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A086873
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Triangle read by rows in which row n >= 1 gives coefficients in expansion of the polynomial Sum((1/n)*binomial(n,k)*binomial(n,k-1)*x^(2k)*(1+x)^(2n-2k),k=1..n) / x^2 in powers of x.
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Row n has 2n-1 terms.
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REFERENCES
| C. Coker, Enumerating a class of lattice paths, Discrete Math., 271 (2003), 13-28.
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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,
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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); (from Sascha Kurz)
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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; ); (from Ray Chandler)
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CROSSREFS
| A059231 gives row sums.
Sequence in context: A158985 A087854 A185041 * A101560 A192456 A010243
Adjacent sequences: A086870 A086871 A086872 * A086874 A086875 A086876
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KEYWORD
| nonn,easy,tabf
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Sep 16 2003
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs) and Ray Chandler (rayjchandler(AT)sbcglobal.net), Sep 17 2003
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