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 A206563 Triangle read by rows: T(n,k) = number of odd/even parts >= k in all partitions of n, if k is odd/even. 18
 1, 2, 1, 5, 1, 1, 8, 4, 1, 1, 15, 5, 3, 1, 1, 24, 11, 5, 3, 1, 1, 39, 15, 9, 4, 3, 1, 1, 58, 28, 13, 9, 4, 3, 1, 1, 90, 38, 23, 12, 8, 4, 3, 1, 1, 130, 62, 33, 21, 12, 8, 4, 3, 1, 1, 190, 85, 51, 29, 20, 11, 8, 4, 3, 1, 1, 268, 131, 73, 48, 28, 20, 11, 8, 4, 3, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Let m and n be two positive integers such that m <= n. It appears that any set formed by m connected shells, or m disconnected shells, or a mixture of both, has the same properties described in the section example. (Cf. A135010, A207031, A207032, A212010). - Omar E. Pol, May 01 2012 LINKS FORMULA It appears that T(n,k) = abs(Sum_{j=k..n} (-1)^j*A181187(n,j)). It appears that A066633(n,k) = T(n,k) - T(n,k+2). - Omar E. Pol, Feb 26 2012 EXAMPLE Calculation for n = 6. Write the partitions of 6 and below the sums of their columns: . .   6 .   3 + 3 .   4 + 2 .   2 + 2 + 2 .   5 + 1 .   3 + 2 + 1 .   4 + 1 + 1 .   2 + 2 + 1 + 1 .   3 + 1 + 1 + 1 .   2 + 1 + 1 + 1 + 1 .   1 + 1 + 1 + 1 + 1 + 1 . ------------------------ .  35, 16,  8,  4,  2,  1  --> Row 6 of triangle A181187. .   |  /|  /|  /|  /|  /| .   | / | / | / | / | / | .   |/  |/  |/  |/  |/  | .  19,  8,  4,  2,  1,  1  --> Row 6 of triangle A066633. . More generally, it appears that the sum of column k is also the total number of parts >= k in all partitions of n. It appears that the first differences of the column sums together with 1 give the number of occurences of k in all partitions of n. On the other hand we can see that the partitions of 6 contain: 24  odd parts >= 1 (the odd parts). 11 even parts >= 2 (the even parts). 5   odd parts >= 3. 3  even parts >= 4. 2   odd parts >= 5. 1  even part  >= 6. Then, using the values of the column sums, it appears that: T(6,1) = 35 - 16 + 8 - 4 + 2 - 1 = 24 T(6,2) =      16 - 8 + 4 - 2 + 1 = 11 T(6,3) =           8 - 4 + 2 - 1 = 5 T(6,4) =               4 - 2 + 1 = 3 T(6,5) =                   2 - 1 = 1 T(6,6) =                       1 = 1 So the 6th row of our triangle gives 24, 11, 5, 3, 1, 1. Finally, for all partitions of 6, we can write: The number of  odd parts      is equal to T(6,1) = 24. The number of even parts      is equal to T(6,2) = 11. The number of  odd parts >= 3 is equal to T(6,3) = 5. The number of even parts >= 4 is equal to T(6,4) = 3. The number of  odd parts >= 5 is equal to T(6,5) = 1. The number of even parts >= 6 is equal to T(6,6) = 1. More generally, we can write the same properties for any positive integer. Triangle begins: 1; 2,    1; 5,    1,  1; 8,    4,  1,  1; 15,   5,  3,  1,  1; 24,  11,  5,  3,  1,  1; 39,  15,  9,  4,  3,  1,  1; 58,  28, 13,  9,  4,  3,  1,  1; 90,  38, 23, 12,  8,  4,  3,  1,  1; 130, 62, 33, 21, 12,  8,  4,  3,  1,  1; CROSSREFS Columns 1-2 give A066897, A066898. Cf. A006128, A066633, A181187, A182703, A207031, A207032. Sequence in context: A175178 A256541 A066421 * A299779 A323954 A143983 Adjacent sequences:  A206560 A206561 A206562 * A206564 A206565 A206566 KEYWORD nonn,tabl AUTHOR Omar E. Pol, Feb 15 2012 EXTENSIONS More terms from Alois P. Heinz, Feb 18 2012 STATUS approved

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Last modified November 21 04:41 EST 2019. Contains 329350 sequences. (Running on oeis4.)