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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 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.)