A188460
Diagonal sums of number triangle A119308.
Original entry on oeis.org
1, 2, 4, 9, 20, 45, 103, 238, 555, 1305, 3090, 7362, 17637, 42460, 102670, 249246, 607256, 1484343, 3639094, 8946260, 22048771, 54467577, 134842844, 334493154, 831296965, 2069573632, 5160747114, 12888640503, 32234749938, 80728619175, 202433907465
Offset: 0
A356261
Partition triangle read by rows, counting irreducible permutations with weakly decreasing Lehmer code, refining triangle A119308.
Original entry on oeis.org
1, 1, 0, 1, 0, 2, 1, 0, 2, 1, 5, 1, 0, 2, 2, 7, 7, 9, 1, 0, 2, 2, 1, 9, 18, 3, 16, 24, 14, 1, 0, 2, 2, 2, 11, 22, 11, 11, 25, 75, 25, 30, 60, 20, 1, 0, 2, 2, 2, 1, 13, 26, 26, 13, 13, 36, 108, 54, 108, 9, 55, 220, 110, 50, 125, 27, 1
Offset: 0
Partition table T(n, k) begins:
[0] 1;
[1] 1;
[2] 0, 1;
[3] 0, 2, 1;
[4] 0, [2, 1], 5, 1;
[5] 0, [2, 2], [7, 7], 9, 1;
[6] 0, [2, 2, 1], [9, 18, 3], [16, 24], 14, 1;
[7] 0, [2, 2, 2], [11, 22, 11, 11], [25, 75, 25], [30, 60], 20, 1;
[8] 0, [2, 2, 2, 1],[13, 26, 26, 13, 13],[36, 108, 54, 108,9],[55, 220, 110],[50, 125], 27, 1;
Summing the bracketed terms reduces the triangle to A119308.
-
# using function perm_red_stats and reducible from A356264
def weakly_decreasing(L: list[int]) -> bool:
return all(x >= y for x, y in zip(L, L[1:]))
@cache
def A356261_row(n: int) -> list[int]:
if n < 2: return [1]
return [0] + [v[1] for v in perm_red_stats(n, irreducible, weakly_decreasing)]
def A356261(n: int, k: int) -> int:
return A356261_row(n)[k]
for n in range(8):
print([n], A356261_row(n))
A188463
Coefficient array of the second column of the inverse of the Riordan array ((1+(r+1)x)/(1+(r+2)x+rx^2), x/(1+(r+2)x+rx^2)).
Original entry on oeis.org
1, 3, 1, 7, 7, 1, 15, 30, 12, 1, 31, 103, 79, 18, 1, 63, 312, 387, 166, 25, 1, 127, 873, 1586, 1085, 305, 33, 1, 255, 2314, 5768, 5719, 2545, 512, 42, 1, 511, 5899, 19261, 25994, 16661, 5285, 805, 52, 1, 1023, 14604, 60337, 106009, 92008, 41881, 10038, 1204, 63, 1
Offset: 0
Triangle begins
1,
3, 1,
7, 7, 1,
15, 30, 12, 1,
31, 103, 79, 18, 1,
63, 312, 387, 166, 25, 1,
127, 873, 1586, 1085, 305, 33, 1,
255, 2314, 5768, 5719, 2545, 512, 42, 1,
511, 5899, 19261, 25994, 16661, 5285, 805, 52, 1
Showing 1-3 of 3 results.
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