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a(n) equals sum of first n terms of A(x)^n for n>=1, with a(0)=1.
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%I #24 Sep 13 2024 09:56:23

%S 1,1,3,16,127,1321,16680,244518,4049199,74404069,1498276873,

%T 32764372213,772675039936,19541627299052,527590805816280,

%U 15146369004674536,460804123171138079,14811876349937896743,501663013214822053815,17858867621856721343253,666744417234185576463077

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

%H Vaclav Kotesovec, <a href="/A088358/b088358.txt">Table of n, a(n) for n = 0..400</a>

%F 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 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 G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 127*x^4 + 1321*x^5 + 16680*x^6 +...

%e The coefficients in A(x)^n begin:

%e n=1: [1, 1, 3, 16, 127, 1321, 16680, 244518, 4049199, ...];

%e n=2: [1, 2, 7, 38, 295, 2992, 37020, 534386, 8745915, ...];

%e n=3: [1, 3, 12, 67, 513, 5088, 61716, 877053, 14181891, ...];

%e n=4: [1, 4, 18, 104, 791, 7696, 91582, 1281160, 20462071, ...];

%e n=5: [1, 5, 25, 150, 1140, 10916, 127565, 1756710, 27706465, ...];

%e n=6: [1, 6, 33, 206, 1572, 14862, 170761, 2315256, 36052245, ...];

%e n=7: [1, 7, 42, 273, 2100, 19663, 222432, 2970108, 45656093, ...];

%e n=8: [1, 8, 52, 352, 2738, 25464, 284024, 3736560, 56696823, ...];

%e n=9: [1, 9, 63, 444, 3501, 32427, 357186, 4632138, 69378300, ...]; ...

%e where the initial terms are derived from the above coefficients like so:

%e a(1) = 1 = 1;

%e a(2) = 1 + 2 = 3;

%e a(3) = 1 + 3 + 12 = 16;

%e a(4) = 1 + 4 + 18 + 104 = 127;

%e a(5) = 1 + 5 + 25 + 150 + 1140 = 1321;

%e a(6) = 1 + 6 + 33 + 206 + 1572 + 14862 = 16680; ...

%e RELATED EXPANSIONS.

%e The series B(x) = Series_Reversion(x/A(x)) begins:

%e B(x) = x + x^2 + 4*x^3 + 26*x^4 + 228*x^5 + 2477*x^6 + 31776*x^7 +...

%e such that A(x) = 1 + x*B'(x)/(1 - B(x)); also,

%e B(x) = Sum_{n>=1} b(n)*x^n where b(n) = [x^(n-1)] A(x)^n/n for n>=1:

%e [1/1, 2/2, 12/3, 104/4, 1140/5, 14862/6, 222432/7, 3736560/8, ...].

%o (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 for(n=0,25,print1(a(n),", "))

%o (PARI) /* A(x) = 1 + x*B'(x)/(1 - B(x)) where B(x/A(x)) = x: */

%o {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 for(n=0,25,print1(a(n),", "))

%Y Cf. A233436.

%K nonn

%O 0,3

%A _Michael Somos_ and _Paul D. Hanna_, Sep 27 2003