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A339033 Triangle read by rows, T(n, k) for 0 <= k <= n. T(n, 0) = 0^n; T(n, n) = n!; otherwise T(n, k) = (n + 1 - k)*(k - 1)!.

%I A339033 #22 May 31 2024 14:43:54
%S A339033 1,0,1,0,2,2,0,3,2,6,0,4,3,4,24,0,5,4,6,12,120,0,6,5,8,18,48,720,0,7,
%T A339033 6,10,24,72,240,5040,0,8,7,12,30,96,360,1440,40320,0,9,8,14,36,120,
%U A339033 480,2160,10080,362880,0,10,9,16,42,144,600,2880,15120,80640,3628800
%N A339033 Triangle read by rows, T(n, k) for 0 <= k <= n. T(n, 0) = 0^n; T(n, n) = n!; otherwise T(n, k) = (n + 1 - k)*(k - 1)!.
%C A339033 Related to the multinomial that is called M2 in Abramowitz and Stegun, p. 831.
%H A339033 Paolo Xausa, <a href="/A339033/b339033.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of the triangle, flattened).
%H A339033 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, page 831.
%F A339033 T(n, k) = n! / A092271(n, k) for k > 0.
%e A339033 Triangle starts:
%e A339033 0: [1]
%e A339033 1: [0, 1]
%e A339033 2: [0, 2, 2]
%e A339033 3: [0, 3, 2,  6]
%e A339033 4: [0, 4, 3,  4, 24]
%e A339033 5: [0, 5, 4,  6, 12, 120]
%e A339033 6: [0, 6, 5,  8, 18,  48, 720]
%e A339033 7: [0, 7, 6, 10, 24,  72, 240, 5040]
%e A339033 8: [0, 8, 7, 12, 30,  96, 360, 1440, 40320]
%e A339033 9: [0, 9, 8, 14, 36, 120, 480, 2160, 10080, 362880]
%t A339033 A339033[n_, k_] := Which[k == 0, Boole[n == 0], n == k, n!, True, (n+1-k)*(k-1)!];
%t A339033 Table[A339033[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Jan 31 2024 *)
%o A339033 (SageMath)
%o A339033 def  A339033(n, k):
%o A339033     if k == 0: return 0^n
%o A339033     if n == k: return factorial(n)
%o A339033     return (n + 1 - k)*factorial(k - 1)
%o A339033 for n in (0..10): print([A339033(n, k) for k in (0..n)])
%o A339033 def A339033Row(n):
%o A339033     S = [0^n]
%o A339033     for k in range(n, 0, -1):
%o A339033         for p in Partitions(n, max_part=k, inner=[k], length=n+1-k):
%o A339033             S.append(p.aut())
%o A339033     return S
%o A339033 for n in (0..10): print(A339033Row(n))
%Y A339033 Cf. A339034 (row sums), A092271.
%K A339033 nonn,tabl
%O A339033 0,5
%A A339033 _Peter Luschny_, Nov 20 2020