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A384147 Array A(n,k) = n*(A(n-1,k)+A(n-2,k)+...+A(n-k,k)), where A(n,k) = n if n <= k, read by antidiagonals with n >= 1 and k >= 1.

%I A384147 #22 Jun 03 2025 19:08:29
%S A384147 1,1,2,1,2,3,1,8,3,4,1,20,3,4,5,1,56,27,4,5,6,1,152,99,4,5,6,7,1,416,
%T A384147 387,64,5,6,7,8,1,1136,1539,304,5,6,7,8,9,1,3104,6075,1504,125,6,7,8,
%U A384147 9,10,1,8480,24003,7504,725,6,7,8,9,10,11,1,23168,94851,37504,4325,216,7,8,9,10,11,12
%N A384147 Array A(n,k) = n*(A(n-1,k)+A(n-2,k)+...+A(n-k,k)), where A(n,k) = n if n <= k, read by antidiagonals with n >= 1 and k >= 1.
%C A384147 Taking only the triangle where 1<=n<=k and reading by rows yields A002024.
%H A384147 Jason Bard, <a href="/A384147/b384147.txt">Table of n, a(n) for n = 1..5050</a>
%F A384147 A(m,m+1) = m^3 for all m >= 1.
%F A384147 A(m,m+2) = m^4 + m^3 - m^2 for all m >= 1.
%F A384147 A(m,m+3) = m^5 + 2m^4 - 2m^2 for all m >= 1.
%F A384147 A(m,m+4) = m^6 + 3m^5 + 2m^4 - 2m^3 - 3m^2 for all m >= 3.
%F A384147 A(m,m+5) = m^7 + 4m^6 + 5m^5 - 5m^3 - 4m^2 for all m >= 4.
%F A384147 ...
%F A384147 A(m,m+k) ~ O(m^(k+2)) for all m >= k-1 may be derived similarly.
%e A384147 Top left corner of the array:
%e A384147    1  1  1  1  1   1    1    1     1      1      1
%e A384147    2  2  8 20 56 152  416 1136  3104   8480  23168
%e A384147    3  3  3 27 99 387 1539 6075 24003  94851 374787
%e A384147    4  4  4  4 64 304 1504 7504 37504 187264 935104
%e A384147    5  5  5  5  5 125  725 4325 25925 155525 933125
%e A384147    6  6  6  6  6   6  216 1476 10296  72036 504216
%e A384147    7  7  7  7  7   7    7  343  2695  21511 172039
%e A384147    8  8  8  8  8   8    8    8   512   4544  40832
%e A384147    9  9  9  9  9   9    9    9     9    729   7209
%e A384147   10 10 10 10 10  10   10   10    10     10   1000
%e A384147   ...
%t A384147 nmax = 100; AntiDiagonalFlatten[matrix_] := Module[{n = Length@matrix}, Flatten[Table[matrix[[i, s - i]], {s, 2, 2 n}, {i, Max[1, s - n], Min[n, s - 1]}], 1]]; A384147 = AntiDiagonalFlatten[Table[LinearRecurrence[ConstantArray[n, n], ConstantArray[n, n], {1, nmax}], {n, 1, nmax}]][[;; nmax*(nmax + 1)/2]]
%Y A384147 Cf. A000012 (row 1), A080040 (row 2).
%Y A384147 Cf. A000578, A002024, A002260.
%K A384147 nonn,tabl
%O A384147 1,3
%A A384147 _Jason Bard_, May 25 2025