 A346424 A319743 Adjacent sequences: A207394 A207395 A207396 * A207398 A207399 A207400 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 17 2012 STATUS approved

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Last modified December 5 05:37 EST 2023. Contains 367575 sequences. (Running on oeis4.)