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%I #3 Mar 30 2012 18:36:59
%S 1,1,1,1,2,2,1,3,7,7,1,4,15,34,34,1,5,26,94,214,214,1,6,40,200,726,
%T 1652,1652,1,7,57,365,1831,6645,15121,15121,1,8,77,602,3865,19388,
%U 70361,160110,160110,1,9,100,924,7239,46481,233154,846144,1925442,1925442,1
%N Rectangular table, read by antidiagonals, where the g.f. of row n is Sum_{i>=0} F_i(x)^n / 2^(i+1), where F_0(x)=x, F_{n+1}(x) = F_n(x+x^2), for n>=1.
%C A122940(n)/n = Sum_{m=1..n} (-1)^(m-1)*T(m,n-m+1)/m ; where l.g.f. of A122940, L(x), satisfies: L(x+x^2) = 2*L(x) - log(1+x).
%F T(n,k) = [x^k] Sum_{i>=0} F_i(x)^n / 2^(i+1) where F_0(x)=x, F_{n+1}(x) = F_n(x+x^2); a sum involving n-th powers of self-compositions of x+x^2 (cf. A122888).
%e Table begins:
%e 1, 1, 2, 7, 34, 214, 1652, 15121, 160110, 1925442, 25924260, ...;
%e 1, 2, 7, 34, 214, 1652, 15121, 160110, 1925442, 25924260, ...;
%e 1, 3, 15, 94, 726, 6645, 70361, 846144, 11392530, 169785124, ...;
%e 1, 4, 26, 200, 1831, 19388, 233154, 3139200, 46784118, ...;
%e 1, 5, 40, 365, 3865, 46481, 625820, 9326720, 152426170, ...;
%e 1, 6, 57, 602, 7239, 97470, 1452610, 23739936, 422171622, ...;
%e 1, 7, 77, 924, 12439, 185388, 3029782, 53879148, 1035760670, ...;
%e 1, 8, 100, 1344, 20026, 327296, 5820360, 111889248, 2312153223, ...;
%e 1, 9, 126, 1875, 30636, 544824, 10473576, 216432783, 4784414985, ...;
%e 1, 10, 155, 2530, 44980, 864712, 17868995, 395007850, 9301284465, ...;
%e Given that A122940 begins:
%e [1, 1, 4, 17, 106, 796, 7176, 75057, 894100, 11946906, ...],
%e demonstrate A122940(n)/n = Sum_{m=1..n} (-1)^(m-1)*T(m,n-m+1)/m
%e at n=4: A122940(4)/4 = 17/4 = 7/1 - 7/2 + 3/3 - 1/4;
%e at n=5: A122940(5)/5 = 106/5 = 34/1 - 34/2 + 15/3 - 4/4 + 1/5;
%e at n=6: A122940(6)/6 = 796/6 = 214/1 - 214/2 + 94/3 - 26/4 + 5/5 - 1/6.
%o (PARI) /* Get T(n,k) from H(n,), the n-th self-composition of x+x^2: */ {H(n,p)=local(F=x+x^2, G=x+x*O(x^p));if(n==0,G=x,for(i=1,n,G=subst(F,x,G));G)} {T(n,k)=round(polcoeff( sum(i=0,6*n+100,H(i,k+n-1)^n/2^(i+1)),k+n-1))}
%Y Cf. A122940; rows: A122942, A122943, A122944, A122945; related tables: A122888, A122946, A122948, A122951.
%K nonn,tabl
%O 1,5
%A _Paul D. Hanna_, Sep 25 2006
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