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A055858 Coefficient triangle for certain polynomials. 11
1, 1, 2, 4, 9, 6, 27, 64, 48, 36, 256, 625, 500, 400, 320, 3125, 7776, 6480, 5400, 4500, 3750, 46656, 117649, 100842, 86436, 74088, 63504, 54432, 823543, 2097152, 1835008, 1605632, 1404928, 1229312, 1075648, 941192, 16777216, 43046721 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The coefficients of the partner polynomials are found in triangle A055864.

LINKS

Table of n, a(n) for n=0..37.

FORMULA

a(n, m)=0 if n<m; a(0, 0)=1, a(n, 0)= n^n, n >= 1, a(n, m)= n^(m-1)*(n+1)^(n-m+1), n >= m >= 1;

E.g.f. for column m: A(m, x); A(0, x)= 1/(1+W(-x)); A(1, x)= -1-diff(W(-x), x) = -1-W(-x)/((1+W(-x))*x); A(2, x)=A(1, x)-int(A(1, x), x)/x-(1/x+x); recursion: A(m, x) = A(m-1, x)-int(A(m-1, x), x)/x-((m-1)^(m-1))*(x^(m-1))/(m-1)!, m >= 3; W(x) principal branch of Lambert's function.

EXAMPLE

{1}; {1,2}; {4,9,6}; {27,64,48,36};...

Fourth row polynomial (n=3): p(3,x)= 27+64*x+48*x^2+36*x^3

MATHEMATICA

a[n_, m_] /; n < m = 0; a[0, 0] = 1; a[n_, 0] := n^n; a[n_, m_] := n^(m-1)*(n+1)^(n-m+1); Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jun 20 2013 *)

CROSSREFS

Column sequences are A000312(n), n >= 1, A055860 (A000169), A055861 (A053506), A055862-3 for m=0..4, row sums: A045531(n+1)= |A039621(n+1, 2)|, n >= 0.

Sequence in context: A011182 A063507 A241473 * A141389 A133757 A076125

Adjacent sequences:  A055855 A055856 A055857 * A055859 A055860 A055861

KEYWORD

easy,nonn,tabl

AUTHOR

Wolfdieter Lang, Jun 20 2000

STATUS

approved

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Last modified December 3 16:24 EST 2016. Contains 278745 sequences.