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 A027293 Triangular array given by rows: P(n,k) is the number of partitions of n that contain k as a part. 26
 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, 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, 5, 3, 2, 1, 1, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 77 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Triangle read by rows in which row n lists the first n partition numbers A000041 in decreasing order. - Omar E. Pol, Aug 06 2011 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 From Gary W. Adamson, Oct 26 2008: (Start) Row sums = A000070: (1, 2, 4, 7, 12, 19, 30, 45, 67, ...); (this triangle)^2 = triangle A146023. (End) (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 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 LINKS Robert Price, Table of n, a(n) for n = 1..5050 Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012. FORMULA P(n,k) = p(n-k) = A000041(n-k), n>=1, k>=1. - Omar E. Pol, Feb 15 2013 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 From Wolfdieter Lang, Apr 14 2021: (Start) 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). 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. Thanks to Gary W. Adamson for motivating me to look at such matrices and sequences. (End) EXAMPLE The triangle P begins (with offsets 0 it is Pa): n \ k 1 2 3 4 5 6 7 8 9 10 ... 1: 1 2: 1 1 3: 2 1 1 4: 3 2 1 1 5: 5 3 2 1 1 6: 7 5 3 2 1 1 7: 11 7 5 3 2 1 1 8: 15 11 7 5 3 2 1 1 9: 22 15 11 7 5 3 2 1 1 10: 30 22 15 11 7 5 3 2 1 1 ... reformatted by Wolfdieter Lang, Apr 14 2021 MATHEMATICA f[n_] := Block[{t = Flatten[Union /@ IntegerPartitions@n]}, Table[Count[t, i], {i, n}]]; Array[f, 13] // Flatten t[n_, k_] := PartitionsP[n-k]; Table[t[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 24 2014 *) CROSSREFS Every column of P is A000041. Cf. A145975, A010815. Cf. A000070, A146023. Cf. A182700, A027293, A067687, A155002. Cf. A343234 (L-eigen-matrix). Sequence in context: A345418 A194543 A287920 * A104762 A152462 A180360 Adjacent sequences: A027290 A027291 A027292 * A027294 A027295 A027296 KEYWORD nonn,tabl AUTHOR STATUS approved

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Last modified January 31 01:17 EST 2023. Contains 359947 sequences. (Running on oeis4.)