A264437 -id:A264437 - cp's OEIS frontend

A357340 Triangle read by rows. T(n, k) = Sum_{j=0..n-k} binomial(-n, j) * A268438(n - k, j).

1, -1, 1, 2, -2, 1, 0, 12, -3, 1, -56, -120, 28, -4, 1, 0, 1680, -450, 50, -5, 1, 15840, -30240, 10416, -1080, 78, -6, 1, 0, 665280, -317520, 33712, -2100, 112, -7, 1, -17297280, -17297280, 12070080, -1391040, 81648, -3600, 152, -8, 1

Offset: 0

Peter Luschny, Sep 25 2022

			Triangle T(n, k) starts:
[0]         1;
[1]        -1,         1;
[2]         2,        -2,        1;
[3]         0,        12,       -3,        1;
[4]       -56,      -120,       28,       -4,     1;
[5]         0,      1680,     -450,       50,    -5,     1;
[6]     15840,    -30240,    10416,    -1080,    78,    -6,   1;
[7]         0,    665280,  -317520,    33712, -2100,   112,  -7,  1;
[8] -17297280, -17297280, 12070080, -1391040, 81648, -3600, 152, -8, 1;

Cf. A357341 (alternating row sums), A264437, A268438, A357339.

A357340 := proc(n, k) local u; u := n - k; (2*u)!*add(binomial(-n, j) * j! *
  add((-1)^(j+m)*binomial(u+j, u+m)*abs(Stirling1(u+m, m)), m=0..j)/(u +j)!, j=0..u) end: seq(print(seq(A357340(n, k), k=0..n)), n=0..8);

# using function A268438
def A357340(n, k):
    return sum(binomial(-n, i) * A268438(n - k, i) for i in range(n - k + 1))
for n in range(10): print([A357340(n, k) for k in range(n + 1)])

A268432 a(n) = Pochhammer(n+1, n)/Clausen(n, 1) = A001813(n) / A160014(n, 1).

Original entry on oeis.org

1, 1, 2, 60, 56, 15120, 15840, 8648640, 17297280, 8821612800, 10158220800, 14079294028800, 474467051520, 32382376266240000, 582882772792320000, 101421602465863680000, 24659370011308032000, 415017197290314178560000, 72810034612335820800000, 2149789081963827444940800000

Offset: 0

Peter Luschny, Feb 14 2016

nonn

Cf. A001813, A027642, A160014, A264437.

a := proc(n) numtheory[divisors](n); map(i->i+1, %);
iquo(mul(4*k+2,k in (0..n-1)), mul(k,k in select(isprime, %))) end:
seq(a(n), n=0..19);

def A268432(n):
    if n <= 1: return 1
    r = rising_factorial(n+1,n)//bernoulli(n).denominator()
    return r if is_even(n) else r//2
[A268432(n) for n in range(20)]

Let b(n) = Pochhammer(n+1,n)/denominator(Bernoulli(n)) then a(2*n) = b(2*n) for n >= 0 and 2*a(2*n+1) = b(2*n+1) for n >= 1 by the von Staudt-Clausen theorem.

cp's OEIS Frontend

A357340 Triangle read by rows. T(n, k) = Sum_{j=0..n-k} binomial(-n, j) * A268438(n - k, j).

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