Convex Polyomino

Author

Eric W. Weisstein
July 28, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/ConvexPolyomino.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ConvexPolyomino.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Series

N

Sum[(((((-1))^n)(x^n)(q^(Binomial[n+1,2])))/(QPochhammer[q,n]QPochhammer[y q,n])),{n,0,∞}]

(OverHat[N])

Sum[(((((-1))^n)(x^n)(q^(Binomial[n+1,2])))/(QPochhammer[q,n-1]QPochhammer[y q,n])),{n,0,∞}]

(Subscript[p, 2n])

ser=Series[t^4(1-6t^2+11t^4-4t^6)/(1-4t^2)^2-4t^8(1-4t^2)^(-3/2),{t,0,30}]

Rest[First/@Partition[CoefficientList[ser,t],2]]

a=SeriesCoefficient[t^4(1-6t^2+11t^4-4t^6)/(1-4t^2)^2-4t^8(1-4t^2)^(-3/2),{t,0,2k}];

FullSimplify[a,k≥0&&k∈Integers]

InputForm[%]

Table[Piecewise[{{0,k<2},{2,k==3},{1,k==2}},4^(-4+k)*(3+2*k-4*(-1)^k*Binomial[-3/2,-4+k])],{k,15}]

FullSimplify[(4^(-4+k)) (3+2 k-4 (((-1))^k) Binomial[-(3/2),-4+k])/.k->2n+8,n∈Integers&&n>0]

Closed-Form

Table[(2n+11)4^n-4(2n+1)Binomial[2n,n],{n,0,10}]

Simplify[Series[(16^(2+n)) (19+4 n-4 Binomial[-(3/2),4+2 n]),{n,∞,1}],n∈Integers&&n>0]