Search: id:A126343
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%I A126343
%S A126343 1,1,2,7,12,12,85,88,130,152,1071,1140,1665,1845,2430,16891,21786,24501,
%T A126343 32066,36066,45222,363378,450506,509110,631883,718914,866306,991571,
%U A126343 9545369,10821336,13004356,14732096,17438450,19851112,23380260,26447976
%N A126343 Triangle, read by rows, of the limit of coefficients of q in {[x^m] W(x,
               q)} as m grows when arranged into a triangle where row n is multiplied 
               by n! for n>=1.
%e A126343 The function W that satisfies: W(x,q) = exp( q*x*W(q*x,q) ) begins:
%e A126343 W(x,q) = 1 + q*x + (1/2 + q)*q^2*x^2 +
%e A126343 (1/6 + 1*q + 1/2*q^2 + 1*q^3)*q^3*x^3 +
%e A126343 (1/24 + 1/2*q + 1*q^2 + 7/6*q^3 + 1*q^4 + 1/2*q^5 + 1*q^6)*q^4*x^4 +...
%e A126343 Coefficients of q in {[x^n] W(x,q)} tend to a limit when read backwards:
%e A126343 n=1: (1/2 + q)*q^2 read backwards: [1, 1/2];
%e A126343 n=2: (1/6 + 1*q + 1/2*q^2 + 1*q^3)*q^3 read backwards: [1, 1/2, 1, 1/
               6];
%e A126343 n=3: (1/24 + 1/2*q + 1*q^2 + 7/6*q^3 + 1*q^4 + 1/2*q^5 + 1*q^6)*q^4 read 
               backwards: [1, 1/2, 1, 7/6, 1, 1/2, 1/24].
%e A126343 The limit of fractional coefficients may be formed into a triangle:
%e A126343 1,
%e A126343 1/2, 1,
%e A126343 7/6, 2, 2,
%e A126343 85/24, 11/3, 65/12, 19/3,
%e A126343 357/40, 19/2, 111/8, 123/8, 81/4, 16891/720, ...
%e A126343 When row n=1,2,3,.. is multiplied by n! we obtain this integer triangle:
%e A126343 1;
%e A126343 1, 2;
%e A126343 7, 12, 12;
%e A126343 85, 88, 130, 152;
%e A126343 1071, 1140, 1665, 1845, 2430;
%e A126343 16891, 21786, 24501, 32066, 36066, 45222;
%e A126343 363378, 450506, 509110, 631883, 718914, 866306, 991571;
%e A126343 9545369, 10821336, 13004356, 14732096, 17438450, 19851112, 23380260, 
               26447976;
%e A126343 279725995, 316750608, 368695521, 417632601, 484621893, 546334029, 632562585, 
               713249235, 820357488;
%e A126343 9251279911, 10612100290, 11923578775, 13648746400, 15329052835, 17462968972, 
               19598497945, 22282099420, 24949824310, 28305482450; ...
%Y A126343 Cf. A126341, A126342, A126265; A126344 (column 1), A126345 (diagonal), 
               A126346 (row sums).
%Y A126343 Sequence in context: A069748 A064441 A110949 this_sequence A049409 A006143 
               A160455
%Y A126343 Adjacent sequences: A126340 A126341 A126342 this_sequence A126344 A126345 
               A126346
%K A126343 nonn,tabl
%O A126343 1,3
%A A126343 Paul D. Hanna (pauldhanna(AT)juno.com), Dec 25 2006

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