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A053632 Irregular triangle read by rows giving coefficients in expansion of Product_{k=1..n} (1 + x^k). 36
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; text; internal format)
OFFSET

0,11

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

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

Rosa, Alexander; Znam, Stefan, 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.

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

EXAMPLE

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;

...

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 *)

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).

Sequence in context: A178058 A260971 A053258 * A242217 A306734 A124060

Adjacent sequences:  A053629 A053630 A053631 * A053633 A053634 A053635

KEYWORD

tabf,nonn,easy,nice

AUTHOR

N. J. A. Sloane, Mar 22 2000

STATUS

approved

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Last modified July 21 02:02 EDT 2019. Contains 325189 sequences. (Running on oeis4.)