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(PARI) T(n, k) = if ((n-k) % 2, 0, stirling(n+1, k, 2)/binomial(k+1, 2)); \\ Michel Marcus, May 23 2023

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/ 1 \ /1 \ /1 \ /1 \

|-1 1 | |0 1 | |1 1 | |0 1 |

| 1 -3 1 | |1 0 1 | |1 3 1 | = |0 0 1 |

|-1 7 -6 1 | |0 5 0 1 | |1 7 6 1 | |0 1 0 1 |

| 1 -15 25 -10 1| |8 0 15 0 1 | |1 15 25 10 1 | |0 0 5 0 1 |

| ... | |... | |... | |0 1 0 15 0 1 |

| | | | | | |... |

| | | | | | |... |

Let P(n,x) = (1 - x)*(1 - 2*x)*...*(1 - n*x). The g.f. for the k-th column of the triangle is (1/(k*(k + 1)))*x^(k-1)*(1/P(k,x) - 1/P(k,-x)) = (x^k)*(x^k*R(k-1,1/x))/((1 - x^2)*(1 - 4*x^2)*...*(1 - k^2*x^2)), where R(n,x) denotes the n-th row polynomial of A164652. (Since the entries of triangle A164652 are integers, it follows that the entries of the present triangle are also integers.)

k = 1 2 3 4 5 6 7 8 9 10

n = 1: 1

2: 0 1

3: 1 0 1

4: 0 5 0 1

5: 1 0 15 0 1

6: 0 21 0 35 0 1

7: 1 0 161 0 70 0 1

8: 0 85 0 777 0 126 0 1

9: 1 0 1555 0 2835 0 210 0 1

10: 0 341 0 14575 0 8547 0 330 0 1

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for n from 1 to 10 do seq(A362041(n, k), k = 1..n+1) od;

Row sums give A363042. Cf. A008275, A008277, A164652, A129825.

Cf. A008275, A008277, A164652, A129825.

nonn,tabl,easy,changed