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 A318644 G.f. A(x) satisfies: A(x) = Sum_{n>=0} (1+x)^(n*(n+1)/2) * x^n / A(x)^n. 3
 1, 1, 1, 1, 2, 4, 11, 32, 106, 376, 1433, 5782, 24574, 109393, 508026, 2453256, 12285347, 63656731, 340626704, 1879183856, 10672897341, 62323897482, 373748877678, 2299318074357, 14497472040378, 93599428822052, 618278575554155, 4175348680420942, 28806364292660618, 202899326988089615, 1458130019936912105, 10685096640964659318 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..495 FORMULA G.f. A(x) satisfies: (1) A(x) = Sum_{n>=0} x^n * (1+x)^(n*(n+1)/2) / A(x)^n. (2) 1 + x = Sum_{n>=0} x^n * (1+x)^(n*(n-1)/2) / A(x)^n. EXAMPLE G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 11*x^6 + 32*x^7 + 106*x^8 + 376*x^9 + 1433*x^10 + 5782*x^11 + 24574*x^12 + 109393*x^13 + 508026*x^14 + ... such that A(x) = 1 + (1+x)*x/A(x) + (1+x)^3*x^2/A(x)^2 + (1+x)^6*x^3/A(x)^3 + (1+x)^10*x^4/A(x)^4 + (1+x)^15*x^5/A(x)^5 + (1+x)^21*x^6/A(x)^6 + (1+x)^28*x^7/A(x)^7 + ... + (1+x)^(n*(n+1)/2) * x^n / A(x)^n + ... Also 1 + x = 1 + x/A(x) + (1+x)*x^2/A(x)^2 + (1+x)^3*x^3/A(x)^3 + (1+x)^6*x^4/A(x)^4 + (1+x)^10*x^5/A(x)^5 + (1+x)^15*x^6/A(x)^6 + (1+x)^21*x^7/A(x)^7 + ... PROG (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(n=0, #A, (1+x +x*O(x^#A))^(n*(n+1)/2) * x^n/Ser(A)^n ) )[#A] ); A[n+1]} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A320951, A303058. Sequence in context: A156043 A268322 A148171 * A113774 A124504 A056324 Adjacent sequences:  A318641 A318642 A318643 * A318645 A318646 A318647 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 07 2018 STATUS approved

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Last modified August 18 15:50 EDT 2019. Contains 326108 sequences. (Running on oeis4.)