A207556 -id:A207556 - cp's OEIS frontend

A207557 G.f.: Sum_{n>=0} 1/(1+x)^(n^2-n) * Product_{k=1..n} ((1+x)^(2*k-1) - 1).

1, 1, 3, 12, 64, 420, 3276, 29581, 303389, 3483053, 44245695, 616103046, 9330961666, 152700926414, 2685132170466, 50488787588936, 1010864433071206, 21470488933116138, 482176661100286182, 11415700804801064258, 284169548252819022230, 7419733139418740010570

Offset: 0

Paul D. Hanna, Feb 18 2012

nonn

Compare g.f. to: Sum_{n>=0} 1/(1+x)^(n^2) * Product_{k=1..n} ((1+x)^(2*k-1) - 1), which is the g.f. of A179525.

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 64*x^4 + 420*x^5 + 3276*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^3-1)/(1+x)^2 + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)/(1+x)^6 + ((1+x)-1)*((1+x)^3-1)*((1+x)^5-1)*((1+x)^7-1)/(1+x)^20 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..135
Hsien-Kuei Hwang, and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], p. 36, 2019.

Cf. A179525, A207556.

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

Given A(x) is the g.f. of this sequence, note that:
1 + x*A(x) = Sum_{n>=0} 1/(1+x)^(n^2+n) * Product_{k=1..n} ((1+x)^(2*k-1) - 1).
a(n) ~ 2*sqrt(6) * 12^(n+1) * n^(n+1) / (exp(n+Pi^2/24) * Pi^(2*n+3)). - Vaclav Kotesovec, May 07 2014

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

A207571 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(3*k-1) - 1).

Original entry on oeis.org

1, 2, 11, 105, 1390, 23520, 484247, 11742927, 327711230, 10343198878, 364237027076, 14156867852699, 601927703437645, 27790427952836499, 1384496764982434033, 74027620787319243688, 4228343290201028904807, 256946673653717460509502, 16551666142815138743519611

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 + 2*x + 11*x^2 + 105*x^3 + 1390*x^4 + 23520*x^5 + 484247*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)^2-1) + ((1+x)^2-1)*((1+x)^5-1) + ((1+x)^2-1)*((1+x)^5-1)*((1+x)^8-1) + ((1+x)^2-1)*((1+x)^5-1)*((1+x)^8-1)*((1+x)^11-1) +...

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, A207570.

CoefficientList[Series[Sum[Product[(1+x)^(3*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)^(3*k-1)-1) +x*O(x^n)),n)}
for(n=0,25,print1(a(n),", "))

a(n) ~ GAMMA(1/3) * 2^(2*n+1/3) * 3^(2*n+7/6) * n^(n+1/6) / (exp(n+Pi^2/72) * Pi^(2*n+11/6)). - Vaclav Kotesovec, May 06 2014

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A207557 G.f.: Sum_{n>=0} 1/(1+x)^(n^2-n) * Product_{k=1..n} ((1+x)^(2*k-1) - 1).

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

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A207571 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(3*k-1) - 1).

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Mathematica

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