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A341376
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G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(5*n)/(1 - x*A(x)^n) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(5*n+4)).
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4
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1, 1, 6, 63, 790, 10896, 159783, 2445499, 38627339, 625074945, 10310825610, 172747624244, 2931749347058, 50298296409267, 870968173547912, 15202903269996956, 267229314929729201, 4726210689606805342, 84045293131256519705, 1501874060469604610959
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OFFSET
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0,3
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COMMENTS
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Equals row k = 5 of rectangular table A340940.
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LINKS
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FORMULA
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Given g.f. A(x), the following sums are all equal:
(1) B(x) = Sum_{n>=0} x^n*A(x)^(5*n)/(1 - x*A(x)^n),
(2) B(x) = Sum_{n>=0} x^n*A(x)^(4*n)/(1 - x*A(x)^(5*n+1)),
(3) B(x) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(5*n+4)),
(4) B(x) = Sum_{n>=0} x^n/(1 - x*A(x)^(n+5)),
(5) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2+5*n) * (1 - x^2*A(x)^(2*n+5)) / ((1 - x*A(x)^n)*(1 - x*A(x)^(n+5))),
(6) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(5*n^2+5*n) * (1 - x^2*A(x)^(10*n+5)) / ((1 - x*A(x)^(5*n+1))*(1 - x*A(x)^(5*n+4)));
see the example section for the value of B(x).
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EXAMPLE
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G.f.: A(x) = 1 + x + 6*x^2 + 63*x^3 + 790*x^4 + 10896*x^5 + 159783*x^6 + 2445499*x^7 + 38627339*x^8 + 625074945*x^9 + 10310825610*x^10 + ...
such that the following sums are all equal:
B(x) = 1/(1-x) + x*A(x)^5/(1 - x*A(x)) + x^2*A(x)^10/(1 - x*A(x)^2) + x^3*A(x)^15/(1 - x*A(x)^3) + x^4*A(x)^20/(1 - x*A(x)^4) + ...
and
B(x) = 1/(1-x*A(x)) + x*A(x)^4/(1 - x*A(x)^6) + x^2*A(x)^8/(1 - x*A(x)^11) + x^3*A(x)^12/(1 - x*A(x)^16) + x^4*A(x)^16/(1 - x*A(x)^21) + ...
also
B(x) = 1/(1-x*A(x)^4) + x*A(x)/(1 - x*A(x)^9) + x^2*A(x)^2/(1 - x*A(x)^14) + x^3*A(x)^3/(1 - x*A(x)^19) + x^4*A(x)^4/(1 - x*A(x)^24) + ...
further
B(x) = 1/(1-x*A(x)^5) + x/(1 - x*A(x)^6) + x^2/(1 - x*A(x)^7) + x^3/(1 - x*A(x)^8) + x^4/(1 - x*A(x)^9) + x^5/(1 - x*A(x)^10) + ...
where
B(x) = 1 + 2*x + 8*x^2 + 60*x^3 + 640*x^4 + 8085*x^5 + 112116*x^6 + 1649968*x^7 + 25311223*x^8 + 400396030*x^9 + 6485530349*x^10 + ...
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PROG
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(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); H=A; A=concat(A, 0);
H[#A-1] = -polcoeff( sum(m=0, #A, x^m/(1 - x*Ser(A)^(m+5)) ) - sum(m=0, #A, x^m*Ser(A)^m/(1 - x*Ser(A)^(5*m+4)) ), #A)/4; A=H); W=A; A[n+1] }
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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