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A094557 a(2*n) equals coefficient of x^n in A(x)^(n+1) and a(2*n+1) equals coefficient of x^n in A(x)^(n+2), for n>=0.

%I A094557 #17 Jan 14 2025 06:02:28
%S A094557 1,1,2,3,9,14,40,65,210,339,1080,1764,5775,9448,30992,50931,168849,
%T A094557 277920,925240,1525887,5106288,8431260,28309440,46796334,157627548,
%U A094557 260788843,880639004,1458096900,4934715105,8175734400,27721876064
%N A094557 a(2*n) equals coefficient of x^n in A(x)^(n+1) and a(2*n+1) equals coefficient of x^n in A(x)^(n+2), for n>=0.
%F A094557 a(2*n) = (n+1)*A094558(n).
%F A094557 G.f. satisfies: A(x) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A094558.
%F A094557 G.f. satisfies: A(x) = F'(x^2)*(1 + F(x^2)/x) where F(x) = Series_Reversion(x/A(x)) and F(x)/x is the g.f. of A094558.
%e A094557 G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 9*x^4 + 14*x^5 + 40*x^6 + 65*x^7 +...
%e A094557 Terms are produced by main and secondary diagonals in the table of successive self-convolutions of this sequence:
%e A094557   [(1), 1, 2, 3, 9, 14, 40, 65, 210, 339, 1080, ...];
%e A094557   [(1),(2), 5, 10, 28, 58, 153, 320, 875, 1850, ...];
%e A094557   [1, (3),(9), 22, 63, 153, 410, 978, 2607, 6222, ...];
%e A094557   [1, 4, (14),(40), 121, 328, 918, 2392, 6504, 16708, ...];
%e A094557   [1, 5, 20, (65),(210), 621, 1830, 5110, 14395, 39085, ...];
%e A094557   [1, 6, 27, 98, (339),(1080), 3356, 9942, 29163, 83008, ...];
%e A094557   [1, 7, 35, 140, 518, (1764),(5775), 18040, 55160, 163863, ...];
%e A094557   [1, 8, 44, 192, 758, 2744, (9448),(30992), 98729, 305240, ...];
%e A094557   [1, 9, 54, 255, 1071, 4104, 14832, (50931),(168849), 542164, ...];
%e A094557   [1, 10, 65, 330, 1470, 5942, 22495, 80660, (277920),(925240), ...]; ...
%e A094557 from which A094558 may be formed from the main diagonal:
%e A094557   [1/1, 2/2, 9/3, 40/4, 210/5, 1080/6, 5775/7, 30992/8, 168849/9, 925240/10,...].
%e A094557 Let G(x) be the g.f. of A094558:
%e A094557 G(x) = 1 + x + 3*x^2 + 10*x^3 + 42*x^4 + 180*x^5 + 825*x^6 + 3874*x^7 +...
%e A094557 then the coefficients of G(x)^2 generates the secondary diagonal:
%e A094557 [1*2/2, 3*2/3, 14*2/4, 65*2/5, 339*2/6, 1764*2/7, 9448*2/8, 50931*2/9,...]
%e A094557 and may be derived from the odd-indexed terms of this sequence.
%o A094557 (PARI) {a(n)=local(A=1+x,G);for(i=1,n,G=serreverse(x/A+x*O(x^n));A=subst(deriv(G),x,x^2)+subst(deriv(G^2/2),x,x^2)/x);polcoeff(A,n)}
%o A094557 for(n=0,30,print1(a(n),", "))
%Y A094557 Cf. A094558, A094600.
%K A094557 nonn
%O A094557 0,3
%A A094557 _Paul D. Hanna_, May 11 2004
%E A094557 Entry revised by _Paul D. Hanna_, Apr 17 2013