A179086 Triangle T(n,k) = |Re|+|Im| where Re+i*Im is the complex coefficient of [x^k] of the series (1-x)^(n+1) * Sum_{k>=0} ((1+i)*k+i)^n *x^k and i the imaginary unit, row n and column k.

1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 36, 74, 36, 1, 1, 85, 478, 478, 85, 1, 1, 168, 2331, 3576, 2331, 168, 1, 1, 315, 8519, 36451, 36451, 8519, 315, 1, 1, 872, 30076, 257128, 341254, 257128, 30076, 872, 1, 1, 1927, 137796, 1239756, 4807694, 4807694, 1239756

Offset

0,5

Comments

Row sums are 1, 2, 6, 28, 148, 1128, 8576, 90572, 917408, 12374348, 161499532,....

Links

Table of n, a(n) for n=0..51.

Formula

p(x,n)= (1 - x)^(n + 1)*Sum[(((1 + I)*k + I)^n)*x^k, {k, 0, Infinity}];

p(x,n)=(1 - x)^(1 + n)* (I^n + (1 + I)^n* x LerchPhi[x, -n, 3/2 + I/2]);

t(n,m)=Abs(RealCoefficients(p(x,n)))+Abs(ImaginaryCoefficients(p(x,n)))

Example

1;
1, 1;
1, 4, 1;  # represents -1+4*i*x+x^2+....
1, 13, 13, 1; # represents -i +(-11+2*i)*x + (-2+11*i)*x^2 + x^3 +....
1, 36, 74, 36, 1;
1, 85, 478, 478, 85, 1;
1, 168, 2331, 3576, 2331, 168, 1;
1, 315, 8519, 36451, 36451, 8519, 315, 1;
1, 872, 30076, 257128, 341254, 257128, 30076, 872, 1;
1, 1927, 137796, 1239756, 4807694, 4807694, 1239756, 137796, 1927, 1;
1, 3364, 455579, 8791264, 44359978, 54279160, 44359978, 8791264, 455579, 3364, 1;

Maple

A179086 := proc(n, k)
    (1-x)^(n+1) *add((I+j*(I+1))^n*x^j, j=0..n);
    coeftayl(%, x=0, k) ;
    abs(Re(%))+abs(Im(%)) ;
end proc: # R. J. Mathar, Jun 16 2015

Mathematica

p[x_, n_] = (1 - x)^(n + 1)*Sum[(((1 + I)*k + I)^n)*x^k, {k, 0, Infinity}]
Flatten[Table[Abs[Re[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]]]
+ Abs[Im[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]]], {n, 0, 10}]]

Crossrefs

Sequence in context: A080248 A139382 A157180 * A255494 A146956 A152613

Adjacent sequences: A179083 A179084 A179085 * A179087 A179088 A179089

Keyword

nonn,tabl

Author

Roger L. Bagula, Jun 28 2010

Status

approved