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A285898 Triangle read by row: T(n,k) = number of partitions of n into exactly k consecutive parts (1 <= k <= n). 6
1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,29

COMMENTS

To partition n into k parts, we see if m exists such that m + (m + 1) + ... + (m + k - 1) = k*m + binomial(k, 2) = n exists. a(n) = 1 if and only if (n - binomial(k, 2)) / k is integer and larger than 0. - David A. Corneth, Apr 28 2017

It appears that this a full version of the irregular triangle A237048. - Omar E. Pol, Apr 28 2017

LINKS

Table of n, a(n) for n=1..136.

FORMULA

Sigma(n) = Sum_{k=1..n} (-1)^(k-1) * ((Sum_{j=k..n} T(j,k))^2 - (Sum_{j=k..n} T(j-1,k))^2), assuming that T(k-1,k) = 0. - Omar E. Pol, Oct 10 2018

EXAMPLE

Triangle begins:

1;

1, 0;

1, 1, 0;

1, 0, 0, 0;

1, 1, 0, 0, 0;

1, 0, 1, 0, 0, 0;

1, 1, 0, 0, 0, 0, 0;

1, 0, 0, 0, 0, 0, 0, 0;

1, 1, 1, 0, 0, 0, 0, 0, 0;

1, 0, 0, 1, 0, 0, 0, 0, 0, 0;

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0;

1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0;

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

...

For n = 15 there are four partitions of 15 into consecutive parts: [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1]. These partitions are formed by 1, 2, 3 and 5 consecutive parts respectively, so the 15th row of the triangle is [1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0].

MAPLE

A285898 := proc(n)

    corn := (n-binomial(k, 2))/k ;

    if type(corn, 'integer') then

        if corn > 0 then

            1 ;

        else

            0;

        end if;

    else

        0 ;

    end if;

end proc: # R. J. Mathar, Apr 30 2017

MATHEMATICA

Table[Function[t, Function[s, ReplacePart[s, Map[# -> 1 &, t]]]@ ConstantArray[0, n]]@ Map[Length, Select[IntegerPartitions@ n, Length@ # == 1 || Union@ Differences@ # == {-1} &]], {n, 15}] // Flatten (* Michael De Vlieger, Apr 28 2017 *)

PROG

(PARI) T(n, k) = n-=binomial(k, 2); if(n>0, n%k==0) \\ David A. Corneth, Apr 28 2017

(Python)

from sympy import binomial

def T(n, k):

    n=n - binomial(k, 2)

    if n>0:

        return 1 if n%k==0 else 0

    return 0

for n in xrange(1, 21): print [T(n, k) for k in xrange(1, n + 1)] # Indranil Ghosh, Apr 28 2017

CROSSREFS

Row sums give A001227.

Cf. A000203, A038547, A067742, A082647, A131576, A204217, A237048, A237593, A245579, A261699, A279387, A281009, A204217.

Sequence in context: A279760 A287457 A324917 * A185118 A240332 A156297

Adjacent sequences:  A285895 A285896 A285897 * A285899 A285900 A285901

KEYWORD

nonn,tabl,easy

AUTHOR

Omar E. Pol and N. J. A. Sloane, Apr 28 2017

STATUS

approved

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Last modified October 19 21:28 EDT 2019. Contains 328244 sequences. (Running on oeis4.)