A181609
Kendell-Mann numbers in terms of Mahonian distribution.
Original entry on oeis.org
2, 3, 7, 23, 108, 604, 3980, 30186, 258969, 2479441, 26207604, 303119227, 3807956707, 51633582121, 751604592219, 11690365070546, 193492748067369, 3395655743755865, 62980031819261211, 1230967683216803500
Offset: 2
M(2)=2, M(3)=3, M(4)=7,...
A186860
Largest coefficient of (1)(1+2x)(1+2x+3x^2)*...*(1+2x+3x^2+...+(n+1)*x^n).
Original entry on oeis.org
1, 2, 7, 49, 562, 9132, 207915, 6296448, 239972192, 11427298486, 661227186254, 45688884832738, 3716852205228166, 351101915633367990, 38275029480566516322, 4750162039324230600200, 666311679640315952033655, 105085327413072323807645048
Offset: 1
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f[n_] := Max@ CoefficientList[ Expand@ Product[ Sum[(i + 1)*x^i, {i, 0, j}], {j, n - 1}], x]; Array[f, 18]
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def A186860(n):
p = prod(sum(i*x^(i-1) for i in (1..k)) for k in (1..n))
return Integer(max(p.coefficients())[0]) # D. S. McNeil, Feb 28 2011
A316775
a(n) is the number of permutations of [1..n] that have the same number of inversions as non-inversions.
Original entry on oeis.org
1, 1, 0, 0, 6, 22, 0, 0, 3836, 29228, 0, 0, 25598186, 296643390, 0, 0, 738680521142, 11501573822788, 0, 0, 62119523114983224, 1214967840930909302, 0, 0, 12140037056605135928410, 285899248139692651257566, 0, 0, 4759461354691529363949651814
Offset: 0
Consider a permutation 1432. It has exactly three pairs of numbers, the first of them is 1, that are in increasing order. The other three pairs are in decreasing order. The other 5 permutations of size 4 with this property are 2341, 2413, 3142, 3214, 4123. Thus a(4) = 6.
A369773
Maximal coefficient of (1 + x) * (1 + x - x^2) * ... * (1 + x - x^2 + ... - (-x)^n).
Original entry on oeis.org
1, 1, 2, 3, 4, 6, 19, 59, 233, 1189, 7046, 45326, 356517, 3108808, 30028121, 325635647, 3830546752, 49403859787, 685063715374, 10162709827329, 162776892315940, 2754021620252692, 49463507801582609, 940216720983170113, 18786988751008626812
Offset: 0
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Table[Max[CoefficientList[Product[(1 - Sum[(-x)^j, {j, 1, i}]), {i, 1, n}], x]], {n, 0, 24}]
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from collections import Counter
def A369773(n):
c = {0:1}
for k in range(1,n+1):
d = Counter(c)
for j in c:
a = c[j]
for i in range(1,k+1):
d[j+i] += (a if i&1 else -a)
c = d
return max(c.values()) # Chai Wah Wu, Feb 01 2024
A369774
Maximal coefficient of (1 - x) * (1 - x - x^2) * ... * (1 - x - x^2 - ... - x^n).
Original entry on oeis.org
1, 1, 1, 2, 3, 8, 13, 63, 167, 1227, 5240, 46958, 297080, 3108808, 26714243, 325635647, 3535022425, 49403859787, 646713449897, 10221697892707, 156049674957354, 2756431502525358, 48028121269507891, 940216720983170113, 18359095114316009613
Offset: 0
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Table[Max[CoefficientList[Product[(1 - Sum[x^j, {j, 1, i}]), {i, 1, n}], x]], {n, 0, 24}]
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a(n) = vecmax(Vec(prod(k=1, n, 1 - sum(i=1, k, x^i)))); \\ Michel Marcus, Feb 01 2024
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from collections import Counter
def A369774(n):
c = {0:1}
for k in range(1,n+1):
d = Counter(c)
for j in c:
a = c[j]
for i in range(1,k+1):
d[j+i] -= a
c = d
return max(c.values()) # Chai Wah Wu, Feb 01 2024
A141473
Number of 3-equitable permutations: permutations on n letters equally avoiding each permutation of S_3.
Original entry on oeis.org
6, 2, 2, 0, 0, 4, 2, 0, 0
Offset: 3
Sunil Abraham (sunil.abraham(AT)lmh.ox.ac.uk), Aug 08 2008
The only 3-equitable permutations in S_4: [3, 1, 4, 2], [2, 4, 1, 3].
