A064984 Triangle of coefficients T[n,m] of polynomials n, n^2, (n+2n^3)/3, n^2(2+n^2)/3, n(3+10n^2+2n^4)/15, etc. after multiplication by the denominators (A049606).

1, 0, 1, 1, 0, 2, 0, 2, 0, 1, 3, 0, 10, 0, 2, 0, 23, 0, 20, 0, 2, 45, 0, 196, 0, 70, 0, 4, 0, 132, 0, 154, 0, 28, 0, 1, 315, 0, 1636, 0, 798, 0, 84, 0, 2, 0, 5067, 0, 7180, 0, 1806, 0, 120, 0, 2, 14175, 0, 83754, 0, 50270, 0, 7392, 0, 330, 0, 4, 0, 146430, 0, 239327, 0, 74800, 0

Offset: 1

Author

Wouter Meeussen, Oct 30 2001

Keywords

• nonn
• tabl

Comments

These polynomials are P(1, n) = 2*Sum[k, {k,1,n-1}] + n, counting up to n and down again; P(2, m) = 2*Sum[P(1,n), {n,1,m-1}] + P(1,m), meaning up and down to n and this for n from 1 up to m and down again; etc.

Examples

			1+2+3+2+1 = 3^2, (1)+(1+2+1)+(1+2+3+2+1)+(1+2+1)+(1) = (n+2n^3)/3.
		

Crossrefs

Row sums give A049606 again, final entry in each row seems to give A048896.

Programs

Mathematica

CoefficientList[ #, n ]&/@(NestList[ ((2*Sum[ #, {n, k-1} ]+(#/. n->k)//Simplify)/.k->n)&, n, -1+16 ] Denominator[ 2^#/#!&/@Range[ 16 ] ])