A354138
a(n) is the numerator of Sum_{k=0..n} (-1)^k / (2*k)!.
Original entry on oeis.org
1, 1, 13, 389, 4357, 1960649, 258805669, 47102631757, 11304631621681, 691843455246877, 1314502564969066301, 607300185015708631061, 335229702128671164345673, 217899306383636256824687449, 32946375125205802031892742289, 848027998784883070051677094421
Offset: 0
1, 1/2, 13/24, 389/720, 4357/8064, 1960649/3628800, 258805669/479001600, ...
Cf.
A010050,
A049470,
A053557,
A061354,
A103816,
A120265,
A143382,
A354211,
A354332,
A354334,
A354378 (denominators).
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Table[Sum[(-1)^k/(2 k)!, {k, 0, n}], {n, 0, 15}] // Numerator
nmax = 15; CoefficientList[Series[Cos[Sqrt[x]]/(1 - x), {x, 0, nmax}], x] // Numerator
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a(n) = numerator(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))
A123438
Numbers k such that the denominator of k!/!k (= A000142(k)/A000166(k)) is prime.
Original entry on oeis.org
4, 5, 6, 7, 10, 11, 15, 16, 34, 44, 63, 66, 168, 427, 575, 928, 1094, 1218, 1363, 1713, 5278, 10814
Offset: 1
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a[n_] := Sum[(-1)^k/k!, {k, 0, n}]; Select[Range[100], PrimeQ[Numerator[a[#]]] &] (* G. C. Greubel, Oct 31 2017 *)
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isok(n) = isprime(numerator(sum(k=0, n, (-1)^k/k!))); \\ Michel Marcus, Aug 28 2013
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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