A327000
A(n, k) = A309522(n, k) - A327001(n, k) for n >= 0 and k >= 3, square array read by ascending antidiagonals.
Original entry on oeis.org
1, 1, 6, 3, 9, 26, 10, 117, 68, 100, 35, 2574, 4500, 517, 365, 126, 70005, 748616, 199155, 4163, 1302, 462, 2082759, 192426260, 282846568, 10499643, 36180, 4606, 1716, 65061234, 59688349943, 799156187475, 141482705378, 663488532, 341733, 16284
Offset: 0
Array starts:
n\k [ 3 4 5 6 7 ]
[0] 1, 6, 26, 100, 365, ... [A125107]
[1] 1, 9, 68, 517, 4163, ... [A048742]
[2] 3, 117, 4500, 199155, 10499643, ... [A326995]
[3] 10, 2574, 748616, 282846568, 141482705378, ... [A327002]
[4] 35, 70005, 192426260, 799156187475, 4961959681629275, ...
[5] 126, 2082759, 59688349943, 3097220486457142, 278271624962638244163, ...
A001700,
-
ListTools:-Flatten([seq(seq(A309522(n-k, k) - A327001(n-k, k), k=3..n), n=3..10)]);
A377659
a(n) = Motzkin(n) - 2^(n - 1 + 0^n) = A001006(n) - A011782(n).
Original entry on oeis.org
0, 0, 0, 0, 1, 5, 19, 63, 195, 579, 1676, 4774, 13463, 37739, 105442, 294188, 820699, 2291243, 6405310, 17937140, 50327731, 141498983, 398666071, 1125566111, 3184339189, 9026625285, 25636264044, 72940663938, 207889060481, 593474349373, 1696848600299, 4858687934567
Offset: 0
N: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
A001006: 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, ...
A011782: 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ...
a: 0, 0, 0, 0, 1, 5, 19, 63, 195, 579, ...
.
For n = 5 the 5 Motzkin words of length 4 that have at least one term > 1 are:
1221, 1211, 1210, 1121, 0121.
For n = 6 the 19 Motzkin words of length 5 that have at least one term > 1 are:
12321, 12221, 12211, 12210, 12121, 12111, 12110, 12101, 12100, 11221, 11211, 11210, 11121, 10121, 01221, 01211, 01210, 01121, 00121.
-
gf := (1 - x - (1-2*x-3*x^2)^(1/2)) / (2*x^2) - (1 - x) / (1 - 2*x):
ser := series(gf, x, 35): seq(coeff(ser, x, n), n = 0..30);
# Alternative:
a := n -> hypergeom([-n/2 + 1/2, -n/2], [2], 4) - 2^(n - 1 + 0^n);
seq(simplify(a(n)), n = 0..29);
-
A377659[n_] := If[n < 4, 0, HypergeometricPFQ[{-n/2, -n/2 + 1/2}, {2}, 4] - 2^(n - 1)];
Array[A377659, 50, 0] (* Paolo Xausa, Dec 04 2024 *)
-
from itertools import islice
show = lambda f, n: print(list(islice(f(), n)))
def aGen():
a, b, n, z = 1, 2, 2, 1
yield 0
while True:
yield b//n - z
n += 1; z *= 2
a, b = b, (3*(n-1)*n*a + (2*n-1)*n*b)//((n+1)*(n-1))
show(aGen, 31)
-
# Generates Motzkin words (for illustration only).
def motzkin_words(n):
return IntegerListsLex(length=n+1, min_slope=-1, max_slope=1,
ceiling=[0]+[+oo for i in range(n-1)]+[0])
def MWList(n, show=True):
c = 0
for w in motzkin_words(n):
if any(p > 1 for p in w):
c += 1
if show: print(''.join(map(str, w[1:-1])))
return c
for n in range(8): print(f"[{n}] -> {MWList(n)}")
A326995
a(n) = A002105(n+1) - A005046(n), reduced tangent numbers minus the number of partitions of a 2*n-set into even blocks.
Original entry on oeis.org
0, 0, 0, 3, 117, 4500, 199155, 10499643, 663488532, 50115742365, 4497657826905, 476074241776188, 58963860817626567, 8475738174076417335, 1402598717609785850700, 265126817539686778513113, 56822367893441673215117997, 13712983199783483607459996660, 3702793973661590950848375537915
Offset: 0
-
B := BellMatrix(n -> modp(n,2), 37): # defined in A264428.
b := n -> add(k, k in B[2*n+1]):
seq(euler(2*n+1, 0)*(-2)^(n+1) - b(n), n=0..18);
Showing 1-3 of 3 results.
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