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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, 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
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
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