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A335625
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G.f. A(x) satisfies: Sum_{n>=0} A(x)^(n*(n+1)/2) * x^n = Sum_{n>=0} A(x)^((n+1)^2) * x^(2*n).
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1
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1, 1, 1, 1, 3, 16, 75, 332, 1509, 7134, 34733, 173318, 885600, 4628518, 24712222, 134684721, 749048314, 4250280931, 24605542944, 145345793814, 876215822887, 5392139083602, 33880944203692, 217412563217615, 1424997242686707, 9540582956884093, 65246631216440040, 455733923252320198, 3250425842974512999
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OFFSET
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0,5
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LINKS
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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + x^3 + 3*x^4 + 16*x^5 + 75*x^6 + 332*x^7 + 1509*x^8 + 7134*x^9 + 34733*x^10 + 173318*x^11 + 885600*x^12 + ...
where the following sums are equal:
B(x) = 1 + A(x)*x + A(x)^3*x^2 + A(x)^6*x^3 + A(x)^10*x^4 + A(x)^15*x^5 + A(x)^21*x^6 + A(x)^28*x^7 + ... + A(x)^(n*(n+1)/2)*x^n + ...
also
B(x) = A(x) + A(x)^4*x^2 + A(x)^9*x^4 + A(x)^16*x^6 + A(x)^25*x^8 + A(x)^36*x^10 + A(x)^49*x^12 + ... + A(x)^((n+1)^2) * x^(2*n) + ...
where
B(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 45*x^5 + 164*x^6 + 653*x^7 + 2767*x^8 + 12313*x^9 + 57053*x^10 + 273493*x^11 + 1350298*x^12 + 6846378*x^13 + 35577209*x^14 + 189216369*x^15 + 1029024489*x^16 + 5719246977*x^17 + 32477483024*x^18 + 188419488119*x^19 + 1116859062391*x^20 + ...
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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(n=0, #A, x^n * Ser(A)^(n*(n+1)/2) ) - sum(n=0, #A, x^(2*n) * Ser(A)^((n+1)^2) ), #A-1)); 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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