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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
1, 1, 2, 7, 12, 12, 85, 88, 130, 152, 1071, 1140, 1665, 1845, 2430, 16891, 21786, 24501, 32066, 36066, 45222, 363378, 450506, 509110, 631883, 718914, 866306, 991571, 9545369, 10821336, 13004356, 14732096, 17438450, 19851112, 23380260, 26447976 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Table of n, a(n) for n=1..36.

EXAMPLE

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

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

(1/6 + 1*q + 1/2*q^2 + 1*q^3)*q^3*x^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*x^4 +...

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

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

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

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].

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

1,

1/2, 1,

7/6, 2, 2,

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

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

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

1;

1, 2;

7, 12, 12;

85, 88, 130, 152;

1071, 1140, 1665, 1845, 2430;

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

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

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

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

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

CROSSREFS

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

Sequence in context: A064441 A110949 A226703 * A344951 A174539 A049409

Adjacent sequences:  A126340 A126341 A126342 * A126344 A126345 A126346

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 25 2006

STATUS

approved

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Last modified June 14 19:28 EDT 2021. Contains 345038 sequences. (Running on oeis4.)