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A049323
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Triangle of coefficients of certain polynomials (exponents in increasing order), equivalent to A033842.
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1
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1, 1, 1, 1, 3, 3, 1, 6, 16, 16, 1, 10, 50, 125, 125, 1, 15, 120, 540, 1296, 1296, 1, 21, 245, 1715, 7203, 16807, 16807, 1, 28, 448, 4480, 28672, 114688, 262144, 262144, 1, 36, 756, 10206, 91854, 551124, 2125764, 4782969, 4782969, 1, 45, 1200, 21000, 252000
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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LINKS
| W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
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FORMULA
| a(n, m) = A033842(n, n-m) = binomial(n+1, m+1)*(n+1)^{m-1}, n >= m >= 0, else 0.
p(k-1, -x)/(1-k*x)^k =(-1+1/(1-k*x)^k)/(x*k^2) is for k=1..5 G.f. for A000012, A001792, A036068, A036070, A036083, respectively.
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EXAMPLE
| {1}; {1,1}; {1,3,3}; {1,6,16,16}; {1,10,50,125,125}; .... E.g. third row {1,3,3} corresponds to polynomial p{3,x)= 1+3*x+3*x^2.
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CROSSREFS
| a(n, 0)= A000012 (powers of 1), a(n, 1)= A000217 (triangular numbers), a(n, n)= A000272(n+1), n >= 0 (diagonal), a(n, n-1)= A000272(n+1), n >= 1.
For n = 0..5 the row sequences a(n, m), m >= 0, are the first columns of the triangles A023531 (unit matrix), A030528, A049324, A049325, A049326, A049327, respectively.
Cf. A033842, A046757.
Sequence in context: A199034 A138464 A117279 * A084144 A180735 A116401
Adjacent sequences: A049320 A049321 A049322 * A049324 A049325 A049326
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
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