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A264401 Triangle read by rows: T(n,k) is the number of partitions of n having least gap k. 1
1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 4, 4, 2, 1, 4, 6, 4, 1, 7, 8, 5, 2, 8, 11, 8, 3, 12, 15, 10, 4, 1, 14, 20, 15, 6, 1, 21, 26, 19, 9, 2, 24, 35, 27, 12, 3, 34, 45, 34, 17, 5, 41, 58, 47, 23, 6, 1, 55, 75, 59, 31, 10, 1, 66, 96, 79, 41, 13, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

The "least gap" of a partition is the least positive integer that is not a part of the partition. For example, the least gap of the partition [7,4,2,2,1] is 3.

Sum of entries in row n is A000041(n).

T(n,1) = A002865(n).

Sum_{k>=1} k*T(n,k) = A022567(n).

LINKS

Alois P. Heinz, Rows n = 0..1000, flattened

P. J. Grabner, A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.

FORMULA

G.f.: G(t,x) = Sum_{j>=1} (t^j*x^{j(j-1)/2}*(1-x^j))/Product_{i>=1}(1-x^i).

EXAMPLE

Row n=5 is 2,3,2; indeed, the least gaps of [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], and [1,1,1,1,1] are 1, 2, 1, 2, 3, 3, and 2, respectively (i.e., two 1s, three 2s, and two 3s).

Triangle starts:

1;

0,1;

1,1;

1,1,1;

2,2,1;

2,3,2;

4,4,2,1;

MAPLE

g := (sum(t^j*x^((1/2)*j*(j-1))*(1-x^j), j = 1 .. 80))/(product(1-x^i, i = 1 .. 80)): gser := simplify(series(g, x = 0, 23)): for n from 0 to 30 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 1 .. degree(P[n])) end do; # yields sequence in triangular form

# second Maple program:

b:= proc(n, i) option remember; `if`(n=0, `if`(i=0, [1, 0],

      [0, x]), `if`(i<1, 0, (p-> [0, p[2] +p[1]*x^i])(

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

    end:

T:= n->(p->seq(coeff(p, x, i), i=1..degree(p)))(b(n, n+1)[2]):

seq(T(n), n=0..20);  # Alois P. Heinz, Nov 29 2015

MATHEMATICA

Needs["Combinatorica`"]; {1, 0}~Join~Flatten[Table[Count[Map[If[# == {}, 0, First@ #] &@ Complement[Range@ n, #] &, Combinatorica`Partitions@ n], n_ /; n == k], {n, 17}, {k, n}] /. 0 -> Nothing] (* Michael De Vlieger, Nov 21 2015 *)

CROSSREFS

Cf. A000041, A002865, A022567.

Sequence in context: A022872 A091423 A221914 * A173304 A029251 A263073

Adjacent sequences:  A264398 A264399 A264400 * A264402 A264403 A264404

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Nov 21 2015

STATUS

approved

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Last modified July 22 20:51 EDT 2019. Contains 325226 sequences. (Running on oeis4.)