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A276906
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G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} k^n * x^(2*k-1) ]^n / n ), a power series in x with integer coefficients.
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3
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1, 1, 1, 3, 7, 18, 53, 188, 799, 4001, 24050, 179248, 1639637, 17764040, 227653634, 3550628492, 67513114323, 1519274903363, 40153164845377, 1278514703044023, 49536414234360980, 2279497269454146657, 122986833567853232448, 7942922462379370617039, 622994706862172074402587, 58218522316121110190816538, 6379893924028925326363565894
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OFFSET
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0,4
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LINKS
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FORMULA
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G.f.: exp( Sum_{n>=1} [ Sum_{k=1..n} A008292(n,k) * x^(2*k-1) / (1-x^2)^(n+1) ]^n / n ), where A008292 are the Eulerian numbers.
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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 7*x^4 + 18*x^5 + 53*x^6 + 188*x^7 + 799*x^8 + 4001*x^9 + 24050*x^10 + 179248*x^11 + 1639637*x^12 +...
The logarithm of g.f. A(x) equals the series:
log(A(x)) = Sum_{n>=1} (x + 2^n*x^3 + 3^n*x^5 +...+ k^n*x^(2*k-1) +...)^n/n.
Explicitly,
log(A(x)) = x + x^2/2 + 7*x^3/3 + 17*x^4/4 + 56*x^5/5 + 199*x^6/6 + 890*x^7/7 + 4649*x^8/8 + 27817*x^9/9 + 195946*x^10/10 + 1684398*x^11/11 + 17397323*x^12/12 +...+ A276907(n)*x^n/n +...
This logarithmic series can be written using the Eulerian numbers like so:
log(A(x)) = x/(1-x^2)^2 + (x + x^3)^2/(1-x^2)^6/2 + (x + 4*x^3 + x^5)^3/(1-x^2)^12/3 + (x + 11*x^3 + 11*x^5 + x^7)^4/(1-x^2)^20/4 + (x + 26*x^3 + 66*x^5 + 26*x^7 + x^9)^5/(1-x^2)^30/5 + (x + 57*x^3 + 302*x^5 + 302*x^7 + 57*x^9 + x^11)^6/(1-x^2)^42/6 +...+ [ Sum_{k=1..n} A008292(n,k) * x^(2*k-1) ]^n / (1-x^2)^(n*(n+1))/n +...
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PROG
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(PARI) {a(n) = polcoeff( exp( sum(m=1, n+1, sum(k=1, n+1, k^m*x^(2*k-1) +x*O(x^n))^m/m ) ), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {A008292(n, k) = sum(j=0, k, (-1)^j * (k-j)^n * binomial(n+1, j))}
{a(n) = my(A=1, Oxn=x*O(x^n)); A = exp( sum(m=1, n+1, sum(k=1, m+1, A008292(m, k)*x^(2*k-1)/(1-x^2 +Oxn)^(m+1) )^m / m ) ); polcoeff(A, n)}
for(n=0, 30, 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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