A369766
Maximal coefficient of Product_{i=1..n} Sum_{j=0..i} x^(i*j).
Original entry on oeis.org
1, 1, 1, 2, 6, 24, 115, 662, 4456, 34323, 298220, 2885156, 30760556, 358379076, 4530375092, 61762729722, 903311893770, 14108704577103, 234387946711329, 4127027097703638, 76774080851679152, 1504640319524566870, 30986929089570280955, 669023741837953551188
Offset: 0
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a:= n-> max(coeffs(expand(mul(add(x^(i*j), j=0..i), i=1..n)))):
seq(a(n), n=0..23); # Alois P. Heinz, Jan 31 2024
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Table[Max[CoefficientList[Product[Sum[x^(i j), {j, 0, i}], {i, 1, n}], x]], {n, 0, 23}]
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a(n) = vecmax(Vec(prod(i=1, n, sum(j=0, i, x^(i*j))))); \\ Michel Marcus, Jan 31 2024
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from collections import Counter
def A369766(n):
c = {0:1,1:1}
for i in range(2,n+1):
d = Counter()
for k in c:
for j in range(0,i*i+1,i):
d[j+k] += c[k]
c = d
return max(c.values()) # Chai Wah Wu, Jan 31 2024
A369775
Maximal coefficient of (1 + x^2) * (1 + x^2 + x^3) * (1 + x^2 + x^3 + x^5) * ... * (1 + x^2 + ... + x^prime(n)).
Original entry on oeis.org
1, 1, 2, 5, 16, 65, 293, 1807, 12946, 106475, 972260, 9858553, 109451903, 1323071345, 17398667717, 247055196932, 3753507625272, 60680317203979, 1043036844360792, 18969267205680868, 364107881070036688, 7366172106829696356, 156467911373737550264
Offset: 0
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Table[Max[CoefficientList[Product[(1 + Sum[x^Prime[j], {j, 1, i}]), {i, 1, n}], x]], {n, 0, 22}]
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a(n) = vecmax(Vec(prod(k=1, n, 1 + sum(i=1, k, x^prime(i))))); \\ Michel Marcus, Feb 01 2024
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from collections import Counter
from sympy import prime, primerange
def A369775(n):
if n == 0: return 1
c, p = {0:1}, list(primerange(prime(n)+1))
for k in range(1,n+1):
d = Counter(c)
for j in c:
a = c[j]
for i in p[:k]:
d[j+i] += a
c = d
return max(c.values()) # Chai Wah Wu, Feb 01 2024
A349404
The maximal coefficient in the expansion of x_1(x_1 + x_2)...(x_1 + x_2 + ... + x_n).
Original entry on oeis.org
1, 1, 1, 2, 4, 9, 27, 96, 384, 1536, 7500, 37500, 194400, 1166400, 7563150, 52942050, 385351680, 3082813440, 24998984640, 224990861760, 2024917755840, 19051200000000, 190512000000000, 1944663768432000, 21391301452752000
Offset: 0
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Max/@Table[Values@CoefficientRules[Times@@Array[Total@Array[x,#]&,n]],{n,0,12}] (* Giorgos Kalogeropoulos, Nov 16 2021 *)
A369791
a(n) is the maximal coefficient of (1 + x^a(1)) * (1 + x^a(1) + x^a(2)) * ... * (1 + x^a(1) + x^a(2) + ... + x^a(n-1)).
Original entry on oeis.org
1, 1, 1, 3, 8, 22, 70, 262, 1088, 5076, 26490, 146542, 896402, 5662622, 39826304, 279072864, 2232912264, 17866212198, 153323343990, 1379920982310, 13115759159982, 131158174385100
Offset: 0
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a[n_] := a[n] = Max[CoefficientList[Product[(1 + Sum[x^a[j], {j, 1, i}]), {i, 1, n - 1}], x]]; Table[a[n], {n, 0, 15}]
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from itertools import islice
from collections import Counter
def A369791_gen(): # generator of terms
c, a = {0:1}, []
while True:
a.append(max(c.values()))
yield a[-1]
d = Counter(c)
for k in c:
for b in a:
d[k+b] += c[k]
c = d
A369791_list = list(islice(A369791_gen(),10)) # Chai Wah Wu, Feb 01 2024
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