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 A211195 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. 1
 1, 2, 6, 32, 240, 2232, 24080, 290048, 3809088, 53691840, 803569184, 12670027776, 209244552192, 3603569846912, 64493380379520, 1196207964360704, 22942371004144640, 454160262238341120, 9265017815023565312, 194524772488764702720, 4198521645139971459072 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Table of n, a(n) for n=0..20. FORMULA 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 a(n) is divisible by (n+1); A211196(n+1) = a(n)/(n+1). EXAMPLE G.f.: A(x) = 1 + 2*x + 6*x^2 + 32*x^3 + 240*x^4 + 2232*x^5 + 24080*x^6 +... The table of coefficients in A(x)^n begins: n=1: [1, 2, 6, 32, 240, 2232, 24080, 290048, 3809088, ...]; n=2: [1, 4, 16, 88, 644, 5808, 60992, 718560, 9267776, ...]; n=3: [1, 6, 30, 176, 1284, 11328, 116136, 1339200, 16962240, ...]; n=4: [1, 8, 48, 304, 2248, 19584, 196800, 2224256, 27672720, ...]; n=5: [1, 10, 70, 480, 3640, 31592, 312640, 3470080, 42432080, ...]; n=6: [1, 12, 96, 712, 5580, 48624, 476224, 5203680, 62599152, ...]; n=7: [1, 14, 126, 1008, 8204, 72240, 703640, 7590592, 89949552, ...]; ... where a(n) is obtained from the antidiagonals in the above table like so: a(1) = 2*(1*1/1); a(2) = 2*(1*2/1 + 2*1/2); a(3) = 2*(1*6/1 + 4*4/2 + 6*1/3); a(4) = 2*(1*32/1 + 6*16/2 + 16*6/3 + 32*1/4); a(5) = 2*(1*240/1 + 8*88/2 + 30*30/3 + 88*8/4 + 240*1/5); a(6) = 2*(1*2232/1 + 10*644/2 + 48*176/3 + 176*48/4 + 644*10/5 + 2232*1/6); a(7) = 2*(1*24080/1 + 12*5808/2 + 70*1284/3 + 304*304/4 + 1284*70/5 + 5808*12/6 + 24080*1/7); ... PROG (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))) for(n=0, 25, print1(a(n), ", ")) (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) for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 21 2013 CROSSREFS Cf. A161881, A211196. Sequence in context: A009686 A012318 A012521 * A346452 A012324 A121676 Adjacent sequences: A211192 A211193 A211194 * A211196 A211197 A211198 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 03 2013 STATUS approved

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Last modified December 7 12:56 EST 2023. Contains 367656 sequences. (Running on oeis4.)