A207571 -id:A207571 - cp's OEIS frontend

A207569 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(2*k-1) - 1).

1, 1, 3, 18, 151, 1640, 21825, 343763, 6253234, 128993019, 2975165831, 75866604098, 2119310099700, 64361149952242, 2111222815441491, 74391641880144734, 2802300974537717340, 112379709083552152423, 4780136025081921948194, 214954914688567198802759

Offset: 0

Paul D. Hanna, Feb 18 2012

Compare g.f. to Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1), which is the g.f. of A179525.

Compare g.f. to Sum_{n>=0} Product_{k=1..n} (1 - (1 - x)^(2*k-1)), which is the g.f. of A158691. - Peter Bala, Dec 04 2020

			G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 151*x^4 + 1640*x^5 + 21825*x^6 + ...
such that, by definition,
A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^3-1) + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1) + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)*((1+x)^7-1) + ...

Michael De Vlieger, Table of n, a(n) for n = 0..200
Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A179525, A207557, A207570, A207571, A215066, A053250, A158691.

CoefficientList[Series[Sum[Product[(1+x)^(2*k-1)-1, {k, 1, n}], {n, 0, 20}], {x, 0, 20}], x] (* Vaclav Kotesovec, May 06 2014 *)

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1+x)^(2*k-1)-1) +x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

a(n) ~ sqrt(12) * 24^n * n^n / (exp(n+Pi^2/48) * Pi^(2*n+1)). - Vaclav Kotesovec, May 06 2014
G.f.: 1/2*( 1 + Sum_{n>=0} (1 + x)^(2*n+1) * Product_{k = 1..n} ((1 + x)^(2*k-1) - 1) ). Cf. A053250 and A215066. - Peter Bala, May 15 2017
Conjectural g.f.: Sum_{n>=0} (-1)^n*Product_{k = 1..n} 1 + ( -1/(1 + x) )^k. - Peter Bala, Dec 04 2020
From Peter Bala, Jan 29 2021: (Start)
Conjectural g.f.s: Sum_{n >= 0} (-1)^n*(1 + x)^(n+1)*Product_{k = 1..n} (1 + (-1)^k*(1 + x)^k)^2. Also
(1/2)*( 1 + Sum_{n >= 0} 1/(1 + x)^(n+1)*Product_{k = 1..n} (1 + (-1)^k/(1 + x)^k) ). (End)

A207570 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(3*k-2) - 1).

Original entry on oeis.org

1, 1, 4, 34, 410, 6455, 125251, 2888305, 77157780, 2342972405, 79701049425, 3002132647515, 124039845584382, 5577660227565634, 271162541308698623, 14172237715785139175, 792418822364402364530, 47198077739119663907870, 2983413619934353599892285

Offset: 0

Paul D. Hanna, Feb 18 2012

nonn

Compare g.f. to: Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1), which is the g.f. of A179525.

			G.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 410*x^4 + 6455*x^5 + 125251*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^4-1) + ((1+x)-1)*((1+x)^4-1)*((1+x)^7-1) + ((1+x)-1)*((1+x)^4-1)*((1+x)^7-1)*((1+x)^10-1) +...

Michael De Vlieger, Table of n, a(n) for n = 0..200
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A179525, A207556, A207569, A207571.

Join[{1},Rest[With[{nn=20},CoefficientList[Series[Sum[Product[ (1+x)^(3k-2)-1,{k,n}],{n,nn}],{x,0,nn}],x]]]] (* Harvey P. Dale, Aug 20 2012 *)

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1+x)^(3*k-2)-1) +x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

a(n) ~ GAMMA(2/3) * 2^(2*n-1/3) * 3^(2*n+5/6) * n^(n-1/6) / (exp(n+Pi^2/72) * Pi^(2*n+7/6)). - Vaclav Kotesovec, May 06 2014

cp's OEIS Frontend

A207569 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(2*k-1) - 1).

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