A086873 Triangle read by rows in which row n >= 1 gives coefficients in expansion of the polynomial Sum_{k=1..n} (1/n)*binomial(n,k)*binomial(n,k-1)*x^(2k)*(1+x)^(2n-2k) / x^2 in powers of x.

1, 1, 2, 2, 1, 4, 9, 10, 5, 1, 6, 21, 44, 57, 42, 14, 1, 8, 38, 116, 240, 336, 308, 168, 42, 1, 10, 60, 240, 680, 1392, 2060, 2160, 1530, 660, 132, 1, 12, 87, 430, 1545, 4152, 8449, 13014, 14985, 12540, 7227, 2574, 429, 1, 14, 119, 700, 3045, 10122, 26173, 53048

Offset: 1

Author

N. J. A. Sloane, Sep 16 2003

Keywords

• nonn
• easy
• tabf

Comments

Row n has 2n-1 terms.

Examples

			For n=3 the polynomial is 1 + 4x + 9x^2 + 10x^3 + 5x^4.
  1;
  1,  2,  2;
  1,  4,  9,  10,    5;
  1,  6, 21,  44,   57,   42,   14;
  1,  8, 38, 116,  240,  336,  308,   168,    42;
  1, 10, 60, 240,  680, 1392, 2060,  2160,  1530,   660,  132;
  1, 12, 87, 430, 1545, 4152, 8449, 13014, 14985, 12540, 7227, 2574, 429;
		

Links

• C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28.

Crossrefs

A059231 gives row sums.

Programs

Maple

j := 0:f := n->sum(binomial(n,k)*binomial(n,k-1)/n*x^(2*k)*(1+x)^(2*n-2*k),k=1..n): for n from 1 to 15 do p := expand(f(n)/x^2):for l from 0 to 2*n-2 do j := j+1: a[j] := coeff(p,x,l):od:od:seq(a[l],l=1..j); # Sascha Kurz

PARI

for(n=1,8,p=sum(k=1,n,(1/n)*binomial(n,k)*binomial(n,k-1)*x^(2*k)*(1+x)^(2*n-2*k))/x^2; for(i=1,2*n-1,print1(polcoeff(p,i-1) ","); ); print; ); \\ Ray Chandler, Sep 17 2003

Extensions

More terms from Vladeta Jovovic and Ray Chandler, Sep 17 2003