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A361682 Array read by descending antidiagonals. A(n, k) is the number of multiset combinations of {0, 1} whose type is defined in the comments. Also A(n, k) = hypergeom([-k, -2], [1], n). 3
1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 13, 7, 1, 1, 15, 25, 22, 9, 1, 1, 21, 41, 46, 33, 11, 1, 1, 28, 61, 79, 73, 46, 13, 1, 1, 36, 85, 121, 129, 106, 61, 15, 1, 1, 45, 113, 172, 201, 191, 145, 78, 17, 1, 1, 55, 145, 232, 289, 301, 265, 190, 97, 19, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
A combination of a multiset M is an unordered selection of k objects of M, where every object can appear at most as many times as it appears in M.
A(n, k) = Cardinality(Union_{j=0..k} Combination(MultiSet(1^[j*n], 0^[(k-j)*n]))), where MultiSet(r^[s], u^[v]) denotes a set that contains the element r with multiplicity s and the element u with multiplicity v; thus the multisets under consideration have n*k elements. Since the base set is {1, 0} the elements can be represented as binary strings. Applying the combination operator to the multisets results in a set of binary strings where '0' resp. '1' can appear at most j*n resp. (k-j)*n times. 'At most' means that they do not have to appear; in other words, the resulting set always includes the empty string ''.
In contrast to the procedure in A361045 we consider here the cardinality of the set union and not the sum of the individual cardinalities. If you want to exclude the empty string, you will find the sequences listed in A361521. The same construction with multiset permutations instead of multiset combinations results in A361043.
A different view can be taken if one considers the hypergeometric representation, hypergeom([-k, -m], [1], n). This is a family of arrays that includes the 'rascal' triangle: the all 1's array A000012 (m = 0), the rascal array A077028 (m = 1), this array (m = 2), and A361731 (m = 3).
LINKS
FORMULA
A(n, k) = 1 + n*k*(4 + n*(k - 1))/2.
T(n, k) = 1 + k*(n - k)*(4 + k*(n - k - 1))/2.
A(n, k) = [x^k] (1 + (n - 1)*x)^2 / (1 - x)^3.
A(n, k) = hypergeom([-k, -2], [1], n).
A(n, k) = A361521(n, k) + 1.
EXAMPLE
Array A(n, k) starts:
[0] 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
[1] 1, 3, 6, 10, 15, 21, 28, 36, ... A000217
[2] 1, 5, 13, 25, 41, 61, 85, 113, ... A001844
[3] 1, 7, 22, 46, 79, 121, 172, 232, ... A038764
[4] 1, 9, 33, 73, 129, 201, 289, 393, ... A081585
[5] 1, 11, 46, 106, 191, 301, 436, 596, ... A081587
[6] 1, 13, 61, 145, 265, 421, 613, 841, ... A081589
[7] 1, 15, 78, 190, 351, 561, 820, 1128, ... A081591
000012 | A028872 | A239325 |
.
Triangle T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 1, 3, 1;
[3] 1, 6, 5, 1;
[4] 1, 10, 13, 7, 1;
[5] 1, 15, 25, 22, 9, 1;
[6] 1, 21, 41, 46, 33, 11, 1;
[7] 1, 28, 61, 79, 73, 46, 13, 1;
[8] 1, 36, 85, 121, 129, 106, 61, 15, 1;
[9] 1, 45, 113, 172, 201, 191, 145, 78, 17, 1.
.
Row 4 of the triangle:
A(0, 4) = 1 = card('').
A(1, 3) = 10 = card('', 0, 00, 000, 1, 10, 100, 11, 110, 111).
A(2, 2) = 13 = card('', 0, 00, 000, 0000, 1, 10, 100, 11, 110, 1100, 111, 1111).
A(3, 1) = 7 = card('', 0, 00, 000, 1, 11, 111).
A(4, 0) = 1 = card('').
MAPLE
A := (n, k) -> 1 + n*k*(4 + n*(k - 1))/2:
for n from 0 to 7 do seq(A(n, k), k = 0..7) od;
# Alternative:
ogf := n -> (1 + (n - 1)*x)^2 / (1 - x)^3:
ser := n -> series(ogf(n), x, 12):
row := n -> seq(coeff(ser(n), x, k), k = 0..9):
seq(print(row(n)), n = 0..7);
PROG
(SageMath)
def A(m: int, steps: int) -> int:
if m == 0: return 1
size = m * steps
cset = set()
for a in range(0, size + 1, m):
S = [str(int(i < a)) for i in range(size)]
C = Combinations(S)
cset.update("".join(i for i in c) for c in C)
return len(cset)
def ARow(n: int, size: int) -> list[int]:
return [A(n, k) for k in range(size + 1)]
for n in range(8): print(ARow(n, 7))
CROSSREFS
Cf. A239592 (main diagonal), A239331 (transposed array).
Sequence in context: A335256 A102036 A121524 * A103141 A085478 A129818
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 21 2023
STATUS
approved

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Last modified June 27 02:41 EDT 2024. Contains 373723 sequences. (Running on oeis4.)