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A053632 Irregular triangle read by rows giving coefficients in expansion of Product_{k=1..n} (1 + x^k).

%I A053632 #84 Oct 20 2023 12:34:09
%S A053632 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,
%T A053632 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,
%U A053632 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
%N A053632 Irregular triangle read by rows giving coefficients in expansion of Product_{k=1..n} (1 + x^k).
%C A053632 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
%C A053632 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
%C A053632 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
%C A053632 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
%C A053632 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
%C A053632 From _Gus Wiseman_, Jan 02 2023: (Start)
%C A053632 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:
%C A053632   6 15 24 33  42  51  141  231  321  411  1311 2211  3111  12111 21111 111111
%C A053632           114 123 132 222  312  1131 1221 2121 11121 11211
%C A053632                   213 1113 1122 1212 2112 1111
%C A053632 (End)
%D A053632 A. V. Yurkin, New binomial and new view on light theory, (book), 2013, 78 pages, no publisher listed.
%H A053632 Alois P. Heinz, <a href="/A053632/b053632.txt">Rows n = 0..40, flattened</a>
%H A053632 S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Signum equations and extremal coefficients</a>.
%H A053632 Steven R. Finch, <a href="/A000980/a000980.pdf">Signum equations and extremal coefficients</a>, February 7, 2009. [Cached copy, with permission of the author]
%H A053632 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000008/">The major index of an integer composition</a>
%H A053632 Alexander Rosa and Štefan Znám, <a href="/A002703/a002703.pdf">A combinatorial problem in the theory of congruences. (Russian)</a>, Mat.-Fys. Casopis Sloven. Akad. Vied 15 1965 49-59. [Annotated scanned copy] See Table 1.
%H A053632 F. Wilcoxon, <a href="http://www.jstor.org/stable/3001968">Individual Comparisons by Ranking Methods</a>, Biometrics Bulletin, v. 1, no. 6 (1945), pp. 80-83.
%H A053632 A. V. Yurkin, <a href="http://www.mce.biophys.msu.ru/eng/archive/abstracts/mce19/sect1138/doc150220/">On similarity of systems of geometrical and arithmetic triangles</a>, in Mathematics, Computing, Education Conference XIX, 2012.
%H A053632 A. V. Yurkin, <a href="http://arxiv.org/abs/1302.6287">New view on the diffraction discovered by Grimaldi and Gaussian beams</a>, arXiv preprint arXiv:1302.6287 [physics.optics], 2013.
%H A053632 A. V. Yurkin, <a href="http://www.mce.biophys.msu.ru/eng/archive/abstracts/mce22/sect1138/doc216454/">About the evident description of distribution of beams and "wavy geometrical trajectories" in long thin pipes</a>, 2014 (original in Russian).
%H A053632 A. V. Yurkin, <a href="https://www.researchgate.net/profile/Alexander_Yurkin/publication/274072415">Symmetric triangle of Pascal and non-linear arithmetic parallelepiped</a>, Book Manuscript, Research Gate 2015.
%F A053632 From _Mitch Harris_, Mar 23 2006: (Start)
%F A053632 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).
%F A053632 G.f.: (1+x)*(1+x^2)*...*(1+x^n). (End)
%F A053632 Sum_{k>=0} k * T(n,k) = A001788(n). - _Alois P. Heinz_, Feb 09 2017
%F A053632 max_{k>=0} T(n,k) = A025591(n). - _Alois P. Heinz_, Jan 20 2023
%e A053632 Triangle begins:
%e A053632   1;
%e A053632   1, 1;
%e A053632   1, 1, 1, 1;
%e A053632   1, 1, 1, 2, 1, 1, 1;
%e A053632   1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1;
%e A053632   1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1;
%e A053632   1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 4, 4, 4, 3, 2, 2, 1, 1, 1;
%e A053632   ...
%e A053632 Row n = 4 counts the following binary words, where k = sum of positions of zeros:
%e A053632   1111  0111  1011  0011  0101  0110  0001  0010  0100  1000  0000
%e A053632                     1101  1110  1001  1010  1100
%e A053632 Row n = 5 counts the following strict partitions of k with all parts <= n (0 is the empty partition):
%e A053632   0  1  2  3  4  5  42  43  53  54  532  542  543  5431 5432 54321
%e A053632            21 31 32 51  52  431 432 541  5321 5421
%e A053632                  41 321 421 521 531 4321
%p A053632 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
%p A053632 # second Maple program:
%p A053632 g:= proc(n) g(n):= `if`(n=0, 1, expand(g(n-1)*(1+x^n))) end:
%p A053632 T:= n-> seq(coeff(g(n), x, k), k=0..degree(g(n))):
%p A053632 seq(T(n), n=0..10);  # _Alois P. Heinz_, Nov 19 2012
%t A053632 Table[CoefficientList[ Series[Product[(1 + t^i), {i, 1, n}], {t, 0, 100}], t], {n, 0, 8}] // Grid (* _Geoffrey Critzer_, May 16 2010 *)
%Y A053632 Cf. A053633, A068009.
%Y A053632 Rows reduced modulo 2 and interpreted as binary numbers: A068052, A068053. Rows converge towards A000009.
%Y A053632 Row sums give A000079.
%Y A053632 Cf. A001788, A028362.
%Y A053632 Cf. A285101 (multiplicative encoding of each row), A285103 (number of odd terms on row n), A285105 (number of even terms).
%Y A053632 Row lengths are A000124.
%Y A053632 A reciprocal version is (A033999, A219977, A291983, A291984, A291985, ...).
%Y A053632 A negative version is A231599.
%Y A053632 A version for partitions is A358194, reversed partitions A264034.
%Y A053632 Cf. A025591, A029931, A063865, A152947, A318283, A359042.
%K A053632 tabf,nonn,easy,nice
%O A053632 0,11
%A A053632 _N. J. A. Sloane_, Mar 22 2000