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A144505
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Triangle read by rows: coefficients of polynomials arising from the recurrence A[n](x) = A[n-1]'(x)/(1-x) with A[0] = exp(x).
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8
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1, 1, -1, 2, 1, -5, 7, -1, 9, -30, 37, 1, -14, 81, -229, 266, -1, 20, -175, 835, -2165, 2431, 1, -27, 330, -2330, 9990, -24576, 27007, -1, 35, -567, 5495, -34300, 137466, -326515, 353522, 1, -44, 910, -11522, 97405, -561386, 2148139, -4976315, 5329837
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OFFSET
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0,4
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LINKS
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Seiichi Manyama, Rows n = 0..140, flattened
N. J. A. Sloane, Rows 0 through 25 of the triangle, together with the corresponding polynomials P[n](x).
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FORMULA
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Let A[0](x) = exp(x), A[n](x) = A[n-1]'(x)/(1-x) for n>0 and let P[n](x) = A[n](x)*(1-x)^(2n-1)/exp(x). Row n of triangle gives coefficients of P[n](x) with exponents of x in decreasing order.
P[n] = Sum((n+k)!/(n-k)!/k!/2^k*(1-x)^(n-k),k=0..n). E.g.f.: exp((1-x)*(1-sqrt(1-2*y)))/sqrt(1-2*y). [Vladeta Jovovic, Dec 15 2008]
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EXAMPLE
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The first few polynomials P[n] (n >= 0) are:
1
1
-x+2
x^2-5*x+7
-x^3+9*x^2-30*x+37
x^4-14*x^3+81*x^2-229*x+266
-x^5+20*x^4-175*x^3+835*x^2-2165*x+2431
x^6-27*x^5+330*x^4-2330*x^3+9990*x^2-24576*x+27007
-x^7+35*x^6-567*x^5+5495*x^4-34300*x^3+137466*x^2-326515*x+353522
x^8-44*x^7+910*x^6-11522*x^5+97405*x^4-561386*x^3+2148139*x^2-4976315*x+5329837
...
Triangle of coefficients begins:
1;
1;
-1, 2;
1, -5, 7;
-1, 9, -30, 37;
1, -14, 81, -229, 266;
-1, 20, -175, 835, -2165, 2431;
1, -27, 330, -2330, 9990, -24576, 27007;
-1, 35, -567, 5495, -34300, 137466, -326515, 353522;
1, -44, 910, -11522, 97405, -561386, 2148139, -4976315, 5329837;
...
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MAPLE
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A[0]:=exp(x);
P[0]:=1;
for n from 1 to 12 do
A[n]:=sort(simplify( diff(A[n-1], x)/(1-x)));
P[n]:=sort(simplify(A[n]*(1-x)^(2*n-1)/exp(x)));
t1:=simplify(x^(degree(P[n], x))*subs(x=1/x, P[n]));
t2:=series(t1, x, 2*n+3);
lprint(P[n]);
lprint(seriestolist(t2));
od:
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CROSSREFS
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Columns give A001515 (really A144301), A144498, A001514, A144506, A144507. Row sums give A001147. Alternating row sums give A043301.
Sequence in context: A129321 A064642 A193630 * A059039 A332022 A109261
Adjacent sequences: A144502 A144503 A144504 * A144506 A144507 A144508
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KEYWORD
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sign,tabf
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AUTHOR
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N. J. A. Sloane, Dec 14 2008
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STATUS
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approved
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