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A053632
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Array giving coefficients in expansion of Product_{k=1..n} (1+x^k).
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12
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 4, 4, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8, 7, 7, 6, 5, 5, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,11
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COMMENTS
| Or, triangle T(n,k) read by rows, giving number of subsets of {1,2,...,n} with sum k. - Roger CUCULIERE (cuculier(AT)imaginet.fr), Nov 19 2000
Row n consists of A000124[n] terms. These are also the successive vectors (their nonzero elements) when one starts with the infinite vector (of zeros) with 1 inserted somewhere and then shifts it one step (right or left) and adds to the original, then shifts the result two steps and adds, three steps and adds, et cetera.
T(n,k) = number of partitions of k into distinct parts <= n. Triangle of distribution of Wilcoxon's signed rank statistic. - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Mar 23 2006
T(n,k)=number of binary words of length n in which the sum of the positions of the 0's is k. Example: T(4,5)=2 because we have 0110 (sum of the positions of the 0's is 1+4=5) and 1001 (sum of the positions of the 0's is 2+3=5). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 23 2006
A fair coin is flipped n times. You receive i dollars for a "success" on the i'th flip,1<=i<=n. T(n,k)/2^n is the probability that you will receive exactly k dollars. Your expectation is n(n+1)/4 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 16 2010]
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REFERENCES
| Wilcoxon, F., Individual Comparisons by Ranking Methods, Biometrics Bulletin, v. 1, no. 6 (1945), p. 80-83.
A. V. Yurkin, On similarity of systems of geometrical and arithmetic triangles, in Mathematics, Computing, Education Conference XIX, 2012. http://www.mce.biophys.msu.ru/eng/archive/abstracts/mce19/sect1138/doc150220/
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LINKS
| S. R. Finch, Signum equations and extremal coefficients.
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FORMULA
| T(n,k) = T(n-1, k) + T(n-1, k-n), T(0,0)=1, T(0,k) = 0, T(n,k) = 0 if k < 0 or k > (n+1 choose 2). g.f. = (1+x)(1+x^2)...(1+x^n). - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Mar 23 2006
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EXAMPLE
| 1; 1,1; 1,1,1,1; 1,1,1,2,1,1,1; 1,1,1,2,2,2,2,2,1,1,1; 1,1,1,2,2,3,3,3,3,3,3,2,2,1,1,1; ...
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MAPLE
| with(gfun, seriestolist); map(op, [seq(seriestolist(series(mul(1+(z^i), i=1..n), z, binomial(n+1, 2)+1)), n=0..10)]);
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MATHEMATICA
| Table[CoefficientList[ Series[Product[(1 + t^i), {i, 1, n}], {t, 0, 100}], t], {n, 0, 8}] // Grid [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), May 16 2010]
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CROSSREFS
| Cf. A053633, A068009. The rows interpreted as binary numbers: A068052, A068053. The rows converge towards A000009.
Sequence in context: A047072 A178058 A053258 * A124060 A140194 A159923
Adjacent sequences: A053629 A053630 A053631 * A053633 A053634 A053635
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KEYWORD
| tabf,nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mar 22 2000
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EXTENSIONS
| Comments and Maple code from Antti Karttunen, Feb 13 2002
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