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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). 116
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 (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
Emmanuel Briand, On partitions with k corners not containing the staircase with one more corner, arXiv:2004.13180 [math.CO], 2020.
Sang June Lee and 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 *)
PROG
(Python)
from math import isqrt
from itertools import count, islice
from sympy.utilities.iterables import partitions
def A116608_gen(): # generator of terms
return (sum(1 for p in partitions(n) if len(p)==k) for n in count(1) for k in range(1, (isqrt((n<<3)+1)-1>>1)+1))
A116608_list = list(islice(A116608_gen(), 30)) # Chai Wah Wu, Sep 14 2023
(Python)
from functools import cache
@cache
def P(n: int, k: int, r: int) -> int:
if n == 0: return 1 if k == 0 else 0
if k == 0: return 0
if r == 0: return 0
return sum(P(n - r * j, k - 1, r - 1)
for j in range(1, n // r + 1)) + P(n, k, r - 1)
def A116608triangle(rows: int) -> list[int]:
return list(filter(None, [P(n, k, n) for n in range(1, rows)
for k in range(1, n + 1)]))
print(A116608triangle(22)) # Peter Luschny, Sep 14 2023, courtesy of Amir Livne Bar-on
CROSSREFS
Cf. A060177 (reflected rows). - Alois P. Heinz, Jan 29 2014
Cf. A274174.
Sequence in context: A289186 A130816 A109951 * A002947 A241605 A128180
KEYWORD
nonn,tabf,look
AUTHOR
Emeric Deutsch, Feb 19 2006
STATUS
approved

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Last modified March 19 09:40 EDT 2024. Contains 370981 sequences. (Running on oeis4.)