This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378579 #8 Jan 09 2025 08:05:39
%S A378579 1,-2,5,-15,54,-226,1041,-5045,25090,-126674,646764,-3335207,17359589,
%T A378579 -91138625,482237135,-2569446532,13774698084,-74245779493,
%U A378579 402105384051,-2187066640025,11941274232967,-65425584835537,359598131529024,-1982178299221646,10955208670488609,-60696056311093958,337040131916813474
%N A378579 G.f. A(x) satisfies x = Sum_{n>=1} ((1 + A(x)^n)^n - 1).
%H A378579 Paul D. Hanna, <a href="/A378579/b378579.txt">Table of n, a(n) for n = 1..500</a>
%F A378579 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A378579 (1) x = Sum_{n>=1} ((1 + A(x)^n)^n - 1).
%F A378579 (2) x = Sum_{n>=1} A(x)^(n^2) / (1 - A(x)^n)^(n+1), from formula by _Seiichi Manyama_ in A318636.
%F A378579 (3) x = Sum_{n>=1} A(x)^n * Sum_{d|n} binomial(n/d,d), from formula by _Ridouane Oudra_ in A318636.
%F A378579 (4) A(x) = Series_Reversion(G(x)), where G(x) = Sum_{n>=1} ((1 + x^n)^n - 1) is the g.f. of A318636.
%e A378579 G.f.: A(x) = x - 2*x^2 + 5*x^3 - 15*x^4 + 54*x^5 - 226*x^6 + 1041*x^7 - 5045*x^8 + 25090*x^9 - 126674*x^10 + 646764*x^11 - 3335207*x^12 + ...
%e A378579 where
%e A378579 x = (1 + A(x))-1 + (1 + A(x)^2)^2-1 + (1 + A(x)^3)^3-1 + (1 + A(x)^4)^4-1 + ...
%e A378579 The expansions of (1 + A(x)^n)^n - 1 begin:
%e A378579 n=1: x - 2*x^2 + 5*x^3 - 15*x^4 + 54*x^5 - 226*x^6 + 1041*x^7 - 5045*x^8 + ...
%e A378579 n=2: 2*x^2 - 8*x^3 + 29*x^4 - 108*x^5 + 430*x^6 - 1848*x^7 + 8484*x^8 + ...
%e A378579 n=3: 3*x^3 - 18*x^4 + 81*x^5 - 336*x^6 + 1395*x^7 - 6048*x^8 + ...
%e A378579 n=4: 4*x^4 - 32*x^5 + 176*x^6 - 848*x^7 + 3934*x^8 - 18416*x^9 + ...
%e A378579 n=5: 5*x^5 - 50*x^6 + 325*x^7 - 1775*x^8 + 9000*x^9 + ...
%e A378579 n=6: 6*x^6 - 72*x^7 + 540*x^8 - 3300*x^9 + 18234*x^10 + ...
%e A378579 n=7: 7*x^7 - 98*x^8 + 833*x^9 - 5635*x^10 + 33761*x^11 + ...
%e A378579 n=8: 8*x^8 - 128*x^9 + 1216*x^10 - 9024*x^11 + 58336*x^12 + ...
%e A378579 ...
%e A378579 the sum of which equals x.
%e A378579 SPECIFIC VALUES.
%e A378579 A(t) = 1/8 at t = 0.16352126551257248889045664875683784263524590236453...
%e A378579 where t = Sum_{n>=1} ((1 + 1/8^n)^n - 1),
%e A378579 also, t = Sum_{n>=1} (1/8)^(n^2) / (1 - 1/8^n)^(n+1).
%e A378579 A(t) = 1/9 at t = 0.14078320572038685935740333771629838603314392626246...
%e A378579 where t = Sum_{n>=1} ((1 + 1/9^n)^n - 1),
%e A378579 also, t = Sum_{n>=1} (1/9)^(n^2) / (1 - 1/9^n)^(n+1).
%e A378579 A(t) = 1/10 at t = 0.12355985214267974666409476695653610216564400778886...
%e A378579 where t = Sum_{n>=1} ((1 + 1/10^n)^n - 1).
%e A378579 A(t) = -1/4 at t = -0.15526284433046589758223569590356891892154738705096...
%e A378579 A(t) = -1/5 at t = -0.13708093574671812870578995929148440226274633630611...
%e A378579 A(1/6) = 0.12685609485901293251324636636937755144064758593774...
%e A378579 where 1/6 = Sum_{n>=1} ((1 + A(1/6)^n)^n - 1).
%e A378579 A(1/7) = 0.11241354571385669088090100601380487815275189296537...
%e A378579 A(1/8) = 0.10094983523585092678474357142194212014408583724977...
%e A378579 A(1/9) = 0.09162346270443389626958872306814641680247795571686...
%e A378579 A(-1/6) = -0.29924902046763454720023815313494776169729752567409...
%e A378579 A(-1/7) = -0.21417128882821263382592721321354392301151580309678...
%e A378579 A(-1/8) = -0.17350785733170913051439226143409362909355792177797...
%o A378579 (PARI) {a(n) = my(A = serreverse( sum(m=1,n, (1 + x^m +x*O(x^n))^m - 1) ));
%o A378579 polcoef(A,n)}
%o A378579 for(n=1,30, print1(a(n),", "))
%Y A378579 Cf. A318636.
%K A378579 sign
%O A378579 1,2
%A A378579 _Paul D. Hanna_, Jan 08 2025