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A078341
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Triangle read by rows: T(n,k) = n*T(n-1,k-1)+k*T(n-1,k) starting with T(0,0)=1.
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1
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1, 0, 1, 0, 1, 2, 0, 1, 7, 6, 0, 1, 18, 46, 24, 0, 1, 41, 228, 326, 120, 0, 1, 88, 930, 2672, 2556, 720, 0, 1, 183, 3406, 17198, 31484, 22212, 5040, 0, 1, 374, 11682, 96040, 295004, 385144, 212976, 40320, 0, 1, 757, 38412, 489298, 2339380, 4965900
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,6
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COMMENTS
| Triangle of coefficients of polynomials P[n]. Let F(t) satisfy dF/dt=exp(x*(exp(F)-1)) and F(0)=0. Then F(t) = sum_{n>=0} P[n]/n! t^n, where P[n] is a polynomial in x of degree n-1. The constant term of the polynomial is zero for n>=2. The coeff. of x is 1 for n>=2. The coeff. of x^n in P[n+1] is n!. The value at 1 is given by sequence A007549.
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FORMULA
| P[1]=1; P[n+1]=x*diff(P[n], x)+x*n*P[n];
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EXAMPLE
| P[1]=1, P[2]=x, P[3]=x+2*x^2, P[4]=x+7*x^2+6*x^3, P[5]=x+18*x^2+46*x^3+24*x^4, P[6]=x+41*x^2+228*x^3+326*x^4+120*x^5.
Rows start 1; 0,1; 0,1,2; 0,1,7,6; 0,1,18,46,24; 0,1,41,228,326,120; ...
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MAPLE
| P[1] := 1; for n from 1 to 10 do P[n+1] := expand(x*diff(P[n], x)+x*n*P[n]) od;
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CROSSREFS
| Cf. A007549, A000142.
Columns include A000007, A057427, A095151, A103768. Diagonals include A000142, A067318. Row sums are A007549.
Sequence in context: A114329 A101371 A154974 * A199459 A065329 A108998
Adjacent sequences: A078338 A078339 A078340 * A078342 A078343 A078344
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Frederic Chapoton (chapoton(AT)math.uqam.ca), Nov 22 2002
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EXTENSIONS
| Additional comments from Henry Bottomley (se16(AT)btinternet.com), Feb 15 2005
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