A211195 - OEIS

%I #14 Jul 19 2016 11:33:30

%S 1,2,6,32,240,2232,24080,290048,3809088,53691840,803569184,

%T 12670027776,209244552192,3603569846912,64493380379520,

%U 1196207964360704,22942371004144640,454160262238341120,9265017815023565312,194524772488764702720,4198521645139971459072

%N a(n) = 2*Sum_{k=0..n-1} {[x^k] A(x)^(n-k)} * {[x^(n-k-1)] A(x)^(k+1)/(k+1)} for n>0, with a(0)=1, where g.f. A(x) = Sum_{n>=0} a(n)*x^n.

%F G.f. satisfies: A(x) = 1 + 2*B(x*G(x)) where B(x) is the g.f. of A211196 such that B'(x) = A(x) and G(x) = A(x*G(x)) so that G(x) = Series_Reversion(x/A(x))/x. - _Paul D. Hanna_, Nov 21 2013

%F a(n) is divisible by (n+1); A211196(n+1) = a(n)/(n+1).

%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 32*x^3 + 240*x^4 + 2232*x^5 + 24080*x^6 +...

%e The table of coefficients in A(x)^n begins:

%e n=1: [1,  2,   6,   32,  240,  2232,  24080,  290048,  3809088, ...];

%e n=2: [1,  4,  16,   88,  644,  5808,  60992,  718560,  9267776, ...];

%e n=3: [1,  6,  30,  176, 1284, 11328, 116136, 1339200, 16962240, ...];

%e n=4: [1,  8,  48,  304, 2248, 19584, 196800, 2224256, 27672720, ...];

%e n=5: [1, 10,  70,  480, 3640, 31592, 312640, 3470080, 42432080, ...];

%e n=6: [1, 12,  96,  712, 5580, 48624, 476224, 5203680, 62599152, ...];

%e n=7: [1, 14, 126, 1008, 8204, 72240, 703640, 7590592, 89949552, ...]; ...

%e where a(n) is obtained from the antidiagonals in the above table like so:

%e a(1) = 2*(1*1/1);

%e a(2) = 2*(1*2/1 + 2*1/2);

%e a(3) = 2*(1*6/1 + 4*4/2 + 6*1/3);

%e a(4) = 2*(1*32/1 + 6*16/2 + 16*6/3 + 32*1/4);

%e a(5) = 2*(1*240/1 + 8*88/2 + 30*30/3 + 88*8/4 + 240*1/5);

%e a(6) = 2*(1*2232/1 + 10*644/2 + 48*176/3 + 176*48/4 + 644*10/5 + 2232*1/6);

%e a(7) = 2*(1*24080/1 + 12*5808/2 + 70*1284/3 + 304*304/4 + 1284*70/5 + 5808*12/6 + 24080*1/7); ...

%o (PARI) a(n)=local(A=1 + sum(j=1, n-1, a(j)*x^j)+x*O(x^n)); if(n==0, 1, 2*sum(k=0, n-1, polcoeff(A^(n-k), k)*polcoeff(A^(k+1)/(k+1), n-k-1)))

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

%o (PARI) a(n)=local(A=1+x); for(i=1, n, A=1+2*subst(intformal(A), x, serreverse(x/A +x*O(x^n)))); polcoeff(A, n)

%o for(n=0, 25, print1(a(n), ", ")) \\ _Paul D. Hanna_, Nov 21 2013

%Y Cf. A161881, A211196.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 03 2013