A362790
a(n) = Sum_{k=0..n} FallingFactorial(n - k, k) * Stirling2(n - k, k), row sums of A362789.
Original entry on oeis.org
1, 0, 1, 2, 5, 22, 95, 450, 2461, 14654, 93851, 647746, 4781801, 37488462, 310842127, 2716308194, 24929090357, 239556785086, 2404139609987, 25139451248418, 273330944247265, 3084182865509966, 36055337388402935, 436016786153035522, 5446585683469420205
Offset: 0
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a := n -> add((-1)^k*pochhammer(k - n, k)*Stirling2(n - k, k), k = 0..iquo(n,2)):
seq(a(n), n = 0..24);
-
def A362790(n):
return sum(falling_factorial(n - k, k) * stirling_number2(n - k, k) for k in range(n//2 + 1))
print([A362790(n) for n in range(12)])
A362788
Triangle read by rows, T(n, k) = RisingFactorial(n - k, k) * Stirling2(n - k, k), for n >= 0 and 0 <= k <= n//2, where '//' denotes integer division.
Original entry on oeis.org
1, 0, 0, 1, 0, 2, 0, 3, 6, 0, 4, 36, 0, 5, 140, 60, 0, 6, 450, 720, 0, 7, 1302, 5250, 840, 0, 8, 3528, 30240, 16800, 0, 9, 9144, 151704, 196560, 15120, 0, 10, 22950, 695520, 1764000, 453600, 0, 11, 56210, 2994750, 13471920, 7761600, 332640
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 0;
[2] 0, 1;
[3] 0, 2;
[4] 0, 3, 6;
[5] 0, 4, 36;
[6] 0, 5, 140, 60;
[7] 0, 6, 450, 720;
[8] 0, 7, 1302, 5250, 840;
[9] 0, 8, 3528, 30240, 16800;
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T := (n, k) -> pochhammer(n - k, k) * Stirling2(n - k, k):
seq(seq(T(n, k), k = 0..iquo(n,2)), n = 0..12);
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def A362788(n, k):
return rising_factorial(n - k, k) * stirling_number2(n - k, k)
for n in range(10):
print([A362788(n, k) for k in range(n//2 + 1)])
A362369
Triangle read by rows, T(n, k) = binomial(n, k) * k! * Stirling2(n-k, k), for n >= 0 and 0 <= k <= n//2, where '//' denotes integer division.
Original entry on oeis.org
1, 0, 0, 2, 0, 3, 0, 4, 12, 0, 5, 60, 0, 6, 210, 120, 0, 7, 630, 1260, 0, 8, 1736, 8400, 1680, 0, 9, 4536, 45360, 30240, 0, 10, 11430, 216720, 327600, 30240, 0, 11, 28050, 956340, 2772000, 831600, 0, 12, 67452, 3993000, 20207880, 13305600, 665280
Offset: 0
Triangle T(n, k) starts:
[0] 1;
[1] 0;
[2] 0, 2;
[3] 0, 3;
[4] 0, 4, 12;
[5] 0, 5, 60;
[6] 0, 6, 210, 120;
[7] 0, 7, 630, 1260;
[8] 0, 8, 1736, 8400, 1680;
[9] 0, 9, 4536, 45360, 30240;
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T := (n, k) -> binomial(n, k) * k! * Stirling2(n-k, k):
seq(seq(T(n, k), k = 0..iquo(n, 2)), n = 0..9);
# second program:
egf := k-> (x*(exp(x)-1))^k / k!:
A362369 := (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A362369(n, k), k=0..iquo(n,2))), n=0..12); # Mélika Tebni, May 10 2023
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def A362369(n, k):
return binomial(n, k) * factorial(k) * stirling_number2(n - k, k)
for n in range(10):
print([A362369(n, k) for k in range(n//2 + 1)])
Showing 1-3 of 3 results.