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A128966
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Triangle read by rows of coefficients of polynomials P[n](x) defined by P[0]=0, P[1]=x+1; for n >= 2, P[n]=(x+1)*P[n-1]+x*P[n-2].
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1
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0, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 10, 6, 1, 1, 8, 20, 20, 8, 1, 1, 10, 34, 50, 34, 10, 1, 1, 12, 52, 104, 104, 52, 12, 1, 1, 14, 74, 190, 258, 190, 74, 14, 1, 1, 16, 100, 316, 552, 552, 316, 100, 16, 1, 1, 18, 130, 490, 1058, 1362, 1058, 490, 130, 18, 1, 1, 20, 164
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| A variant of A008288 (they satsify the same recurrence).
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FORMULA
| P[n](x) = (x+1) * ( ((x+1+sqrt(x^2+6x+1))/2)^n - ((x+1-sqrt(x^2+6x+1))/2)^n ) / sqrt(x^2+6x+1) - Max Alekseyev (maxale(AT)gmail.com), Mar 10 2008
P[n](x) = (x+1) * (sqrt(x)*I)^(n-1) * U[n-1](-I*(x+1)/sqrt(x)/2), where U[n](t) is Chebyshev polynomial of the 2nd kind. - Max Alekseyev (maxale(AT)gmail.com), Mar 10 2008
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EXAMPLE
| Triangle begins:
0
1, 1
1, 2, 1
1, 4, 4, 1
1, 6, 10, 6, 1
1, 8, 20, 20, 8, 1
1, 10, 34, 50, 34, 10, 1
1, 12, 52, 104, 104, 52, 12, 1
1, 14, 74, 190, 258, 190, 74, 14, 1
1, 16, 100, 316, 552, 552, 316, 100, 16, 1
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MAPLE
| P[0]:=0;
P[1]:=x+1;
for n from 2 to 14 do
P[n]:=expand((x+1)*P[n-1]+x*P[n-2]);
lprint(P[n]);
lprint(seriestolist(series(P[n], x, 200)));
od:
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PROG
| (PARI) { T(n, k) = sum(m=0, (n-1)\2, binomial(n, 2*m+1) * sum(j=0, m, binomial(m, j) * binomial(n-2*j, k-j) * 2^(2*j+1-n) ) ) } - Max Alekseyev (maxale(AT)gmail.com), Mar 10 2008
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CROSSREFS
| Sequence in context: A156580 A157528 A132731 * A055907 A172991 A203906
Adjacent sequences: A128963 A128964 A128965 * A128967 A128968 A128969
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KEYWORD
| nonn,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), May 10 2007
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