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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. 5

%I

%S 1,1,2,7,12,12,85,88,130,152,1071,1140,1665,1845,2430,16891,21786,

%T 24501,32066,36066,45222,363378,450506,509110,631883,718914,866306,

%U 991571,9545369,10821336,13004356,14732096,17438450,19851112,23380260,26447976

%N 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 The function W that satisfies: W(x,q) = exp( q*x*W(q*x,q) ) begins:

%e W(x,q) = 1 + q*x + (1/2 + q)*q^2*x^2 +

%e (1/6 + 1*q + 1/2*q^2 + 1*q^3)*q^3*x^3 +

%e (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 Coefficients of q in {[x^n] W(x,q)} tend to a limit when read backwards:

%e n=1: (1/2 + q)*q^2 read backwards: [1, 1/2];

%e n=2: (1/6 + 1*q + 1/2*q^2 + 1*q^3)*q^3 read backwards: [1, 1/2, 1, 1/6];

%e 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 The limit of fractional coefficients may be formed into a triangle:

%e 1,

%e 1/2, 1,

%e 7/6, 2, 2,

%e 85/24, 11/3, 65/12, 19/3,

%e 357/40, 19/2, 111/8, 123/8, 81/4, 16891/720, ...

%e When row n=1,2,3,.. is multiplied by n! we obtain this integer triangle:

%e 1;

%e 1, 2;

%e 7, 12, 12;

%e 85, 88, 130, 152;

%e 1071, 1140, 1665, 1845, 2430;

%e 16891, 21786, 24501, 32066, 36066, 45222;

%e 363378, 450506, 509110, 631883, 718914, 866306, 991571;

%e 9545369, 10821336, 13004356, 14732096, 17438450, 19851112, 23380260, 26447976;

%e 279725995, 316750608, 368695521, 417632601, 484621893, 546334029, 632562585, 713249235, 820357488;

%e 9251279911, 10612100290, 11923578775, 13648746400, 15329052835, 17462968972, 19598497945, 22282099420, 24949824310, 28305482450; ...

%Y Cf. A126341, A126342, A126265; A126344 (column 1), A126345 (diagonal), A126346 (row sums).

%K nonn,tabl

%O 1,3

%A _Paul D. Hanna_, Dec 25 2006

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Last modified January 23 07:07 EST 2020. Contains 331168 sequences. (Running on oeis4.)