A354305
a(n) is the denominator of Sum_{k=0..n} (-1)^k / (k!)^2.
Original entry on oeis.org
1, 1, 4, 9, 192, 1800, 103680, 529200, 232243200, 8230118400, 1463132160000, 39833773056000, 20858412072960000, 1615657835151360000, 584619573580922880000, 1908495817772544000000, 29184209113159670169600000, 3953548328298349068288000000, 185476873609942457647104000000
Offset: 0
1, 0, 1/4, 2/9, 43/192, 403/1800, 23213/103680, 118483/529200, 51997111/232243200, 1842647621/8230118400, ...
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Table[Sum[(-1)^k/(k!)^2, {k, 0, n}], {n, 0, 18}] // Denominator
nmax = 18; CoefficientList[Series[BesselJ[0, 2 Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Denominator
Accumulate[Table[(-1)^k/(k!)^2,{k,0,20}]]//Denominator (* Harvey P. Dale, Apr 25 2023 *)
A373418
Triangle read by rows: T(n,k) is the number of permutations in symmetric group S_n with (n-k) fixed points and an odd number of non-fixed point cycles. Equivalent to the number of cycles of n items with cycle type defined by non-unity partitions of k <= n that contain an odd number of parts.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 0, 0, 3, 2, 0, 0, 6, 8, 6, 0, 0, 10, 20, 30, 24, 0, 0, 15, 40, 90, 144, 135, 0, 0, 21, 70, 210, 504, 945, 930, 0, 0, 28, 112, 420, 1344, 3780, 7440, 7420, 0, 0, 36, 168, 756, 3024, 11340, 33480, 66780, 66752, 0, 0, 45, 240, 1260, 6048, 28350, 111600, 333900, 667520, 667485
Offset: 0
Triangle begins:
n: {k<=n}
0: {0}
1: {0, 0}
2: {0, 0, 1}
3: {0, 0, 3, 2}
4: {0, 0, 6, 8, 6}
5: {0, 0, 10, 20, 30, 24}
6: {0, 0, 15, 40, 90, 144, 135}
7: {0, 0, 21, 70, 210, 504, 945, 930}
8: {0, 0, 28, 112, 420, 1344, 3780, 7440, 7420}
9: {0, 0, 36, 168, 756, 3024, 11340, 33480, 66780, 66752}
10: {0, 0, 45, 240, 1260, 6048, 28350, 111600, 333900, 667520, 667485}
T(n,0) = 0 because the sole permutation in S_n with n fixed points, namely the identity permutation, has 0 non-fixed point cycles, not an odd number.
T(n,1) = 0 because there are no permutations in S_n with (n-1) fixed points.
Example:
T(3,3) = 2 since S_3 contains 3 permutations with 0 fixed points and an odd number of non-fixed point cycles, namely the derangements (123) and (132).
Worked Example:
T(7,6) = 945 permutations in S_7 with 1 fixed point and an odd number of non-fixed point cycles;
T(7,6) = 945 possible 6- and (2,2,2)-cycles of 7 items.
N(n,y) = possible y-cycles of n items;
N(n,y) = (n!/(n-k)!) / (M(y) * s(y)).
y = partition of k<=n with q parts = (p_1, p_2, ..., p_i, ..., p_q) such that k = Sum_{i=1..q} p_i.
Or:
y = partition of k<=n with d distinct parts, each with multiplicity m_j = (y_1^m_1, y_2^m_2, ..., y_j^m_j, ..., y_d^m_d) such that k = Sum_{j=1..d} m_j*y_j.
M(y) = Product_{i=1..q} p_i = Product_{j=1..d} y_j^m_j.
s(y) = Product_{j=1..d} m_j! ("symmetry of repeated parts").
Note: (n!/(n-k)!) / s(y) = multinomial(n, {m_j}).
Therefore:
T(7,6) = N(7,y=(6)) + N(7,y=(2^3))
= (7!/6) + (7!/(2^3)/3!)
= 7! * (1/6 + 1/48)
= 5040 * (3/16);
T(7,6) = 945.
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b:= proc(n, t) option remember; `if`(n=0, t, add(expand(`if`(j>1, x^j, 1)*
b(n-j, irem(t+signum(j-1), 2)))*binomial(n-1, j-1)*(j-1)!, j=1..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..10);
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Table[Table[n!/(n-k)!/2 * (Sum[(-1)^j/j!, {j, 0, k}] - ((k - 1)/k!)),{k,1,n}], {n,1,10}]
A354378
a(n) is the denominator of Sum_{k=0..n} (-1)^k / (2*k)!.
Original entry on oeis.org
1, 2, 24, 720, 8064, 3628800, 479001600, 87178291200, 20922789888000, 1280474741145600, 2432902008176640000, 1124000727777607680000, 620448401733239439360000, 403291461126605635584000000, 60977668922342772100300800000, 1569543549184562477137920000000
Offset: 0
1, 1/2, 13/24, 389/720, 4357/8064, 1960649/3628800, 258805669/479001600, ...
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Table[Sum[(-1)^k/(2 k)!, {k, 0, n}], {n, 0, 15}] // Denominator
nmax = 15; CoefficientList[Series[Cos[Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Denominator
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a(n) = denominator(sum(k=0, n, (-1)^k/(2*k)!)); \\ Michel Marcus, May 24 2022
A355266
Triangle read by rows, T(n, k) = (-1)^(n-k)*Bell(k)*Stirling1(n+1, k+1), for 0 <= k <= n.
Original entry on oeis.org
1, 1, 1, 2, 3, 2, 6, 11, 12, 5, 24, 50, 70, 50, 15, 120, 274, 450, 425, 225, 52, 720, 1764, 3248, 3675, 2625, 1092, 203, 5040, 13068, 26264, 33845, 29400, 16744, 5684, 877, 40320, 109584, 236248, 336420, 336735, 235872, 110838, 31572, 4140
Offset: 0
Triangle T(n, k) begins:
[0] 1;
[1] 1, 1;
[2] 2, 3, 2;
[3] 6, 11, 12, 5;
[4] 24, 50, 70, 50, 15;
[5] 120, 274, 450, 425, 225, 52;
[6] 720, 1764, 3248, 3675, 2625, 1092, 203;
[7] 5040, 13068, 26264, 33845, 29400, 16744, 5684, 877;
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T := (n, k) -> (-1)^(n-k)*combinat:-bell(k)*Stirling1(n+1, k+1):
seq(seq(T(n, k), k = 0..n), n = 0..8);
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from functools import cache
@cache
def b(n: int, k=0):
return int(n < 1) or k * b(n - 1, k) + b(n - 1, k + 1)
@cache
def s(n: int) -> list[int]:
if n == 0: return [1]
row = [0] + s(n - 1)
for k in range(1, n): row[k] = row[k] + (n - 1) * row[k + 1]
return row
def A355266_row(n):
return [s * b(k - 1) for k, s in enumerate(s(n + 1))][1:]
for n in range(9): print(A355266_row(n))
Original entry on oeis.org
1, 4, 19, 48, 645, 5346, 9989, 423680, 4936673, 22863284, 717864203, 10398234146, 14778845999, 2318706892436, 41349958502663, 67290481692176, 1273710986008283, 21639017114636720, 1870679510063123381
Offset: 0
a(16) = 1 + 3 + 15 + 29 + 597 + 4701 + 4643 + 413691 + 4512993 + 17926611 + 695000919 + 9680369943 + 4380611853 + 2303928046437 + 39031251610227 + 25940523189513 + 1206420504316107 = 1273710986008283 is prime.
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