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 A144505 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). 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS FORMULA 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). [From Vladeta Jovovic, Dec 15 2008] EXAMPLE 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 ... MAPLE 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: CROSSREFS 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 A109261 A085240 Adjacent sequences:  A144502 A144503 A144504 * A144506 A144507 A144508 KEYWORD sign,tabf,changed AUTHOR N. J. A. Sloane, Dec 14 2008 STATUS approved

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