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A116608 Triangle read by rows: T(n,k) is number of partitions of n having k distinct parts (n>=1, k>=1). 50
1, 2, 2, 1, 3, 2, 2, 5, 4, 6, 1, 2, 11, 2, 4, 13, 5, 3, 17, 10, 4, 22, 15, 1, 2, 27, 25, 2, 6, 29, 37, 5, 2, 37, 52, 10, 4, 44, 67, 20, 4, 44, 97, 30, 1, 5, 55, 117, 52, 2, 2, 59, 154, 77, 5, 6, 68, 184, 117, 10, 2, 71, 235, 162, 20, 6, 81, 277, 227, 36, 4, 82, 338, 309, 58, 1, 4, 102 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row n has floor([sqrt(1+8n)-1]/2) terms (number of terms increases by one at each triangular number).

Row sums yield the partition numbers (A000041).

Row n has length A003056(n), hence the first element of column k is in row A000217(k). - Omar E. Pol, Jan 19 2014

LINKS

Alois P. Heinz, Rows n = 1..500, flattened

Emmanuel Briand, On partitions with k corners not containing the staircase with one more corner, arXiv:2004.13180 [math.CO], 2020.

Sang June Lee, Jun Seok Oh, On zero-sum free sequences contained in random subsets of finite cyclic groups, arXiv:2003.02511 [math.CO], 2020.

FORMULA

G.f.: -1 + Product_{j=1..infinity} 1 + tx^j/(1-x^j).

T(n,1) = A000005(n) (number of divisors of n).

T(n,2) = A002133(n).

T(n,3) = A002134(n).

Sum_{k>=1} k * T(n,k) = A000070(n-1).

Sum_{k>=0} k! * T(n,k) = A274174(n). - Alois P. Heinz, Jun 13 2016

T(n + A000217(k), k) = A000712(n), for 0 <= n <= k [Briand]. - Álvar Ibeas, Nov 04 2020

EXAMPLE

T(6,2) = 6 because we have [5,1], [4,2], [4,1,1], [3,1,1,1], [2,2,1,1] and [2,1,1,1,1,1] ([6], [3,3], [3,2,1], [2,2,2] and [1,1,1,1,1,1] do not qualify).

Triangle starts:

  1;

  2;

  2,  1;

  3,  2;

  2,  5;

  4,  6, 1;

  2, 11, 2;

  4, 13, 5;

  3, 17, 10;

  4, 22, 15, 1;

  ...

MAPLE

g:=product(1+t*x^j/(1-x^j), j=1..30)-1: gser:=simplify(series(g, x=0, 27)): for n from 1 to 23 do P[n]:=sort(coeff(gser, x^n)) od: for n from 1 to 23 do seq(coeff(P[n], t^j), j=1..floor(sqrt(1+8*n)/2-1/2)) od; # yields sequence in triangular form

# second Maple program:

b:= proc(n, i) option remember; local j; if n=0 then 1

      elif i<1 then 0 else []; for j from 0 to n/i do zip((x, y)

      ->x+y, %, [`if`(j>0, 0, [][]), b(n-i*j, i-1)], 0) od; %[] fi

    end:

T:= n-> subsop(1=NULL, [b(n, n)])[]:

seq(T(n), n=1..30); # Alois P. Heinz, Nov 07 2012

MATHEMATICA

p=Product[1+(y x^i)/(1-x^i), {i, 1, 20}]; f[list_]:=Select[list, #>0&]; Flatten[Map[f, Drop[CoefficientList[Series[p, {x, 0, 20}], {x, y}], 1]]] (* Geoffrey Critzer, Nov 28 2011 *)

Table[Length /@ Split[Sort[Length /@ Union /@ IntegerPartitions@n]], {n, 22}] // Flatten (* Robert Price, Jun 13 2020 *)

CROSSREFS

Cf. A000041, A000005, A000070, A002133, A002134.

Cf. A060177 (reflected rows). - Alois P. Heinz, Jan 29 2014

Cf. A274174.

Sequence in context: A289186 A130816 A109951 * A002947 A241605 A128180

Adjacent sequences:  A116605 A116606 A116607 * A116609 A116610 A116611

KEYWORD

nonn,tabf,look

AUTHOR

Emeric Deutsch, Feb 19 2006

STATUS

approved

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Last modified November 28 23:44 EST 2020. Contains 338755 sequences. (Running on oeis4.)