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A053632
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Irregular triangle read by rows giving coefficients in expansion of Product_{k=1..n} (1 + x^k).
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82
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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
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OFFSET
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0,11
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COMMENTS
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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, etc. - Antti Karttunen, Feb 13 2002
T(n,k) = number of partitions of k into distinct parts <= n. Triangle of distribution of Wilcoxon's signed rank statistic. - Mitch Harris, 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, 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 dollars. - Geoffrey Critzer, May 16 2010
With offset 1, also the number of integer compositions of n whose partial sums add up to k for k = n..n(n+1)/2. For example, row n = 6 counts the following compositions:
6 15 24 33 42 51 141 231 321 411 1311 2211 3111 12111 21111 111111
114 123 132 222 312 1131 1221 2121 11121 11211
213 1113 1122 1212 2112 1111
(End)
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REFERENCES
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A. V. Yurkin, New binomial and new view on light theory, (book), 2013, 78 pages, no publisher listed.
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LINKS
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FORMULA
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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). (End)
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EXAMPLE
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Triangle begins:
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;
...
Row n = 4 counts the following binary words, where k = sum of positions of zeros:
1111 0111 1011 0011 0101 0110 0001 0010 0100 1000 0000
1101 1110 1001 1010 1100
Row n = 5 counts the following strict partitions of k with all parts <= n (0 is the empty partition):
0 1 2 3 4 5 42 43 53 54 532 542 543 5431 5432 54321
21 31 32 51 52 431 432 541 5321 5421
41 321 421 521 531 4321
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MAPLE
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with(gfun, seriestolist); map(op, [seq(seriestolist(series(mul(1+(z^i), i=1..n), z, binomial(n+1, 2)+1)), n=0..10)]); # Antti Karttunen, Feb 13 2002
# second Maple program:
g:= proc(n) g(n):= `if`(n=0, 1, expand(g(n-1)*(1+x^n))) end:
T:= n-> seq(coeff(g(n), x, k), k=0..degree(g(n))):
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MATHEMATICA
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Table[CoefficientList[ Series[Product[(1 + t^i), {i, 1, n}], {t, 0, 100}], t], {n, 0, 8}] // Grid (* Geoffrey Critzer, May 16 2010 *)
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CROSSREFS
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Rows reduced modulo 2 and interpreted as binary numbers: A068052, A068053. Rows converge towards A000009.
Cf. A285101 (multiplicative encoding of each row), A285103 (number of odd terms on row n), A285105 (number of even terms).
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KEYWORD
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tabf,nonn,easy,nice
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AUTHOR
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STATUS
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approved
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