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%I #72 Jun 05 2021 08:39:46
%S 1,1,1,2,1,1,3,2,1,1,5,3,2,1,1,7,5,3,2,1,1,11,7,5,3,2,1,1,15,11,7,5,3,
%T 2,1,1,22,15,11,7,5,3,2,1,1,30,22,15,11,7,5,3,2,1,1,42,30,22,15,11,7,
%U 5,3,2,1,1,56,42,30,22,15,11,7,5,3,2,1,1,77
%N Triangular array given by rows: P(n,k) is the number of partitions of n that contain k as a part.
%C Triangle read by rows in which row n lists the first n partition numbers A000041 in decreasing order. - _Omar E. Pol_, Aug 06 2011
%C A027293 * an infinite lower triangular matrix with A010815 (1, -1, -1, 0, 0, 1, ...) as the main diagonal the rest zeros = triangle A145975 having row sums = [1, 0, 0, 0, ...]. These matrix operations are equivalent to the comment in A010815 stating "when convolved with the partition numbers = [1, 0, 0, 0, ...]. - _Gary W. Adamson_, Oct 25 2008
%C From _Gary W. Adamson_, Oct 26 2008: (Start)
%C Row sums = A000070: (1, 2, 4, 7, 12, 19, 30, 45, 67, ...);
%C (this triangle)^2 = triangle A146023. (End)
%C (1) It appears that P(n,k) is also the total number of occurrences of k in the last k sections of the set of partitions of n (cf. A182703). (2) It appears that P(n,k) is also the difference, between n and n-k, of the total number of occurrences of k in all their partitions (cf. A066633). - _Omar E. Pol_, Feb 07 2012
%C Sequence B is called a reverse reluctant sequence of sequence A, if B is a triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. The present sequence is the reverse reluctant sequence of (A000041(k-1))_{k>=0}. - _Boris Putievskiy_, Dec 14 2012
%H Robert Price, <a href="/A027293/b027293.txt">Table of n, a(n) for n = 1..5050</a>
%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%F P(n,k) = p(n-k) = A000041(n-k), n>=1, k>=1. - _Omar E. Pol_, Feb 15 2013
%F a(n) = A000041(m), where m = (t*t + 3*t + 4)/2 - n, t = floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Dec 14 2012
%F From _Wolfdieter Lang_, Apr 14 2021: (Start)
%F Pa(n, m) = P(n+1, m+1) = A000041(n-m), for n >= m >= 0, and 0 otherwise, gives the Riordan matrix Pa = (P(x), x), of Toeplitz type, with the o.g.f. P(x) of A000041. The o.g.f. of triangle Pa (the o.g.f. of the row polynomials RPa(n, x) = Sum_{m=0..n} Pa(n, m)*x^m) is G(z, x) = P(z)/(1 - x*z).
%F The (infinite) matrix Pa has the 'L-eigen-sequence' B = A067687, that is, Pa*vec(B) = L*vec(B), with the matrix L with elements L(i, j) = delta(i, j-1) (Kronecker's delta symbol). For such L-eigen-sequences see the Bernstein and Sloane links under A155002.
%F Thanks to _Gary W. Adamson_ for motivating me to look at such matrices and sequences. (End)
%e The triangle P begins (with offsets 0 it is Pa):
%e n \ k 1 2 3 4 5 6 7 8 9 10 ...
%e 1: 1
%e 2: 1 1
%e 3: 2 1 1
%e 4: 3 2 1 1
%e 5: 5 3 2 1 1
%e 6: 7 5 3 2 1 1
%e 7: 11 7 5 3 2 1 1
%e 8: 15 11 7 5 3 2 1 1
%e 9: 22 15 11 7 5 3 2 1 1
%e 10: 30 22 15 11 7 5 3 2 1 1
%e ... reformatted by _Wolfdieter Lang_, Apr 14 2021
%t f[n_] := Block[{t = Flatten[Union /@ IntegerPartitions@n]}, Table[Count[t, i], {i, n}]]; Array[f, 13] // Flatten
%t t[n_, k_] := PartitionsP[n-k]; Table[t[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jan 24 2014 *)
%Y Every column of P is A000041.
%Y Cf. A145975, A010815.
%Y Cf. A000070, A146023.
%Y Cf. A182700, A027293, A067687, A155002.
%Y Cf. A343234 (L-eigen-matrix).
%K nonn,tabl
%O 1,4
%A _Clark Kimberling_
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