editing

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Paul D. Hanna, <a href="/A184356/b184356.txt">Table of n, a(n) for n = 0..100</a>

1, 1, 2, 10, 75, 757, 9955, 161608, 3149491, 72294325, 1919933126, 58189667167, 1991123304634, 76201510956909, 3235630545496281, 151399102211450842, 7760065212106661217, 433404831023513573519, 26253103133315432898270, 1717576707472491422233436, 120912301935843736344714288

G.f.: Sum_{n>=0} x^n * (1+x)^(-2n2*n^2 - n) / [Sum_{k>=0..n-1} C(n+k,k)^2*(-x)^k]^n.

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

for(n=0, 30, print1(a(n), ", "))

for(n=0, 30, print1(a(n), ", "))

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editing

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Jan 12 2011

proposed

approved

G.f.: Sum_{n>=0} x^n/[Sum_{k=0..n} C(n,k)^2*(-x)^k]^n.

G.f.: Sum_{n>=0} x^n*(1+x)^(-2n^2-n)/[Sum_{k=0..n-1} C(n+k,k)^2*(-x)^k]^n.

G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 75*x^4 + 757*x^5 + 9955*x^6 +...

The g.f. can also be expressed as:

A(x) = 1 + x*(1+x)^-3/(1 - 2^2*x + 3^2*x^2 - 4^2*x^3 + 5^2*x^4 -+...)

+ x^2*(1+x)^-10/(1 - 3^2*x + 6^2*x^2 - 10^2*x^3 + 15^2*x^4 -+...)^2

+ x^3*(1+x)^-21/(1 - 4^2*x + 10^2*x^2 - 20^2*x^3 + 35^2*x^4 -+...)^3

+ x^4*(1+x)^-36/(1 - 5^2*x + 15^2*x^2 - 35^2*x^3 + 70^2*x^4 -+...)^4

+ x^5*(1+x)^-55/(1 - 6^2*x + 21^2*x^2 - 56^2*x^3 + 126^2*x^4 -+...)^5 +...

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

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

Cf. A184355, A183165, A183166.

allocated for Paul D. Hanna

G.f.: Sum_{n>=0} x^n/[Sum_{k=0..n} C(n,k)^2*(-x)^k]^n.

1, 1, 2, 10, 75, 757, 9955, 161608, 3149491, 72294325, 1919933126, 58189667167, 1991123304634, 76201510956909, 3235630545496281, 151399102211450842, 7760065212106661217, 433404831023513573519, 26253103133315432898270

0,3

G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 75*x^4 + 757*x^5 + 9955*x^6 +...

equals the sum of the series:

A(x) = 1 + x/(1-x) + x^2/(1 - 2^2*x + x^2)^2 +

+ x^3/(1 - 3^2*x + 3^2*x^2 - x^3)^3

+ x^4/(1 - 4^2*x + 6^2*x^2 - 4^2*x^3 + x^4)^4

+ x^5/(1 - 5^2*x + 10^2*x^2 - 10^2*x^3 + 5^2*x^4 - x^5)^5

+ x^6/(1 - 6^2*x + 15^2*x^2 - 20^2*x^3 + 15^2*x^4 - 6^2*x^5 + x^6)^6 +...

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

Cf. A184355, A183165, A183166.

allocated

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jan 12 2011

approved

proposed

allocated for Paul D. Hanna

allocated

approved