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A325059
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G.f.: Sum_{n>=0} x^n * (1 + (1+x)^n)^n / (1 - x*(1+x)^n)^(n+1).
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2
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1, 3, 10, 41, 190, 973, 5413, 32351, 205966, 1387807, 9845083, 73215780, 568757151, 4601092084, 38660287934, 336623442207, 3031260737552, 28178974826871, 269995107206317, 2662508737568260, 26987695641386128, 280844928307623929, 2997258604356945337, 32772404387384205040, 366794262989293809151, 4198563078511314225148, 49113768374394883013208, 586698015175211371037407, 7152213983896219165256687
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: Sum_{n>=0} x^n * (1 + (1+x)^n)^n / (1 - x*(1+x)^n)^(n+1).
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * ( (1+x)^n + (1+x)^k )^(n-k).
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * Sum_{j=0..n-k} binomial(n-k,j) * (1 + x)^((n-j)*(n-k)).
FORMULAS INVOLVING TERMS.
a(n) = Sum_{i=0..n} Sum_{j=0..n-i} Sum_{k=0..n-i-j} binomial(n-i,j) * binomial(n-i-j,k) * binomial((n-i-j)*(n-i-k),i).
a(n) = Sum_{i=0..n} Sum_{j=0..n-i} Sum_{k=0..n-i-j} binomial((n-i-j)*(n-i-k),i) * (n-i)! / ((n-i-j-k)!*j!*k!).
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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 10*x^2 + 41*x^3 + 190*x^4 + 973*x^5 + 5413*x^6 + 32351*x^7 + 205966*x^8 + 1387807*x^9 + 9845083*x^10 + 73215780*x^11 + 568757151*x^12 + ...
such that
A(x) = 1/(1-x) + x*(1 + (1+x))/(1 - x*(1+x))^2 + x^2*(1 + (1+x)^2)^2/(1 - x*(1+x)^2)^3 + x^3*(1 + (1+x)^3)^3/(1 - x*(1+x)^3)^4 + x^4*(1 + (1+x)^4)^4/(1 - x*(1+x)^4)^5 + x^5*(1 + (1+x)^5)^5/(1 - x*(1+x)^5)^6 + x^6*(1 + (1+x)^6)^6/(1 - x*(1+x)^6)^7 + x^7*(1 + (1+x)^7)^7/(1 - x*(1+x)^7)^8 + ...
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PROG
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(PARI) {a(n) = my(A = sum(m=0, n+1, x^m*((1+x +x*O(x^n) )^m + 1)^m/(1 - x*(1+x +x*O(x^n) )^m )^(m+1) )); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))
(PARI) {a(n) = sum(i=0, n, sum(k=0, n-i, sum(j=0, n-i-k, binomial(n-i, k) * binomial(n-i-k, j) * binomial((n-i-k)*(n-i-j), i) )))}
for(n=0, 35, print1(a(n), ", "))
(PARI) {a(n) = sum(i=0, n, sum(j=0, n-i, sum(k=0, n-i-j, binomial((n-i-j)*(n-i-k), i) * (n-i)! / ((n-i-j-k)!*j!*k!) )))}
for(n=0, 35, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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