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A096144 Triangle T(n,k) = number of partitions of n in which the least part occurs exactly k times, k=1..n. 2
1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 4, 1, 1, 0, 1, 4, 3, 2, 1, 0, 1, 7, 3, 2, 1, 1, 0, 1, 8, 6, 2, 3, 1, 1, 0, 1, 12, 5, 6, 2, 2, 1, 1, 0, 1, 14, 11, 5, 4, 3, 2, 1, 1, 0, 1, 21, 11, 8, 5, 4, 2, 2, 1, 1, 0, 1, 24, 17, 11, 9, 4, 5, 2, 2, 1, 1, 0, 1, 34, 20, 15, 9, 8, 4, 4, 2, 2, 1, 1, 0, 1, 41, 30, 18, 14, 9, 7, 5, 4, 2, 2, 1, 1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Reversed rows converge to A002865. - Joerg Arndt, Jul 07 2014

T(n,k) is the number of partitions of n for which the difference between the two largest distinct parts is k (in partitions having only 1 part, we assume that 0 is also a part). This follows easily from the definition by taking the conjugate partitions. Example: T(7,2) = 3 because we have [3,1,1,1,1], [3,3,1], and [4,2,1]. - Emeric Deutsch, Dec 05 2015.

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

FORMULA

G.f. for k-th column: sum(m>=1, x^(k*m)/prod(i>=m+1, 1-x^i ) ).

EXAMPLE

Triangle starts:

01:  1

02:  1 1

03:  2 0 1

04:  2 2 0 1

05:  4 1 1 0 1

06:  4 3 2 1 0 1

07:  7 3 2 1 1 0 1

08:  8 6 2 3 1 1 0 1

09: 12 5 6 2 2 1 1 0 1

10: 14 11 5 4 3 2 1 1 0 1

11: 21 11 8 5 4 2 2 1 1 0 1

12: 24 17 11 9 4 5 2 2 1 1 0 1

13: 34 20 15 9 8 4 4 2 2 1 1 0 1

14: 41 30 18 14 9 7 5 ...

T(7,2)=3 because we have: 5+1+1, 3+2+2, 3+2+1+1. - Geoffrey Critzer, Jun 20 2014

MAPLE

b:= proc(n, i) option remember; `if`(i=1, x^n,

      `if`(irem(n, i, 'k')=0, x^k, 0)+

       add(b(n-i*j, i-1), j=0..(n-1)/i))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):

seq(T(n), n=1..20);  # Alois P. Heinz, Jun 20 2014

MATHEMATICA

nn=20; Table[Take[Map[Drop[#, 1]&, Drop[CoefficientList[Series[Sum[y x^k/(1-y x^k) Product[1/(1- x^j), {j, k+1, nn}], {k, 1, nn}], {x, 0, nn}], {x, y}], 1]][[i]], i], {i, 1, nn}]//Grid (* Geoffrey Critzer, Jun 20 2014 *)

CROSSREFS

Cf. A002865 (first column), A096373 (second column), A000041 (row sums).

T(2n,n) gives A232697(n). - Alois P. Heinz, Jun 20 2014

Sequence in context: A133121 A091602 A035465 * A118401 A147767 A113678

Adjacent sequences:  A096141 A096142 A096143 * A096145 A096146 A096147

KEYWORD

easy,nonn,tabl

AUTHOR

Vladeta Jovovic, Jul 24 2004

STATUS

approved

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Last modified July 6 23:01 EDT 2020. Contains 335484 sequences. (Running on oeis4.)