A207651 -id:A207651 - cp's OEIS frontend

A207652 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1)/(1 - x^k).

1, 1, 3, 10, 45, 249, 1709, 13912, 131168, 1402706, 16757321, 221018769, 3188425939, 49925523804, 843121969923, 15272776193787, 295372123082865, 6073931908657770, 132329525329523223, 3044691799670213778, 73771773281455834427, 1877511491197391256001

Offset: 0

Paul D. Hanna, Feb 19 2012

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 45*x^4 + 249*x^5 + 1709*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)-1)/(1-x) + ((1+x)-1)*((1+x)^2-1)/((1-x)*(1-x^2)) + ((1+x)-1)*((1+x)^2-1)*((1+x)^3-1)/((1-x)*(1-x^2)*(1-x^3)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..210
Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Cf. A207651, A207653, A207654.

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,((1+x)^k-1)/(1-x^k +x*O(x^n)) )),n)}
for(n=0,40,print1(a(n),", "))

From Vaclav Kotesovec, Oct 31 2014: (Start)
a(n) ~ 6*sqrt(2) * 12^n * n! / (exp(Pi^2/24) * Pi^(2*n+2)).
a(n) ~ 12^(n+1) * n^(n+1/2) / (exp(n + Pi^2/24) * Pi^(2*n+3/2)).
(End)

A207653 G.f.: Sum_{n>=0} Product_{k=1..n} (1 - (1-x)^(2k-1))/(1 - x^(2k-1)).

Original entry on oeis.org

1, 1, 4, 16, 77, 460, 3287, 27561, 265307, 2880875, 34821316, 463543454, 6737545832, 106158368798, 1802204594518, 32793160634292, 636683459975767, 13137118248246982, 287070448575006268, 6622644707103106925, 160846900060253917905, 4102379491083664461080

Offset: 0

Paul D. Hanna, Feb 19 2012

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 77*x^4 + 460*x^5 + 3287*x^6 +...
such that, by definition,
A(x) = 1 + (1-(1-x))/(1-x) + (1-(1-x))*(1-(1-x)^3)/((1-x)*(1-x^3)) + (1-(1-x))*(1-(1-x)^3)*(1-(1-x)^5)/((1-x)*(1-x^3)*(1-x^5)) +...

Vaclav Kotesovec, 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. A207651, A207652, A207654.

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,(1-(1-x)^(2*k-1))/(1-x^(2*k-1) +x*O(x^n)) )),n)}
for(n=0,40,print1(a(n),", "))

From Vaclav Kotesovec, Oct 31 2014: (Start)
a(n) ~ 6*sqrt(2) * exp(Pi^2/24) * 12^n * n! / Pi^(2*n+2).
a(n) ~ exp(Pi^2/24) * 12^(n+1) * n^(n+1/2) / (exp(n) * Pi^(2*n+3/2)).
(End)

A207654 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(2k-1) - 1)/(1 - x^(2k-1)).

Original entry on oeis.org

1, 1, 4, 22, 173, 1816, 23659, 367573, 6622465, 135637477, 3111148862, 78984029782, 2198423489832, 66562555228478, 2177861372888738, 76571625673934064, 2878937040339348981, 115260759545001030638, 4895471242828376133806, 219853190410155476470763

Offset: 0

Paul D. Hanna, Feb 19 2012

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 22*x^3 + 173*x^4 + 1816*x^5 + 23659*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)-1)/(1-x) + ((1+x)-1)*((1+x)^3-1)/((1-x)*(1-x^3)) + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)/((1-x)*(1-x^3)*(1-x^5)) +...

Vaclav Kotesovec, 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. A207651, A207652, A207653.

With[{nn=20},CoefficientList[Series[Sum[Product[((1+x)^(2k-1)-1)/(1- x^(2k-1)),{k,n}],{n,0,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Sep 06 2015 *)

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

From Vaclav Kotesovec, Oct 31 2014: (Start)
a(n) ~ sqrt(6) * 24^n * n! / (exp(Pi^2/48) * sqrt(n) * Pi^(2*n+3/2)).
a(n) ~ 2^n * 12^(n+1/2) * n^n / (exp(n + Pi^2/48) * Pi^(2*n+1)).
(End)

A193548 Decimal expansion of exp(Pi^2/12).

Original entry on oeis.org

2, 2, 7, 6, 1, 0, 8, 1, 5, 1, 6, 2, 5, 7, 3, 4, 0, 9, 4, 7, 9, 1, 0, 6, 1, 4, 1, 2, 0, 3, 1, 4, 9, 7, 4, 4, 6, 6, 9, 7, 9, 7, 4, 2, 6, 0, 3, 0, 0, 2, 3, 7, 7, 5, 6, 1, 5, 5, 1, 6, 1, 7, 0, 9, 8, 2, 7, 5, 0, 6, 3, 7, 2, 8, 6, 3, 0, 1, 4, 3, 1, 8, 6, 6, 8, 4, 6, 5, 7

Offset: 1

John M. Campbell, Jul 30 2011

Cf. A001113, A022493, A122214, A122215, A122216, A122217, A138265, A207651, A242153, A242154, A242155, A242156, A242157, A242158, A242159, A242160, A242161, A242162, A242163, A242164.

Product[Product[k^((1/(n+1))*(-1)^(k)*Binomial[n,k-1]*HarmonicNumber[n]),{k,1,n+1}],{n,1,Infinity}]
RealDigits[E^(Pi^2/12), 10, 100]

exp(Pi^2/12) \\ Charles R Greathouse IV, Jul 30 2011

exp(Pi^2/12) = Product_{n>=1} Product_{k=1..n+1} k^(1/(n+1)) * H(n) * (-1)^k * binomial(n, k-1) where H(n) is the n-th harmonic number.

exp(Pi^2/12) = lim_{n -> infinity} Product_{k=1..n} (1 + k/n)^(1/k). - Peter Bala, Feb 14 2015

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A207652 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1)/(1 - x^k).

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A207653 G.f.: Sum_{n>=0} Product_{k=1..n} (1 - (1-x)^(2*k-1))/(1 - x^(2*k-1)).

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A207654 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(2*k-1) - 1)/(1 - x^(2*k-1)).

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A193548 Decimal expansion of exp(Pi^2/12).

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Mathematica

PARI

Formula

A207653 G.f.: Sum_{n>=0} Product_{k=1..n} (1 - (1-x)^(2k-1))/(1 - x^(2k-1)).

A207654 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(2k-1) - 1)/(1 - x^(2k-1)).