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A088358 a(n) equals sum of first n terms of A(x)^n for n>=1, with a(0)=1.

%I A088358 #24 Sep 13 2024 09:56:23
%S A088358 1,1,3,16,127,1321,16680,244518,4049199,74404069,1498276873,
%T A088358 32764372213,772675039936,19541627299052,527590805816280,
%U A088358 15146369004674536,460804123171138079,14811876349937896743,501663013214822053815,17858867621856721343253,666744417234185576463077
%N A088358 a(n) equals sum of first n terms of A(x)^n for n>=1, with a(0)=1.
%H A088358 Vaclav Kotesovec, <a href="/A088358/b088358.txt">Table of n, a(n) for n = 0..400</a>
%F A088358 G.f. satisfies: A(x) = 1 + x*B'(x)/(1 - B(x)) where B(x/A(x)) = x. - _Paul D. Hanna_, Nov 01 2013
%F A088358 a(n) ~ c * n! * n^alpha / LambertW(1)^n, where alpha = (1 + 3*LambertW(1))/(1 + 1/LambertW(1)) and c = 0.192874788982750074134074506494559... - _Vaclav Kotesovec_, Sep 13 2024
%e A088358 G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 127*x^4 + 1321*x^5 + 16680*x^6 +...
%e A088358 The coefficients in A(x)^n begin:
%e A088358 n=1: [1, 1,  3,  16,  127,  1321,  16680,  244518,  4049199, ...];
%e A088358 n=2: [1, 2,  7,  38,  295,  2992,  37020,  534386,  8745915, ...];
%e A088358 n=3: [1, 3, 12,  67,  513,  5088,  61716,  877053, 14181891, ...];
%e A088358 n=4: [1, 4, 18, 104,  791,  7696,  91582, 1281160, 20462071, ...];
%e A088358 n=5: [1, 5, 25, 150, 1140, 10916, 127565, 1756710, 27706465, ...];
%e A088358 n=6: [1, 6, 33, 206, 1572, 14862, 170761, 2315256, 36052245, ...];
%e A088358 n=7: [1, 7, 42, 273, 2100, 19663, 222432, 2970108, 45656093, ...];
%e A088358 n=8: [1, 8, 52, 352, 2738, 25464, 284024, 3736560, 56696823, ...];
%e A088358 n=9: [1, 9, 63, 444, 3501, 32427, 357186, 4632138, 69378300, ...]; ...
%e A088358 where the initial terms are derived from the above coefficients like so:
%e A088358 a(1) = 1 = 1;
%e A088358 a(2) = 1 + 2 = 3;
%e A088358 a(3) = 1 + 3 + 12 = 16;
%e A088358 a(4) = 1 + 4 + 18 + 104 = 127;
%e A088358 a(5) = 1 + 5 + 25 + 150 + 1140 = 1321;
%e A088358 a(6) = 1 + 6 + 33 + 206 + 1572 + 14862 = 16680; ...
%e A088358 RELATED EXPANSIONS.
%e A088358 The series B(x) = Series_Reversion(x/A(x)) begins:
%e A088358 B(x) = x + x^2 + 4*x^3 + 26*x^4 + 228*x^5 + 2477*x^6 + 31776*x^7 +...
%e A088358 such that A(x) = 1 + x*B'(x)/(1 - B(x)); also,
%e A088358 B(x) = Sum_{n>=1} b(n)*x^n where b(n) = [x^(n-1)] A(x)^n/n for n>=1:
%e A088358 [1/1, 2/2, 12/3, 104/4, 1140/5, 14862/6, 222432/7, 3736560/8, ...].
%o A088358 (PARI) {a(n)=local(A); if(n<2,n>=0,A=1+x; for(i=2,n,A+=x^i*subst(Pol((A+O(x^i))^i),x,1)); polcoeff(A,n))}
%o A088358 for(n=0,25,print1(a(n),", "))
%o A088358 (PARI) /* A(x) = 1 + x*B'(x)/(1 - B(x)) where B(x/A(x)) = x: */
%o A088358 {a(n)=local(A=1+x);for(i=1,n,B=serreverse(x/A+x*O(x^n));A=1+x*deriv(B)/(1-B));polcoeff(A,n)}
%o A088358 for(n=0,25,print1(a(n),", "))
%Y A088358 Cf. A233436.
%K A088358 nonn
%O A088358 0,3
%A A088358 _Michael Somos_ and _Paul D. Hanna_, Sep 27 2003