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

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

Offset: 0

Author

N. J. A. Sloane, Mar 22 2000

Keywords

• tabf
• nonn
• easy
• nice

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, 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

From Gus Wiseman, Jan 02 2023: (Start)

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)

Examples

			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
		

References

• A. V. Yurkin, New binomial and new view on light theory, (book), 2013, 78 pages, no publisher listed.

Links

• Alois P. Heinz, Rows n = 0..40, flattened
• S. R. Finch, Signum equations and extremal coefficients.
• Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
• FindStat - Combinatorial Statistic Finder, The major index of an integer composition
• Alexander Rosa and Štefan Znám, A combinatorial problem in the theory of congruences. (Russian), Mat.-Fys. Casopis Sloven. Akad. Vied 15 1965 49-59. [Annotated scanned copy] See Table 1.
• F. Wilcoxon, Individual Comparisons by Ranking Methods, Biometrics Bulletin, v. 1, no. 6 (1945), pp. 80-83.
• A. V. Yurkin, On similarity of systems of geometrical and arithmetic triangles, in Mathematics, Computing, Education Conference XIX, 2012.
• A. V. Yurkin, New view on the diffraction discovered by Grimaldi and Gaussian beams, arXiv preprint arXiv:1302.6287 [physics.optics], 2013.
• A. V. Yurkin, About the evident description of distribution of beams and "wavy geometrical trajectories" in long thin pipes, 2014 (original in Russian).
• A. V. Yurkin, Symmetric triangle of Pascal and non-linear arithmetic parallelepiped, Book Manuscript, Research Gate 2015.

Crossrefs

Cf. A053633, A068009.

Rows reduced modulo 2 and interpreted as binary numbers: A068052, A068053. Rows converge towards A000009.

Row sums give A000079.

Cf. A001788, A028362.

Cf. A285101 (multiplicative encoding of each row), A285103 (number of odd terms on row n), A285105 (number of even terms).

Row lengths are A000124.

A reciprocal version is (A033999, A219977, A291983, A291984, A291985, ...).

A negative version is A231599.

A version for partitions is A358194, reversed partitions A264034.

Cf. A025591, A029931, A063865, A152947, A318283, A359042.

Programs

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)]); # 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))):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 19 2012

Mathematica

Table[CoefficientList[ Series[Product[(1 + t^i), {i, 1, n}], {t, 0, 100}], t], {n, 0, 8}] // Grid (* Geoffrey Critzer, May 16 2010 *)

Formula

From Mitch Harris, Mar 23 2006: (Start)

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)

Sum_{k>=0} k * T(n,k) = A001788(n). - Alois P. Heinz, Feb 09 2017

max_{k>=0} T(n,k) = A025591(n). - Alois P. Heinz, Jan 20 2023