A227934 -id:A227934 - cp's OEIS frontend

A178325 G.f.: A(x) = Sum_{n>=0} x^n/(1-x)^(n^2).

1, 1, 2, 6, 21, 83, 363, 1730, 8889, 48829, 284858, 1755325, 11374092, 77208275, 547261631, 4039201624, 30967330941, 246084049137, 2023030659970, 17175765057532, 150367445873108, 1355528352031358, 12566899017130088

Offset: 0

Paul D. Hanna, Dec 21 2010

nonn

Equals the row sums of triangle A214398.

a(n) is the number of weak compositions of n such that if the first part is equal to k then there are a total of k^2 + 1 parts. A weak composition is an ordered partition of the integer n into nonnegative parts. a(3) = 6 because we have: 1+2, 2+0+0+0+1, 2+0+0+1+0, 2+0+1+0+0, 2+1+0+0+0, 3+0+0+0+0+0+0+0+0+0. - Geoffrey Critzer, Oct 09 2013

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 83*x^5 + 363*x^6 +...
A(x) = 1 + (x-x^2)*((1-x)-x)/((1-x)^3-x) + (x-x^2)^2*((1-x)-x)*((1-x)^5-x)/(((1-x)^3-x)*((1-x)^7-x)) + (x-x^2)^3*((1-x)-x)*((1-x)^5-x)*((1-x)^9-x)/(((1-x)^3-x)*((1-x)^7-x)*((1-x)^11-x)) +...

Paul D. Hanna, Table of n, a(n) for n = 0..250

Cf. A214398, A230050, A227934, A227935.

nn=22;CoefficientList[Series[Sum[x^k/(1-x)^(k^2),{k,0,nn}],{x,0,nn}],x]  (* Geoffrey Critzer, Oct 09 2013 *)

{a(n)=sum(k=0,n,binomial((n-k)^2+k-1,k))}

{a(n)=polcoeff(sum(m=0,n,x^m/(1-x+x*O(x^n))^(m^2)),n)}

{a(n)=polcoeff(sum(m=0,n,(x-x^2)^m*prod(k=1,m,((1-x)^(4*k-3)-x)/((1-x)^(4*k-1)-x +x*O(x^n)))),n)}

a(n) = Sum_{k=0..n} C((n-k)^2 + k-1, k).
G.f.: A(x) = Sum_{n>=0} (x-x^2)^n*Product_{k=1..n} ((1-x)^(4*k-3) - x)/((1-x)^(4*k-1) - x) due to a q-series identity.
Let q = 1/(1-x), then g.f. A(x) equals the continued fraction:
. A(x) = 1/(1- q*x/(1- q*(q^2-1)*x/(1- q^5*x/(1- q^3*(q^4-1)*x/(1- q^9*x/(1- q^5*(q^6-1)*x/(1- q^13*x/(1- q^7*(q^8-1)*x/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
log(a(n)) ~ n*(log(n) - 2) * (1 + log(4*n) - log((log(n) - 2)*log(n))) / log(n). - Vaclav Kotesovec, Jan 10 2023

A230050 G.f.: Sum_{n>=0} x^n / (1-x)^(n^3).

Original entry on oeis.org

1, 1, 2, 10, 65, 564, 6191, 82050, 1295263, 23764278, 499547080, 11892550569, 317112508944, 9392408105655, 306739296397827, 10973970687363844, 427724034697254939, 18073023112616933860, 824247511186225346295, 40415810147764633887442, 2123162727678797736474583

Offset: 0

Paul D. Hanna, Oct 06 2013

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 65*x^4 + 564*x^5 + 6191*x^6 + 82050*x^7 +...
where
A(x) = 1 + x/(1-x) + x^2/(1-x)^8 + x^3/(1-x)^27 + x^4/(1-x)^64 + x^5/(1-x)^125 + x^6/(1-x)^216 + x^7/(1-x)^343 +...

Cf. A230049, A178325, A227934, A227935.

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

{a(n)=sum(k=0,n,binomial(k^3+n-k-1, n-k))}
for(n=0,25,print1(a(n),", "))

a(n) = Sum_{k=0..n} binomial(k^3 + n-k-1, n-k).

Equals row sums of triangle A230049.

A227935 G.f.: Sum_{n>=0} x^n / (1-x)^(n^5).

Original entry on oeis.org

1, 1, 2, 34, 773, 36656, 3001377, 333647780, 58561139773, 13838291852092, 4280413527001849, 1779704699369214238, 931039792575220097699, 604786686422678514970170, 489307443863919174036440087, 478922652139578822529676247092, 560120417434857039499787289137249

Offset: 0

Paul D. Hanna, Oct 06 2013

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 34*x^3 + 773*x^4 + 36656*x^5 + 3001377*x^6 +...
where
A(x) = 1 + x/(1-x) + x^2/(1-x)^32 + x^3/(1-x)^243 + x^4/(1-x)^1024 + x^5/(1-x)^3125 + x^6/(1-x)^7776 +...

Cf. A178325, A230050, A227934.

{a(n)=polcoeff(sum(k=0,n,x^k/(1-x+x*O(x^n))^(k^5)),n)}
for(n=0,20,print1(a(n),", "))

{a(n)=sum(k=0,n,binomial(k^5+n-k-1, n-k))}
for(n=0,20,print1(a(n),", "))

a(n) = Sum_{k=0..n} binomial(k^5 + n-k-1, n-k).

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A178325 G.f.: A(x) = Sum_{n>=0} x^n/(1-x)^(n^2).

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A230050 G.f.: Sum_{n>=0} x^n / (1-x)^(n^3).

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A227935 G.f.: Sum_{n>=0} x^n / (1-x)^(n^5).

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