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A337912
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G.f. A(x) satisfies: Sum_{n>=0} x^n * A(x)^(n*(n-1)/2) = 2 - Sum_{n>=0} (-x)^n * A(x)^(n*(n+1)/2).
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1
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1, 2, 8, 40, 224, 1368, 8992, 62792, 459808, 3493848, 27354144, 219715656, 1805543456, 15147831640, 129520892704, 1127187952712, 9975412520480, 89728147204568, 820171092473888, 7618324242746696, 71921793489841696, 690298016720515416, 6738731451282651936, 66946338886173682504, 677261057358471098400
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OFFSET
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0,2
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COMMENTS
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a(2*n) = 0 (mod 16) for n>1 and a(2*n+1) = 8 (mod 16) for n>0 (conjecture).
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LINKS
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 8*x^2 + 40*x^3 + 224*x^4 + 1368*x^5 + 8992*x^6 + 62792*x^7 + 459808*x^8 + 3493848*x^9 + 27354144*x^10 + ...
where the following series illustrate the definition.
B(x) = 1 + x + x^2*A(x) + x^3*A(x)^3 + x^4*A(x)^6 + x^5*A(x)^10 + x^6*A(x)^15 + x^7*A(x)^21 + x^8*A(x)^28 + ... + x^n*A(x)^(n*(n-1)/2) + ...
2 - B(x) = 1 - x*A(x) + x^2*A(x)^3 - x^3*A(x)^6 + x^4*A(x)^10 - x^5*A(x)^15 + x^6*A(x)^21 - x^7*A(x)^28 +- ... + (-x)^n*A(x)^(n*(n+1)/2) + ...
Explicitly,
B(x) = Sum_{n>=0} x^n * A(x)^(n*(n-1)/2) = 1 + x + x^2 + 3*x^3 + 15*x^4 + 89*x^5 + 577*x^6 + 3979*x^7 + 28815*x^8 + 217201*x^9 + 1692161*x^10 + 13551891*x^11 + 111134095*x^12 + 930739465*x^13 + 7945996609*x^14 + 69062717851*x^15 + ...
2 - B(x) = Sum_{n>=0} (-x)^n * A(x)^(n*(n+1)/2) = 1 - x - x^2 - 3*x^3 - 15*x^4 - 89*x^5 - 577*x^6 - 3979*x^7 - 28815*x^8 - 217201*x^9 - 1692161*x^10 - ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, x^m*Ser(A)^(m*(m-1)/2)) + sum(m=0, #A, (-x)^m*Ser(A)^(m*(m+1)/2) ), #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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