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A240185 Irregular triangular array:  T(n,k) = number of partitions p of n such that k is the number of numbers between min(p) and max(p) that are not parts of p. 2
1, 1, 2, 3, 4, 1, 5, 1, 1, 7, 2, 1, 1, 8, 3, 2, 1, 1, 10, 6, 2, 2, 1, 1, 13, 5, 6, 2, 2, 1, 1, 15, 10, 6, 5, 2, 2, 1, 1, 18, 12, 9, 6, 5, 2, 2, 1, 1, 23, 16, 12, 10, 5, 5, 2, 2, 1, 1, 26, 21, 16, 12, 10, 5, 5, 2, 2, 1, 1, 31, 28, 22, 16, 13, 9, 5, 5, 2, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

T(n,0) = number of partitions of n such that every number between the least part and greatest part is a part, as given by A034296(n); (sum of numbers in row n) = A000041(n), for n >= 0.

Number of entries in row n is n-2 (n>=3). - Emeric Deutsch, Dec 30 2015

Sum(k*T(n,k), k>=0) = A265252(n). - Emeric Deutsch, Dec 30 2015

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000

FORMULA

G.f.: G(t,x) = Sum_{i>=1} x^i/(1-x^i) + Sum_{i>=1} x^{2i+1}/((1-x^i)(1-x^{i+1}) + Sum_{i>=1} Sum(j>=i+2} (x^{i+j}/((1-x^i)(1-x^j)) * Product_{k=i+1..j-1} (t+x^k/(1-x^k))). - Emeric Deutsch, Dec 30 2015

EXAMPLE

First 12 rows:

1

1

2

3

4 ... 1

5 ... 1 ... 1

7 ... 2 ... 1 ... 1

8 ... 3 ... 2 ... 1 ... 1

10 .. 6 ... 2 ... 2 ... 1 ... 1

13 .. 5 ... 6 ... 2 ... 2 ... 1 ... 1

15 .. 10 .. 6 ... 5 ... 2 ... 2 ... 1 ... 1

18 .. 12 .. 9 ... 6 ... 5 ... 2 ... 2 ... 1 ... 1

23 .. 16 .. 12 .. 10 .. 5 ... 5 ... 2 ... 2 ... 1 .. 1

To get row 6 (counting the top row as row 0), let p range through the partitions of 6, with m(p) = numbers between min(p) and max(p) that are missing:

p ....... m(p) ... #m(p)

6 ...... (none) ... 0

51 ...... 2,3,4 ... 3

42 ...... 3 ....... 1

411 ..... 2,3 ..... 2

33 ..... (none) ... 0

321 .... (none) ... 0

3111 .... 2 ....... 1

222 .... (none) ... 0

2211 ... (none) ... 0

21111... (none) ... 0

111111 . (none) ... 0

The column headed #m(p) shows seven 0's, two 1's, one 2, and one 3, so that row 6 of the array is 7 2 1 1.

MAPLE

g := add(x^i/(1-x^i), i = 1 .. 80) + add(x^(2*i+1)/((1-x^i)*(1-x^(i+1))), i = 1 .. 80)+add(add(x^(i+j)*mul(t+x^k/(1-x^k), k = i+1 .. j-1)/((1-x^i)*(1-x^j)), j = i+2 .. 80), i = 1 .. 80): gser := simplify(series(g, x = 0, 25)): for n to 20 do P[n] := sort(coeff(gser, x, n)) end do: 1; 1; 2; for n from 3 to 20 do seq(coeff(P[n], t, q), q = 0 .. n-3) end do; # yields sequence in triangular form, Emeric Deutsch, Dec 30 2015

MATHEMATICA

z = 30; m = Map[Last[Transpose[Tally[Sort[Map[Total[(-Differences[#] /. {{} -> {1}, 0 -> 1}) - 1] &, IntegerPartitions[#]]]]]] &, -1 + Range[z]] (* A240185, array *)

Flatten[m]  (* A240185, sequence *)

g[n_] := Map[Total[(-Differences[#] /. {{} -> {1}, 0 -> 1}) - 1] &, IntegerPartitions[n]]; Table[Count[g[n], 0], {n, 0, z}]  (* A034296 *)

(* Peter J. C. Moses, Apr 03 2014 *)

CROSSREFS

Cf. A034296, A000041, A265252.

Sequence in context: A117742 A117716 A211234 * A292032 A097150 A319826

Adjacent sequences:  A240182 A240183 A240184 * A240186 A240187 A240188

KEYWORD

nonn,tabf,easy

AUTHOR

Clark Kimberling, Apr 03 2014

STATUS

approved

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Last modified October 21 14:42 EDT 2021. Contains 348155 sequences. (Running on oeis4.)