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

1,3

Comments

Row n has 2n-1 terms.

Links

Table of n, a(n) for n=1..57.

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

Example

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;

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

Prog

(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

Crossrefs

A059231 gives row sums.

Sequence in context: A295687 A087854 A185041 * A101560 A307456 A291260

Adjacent sequences: A086870 A086871 A086872 * A086874 A086875 A086876

Keyword

nonn,easy,tabf

Author

N. J. A. Sloane, Sep 16 2003

Extensions

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

Status

approved