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.

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, 387, 64, 5, 6, 7, 8, 1, 1136, 1539, 304, 5, 6, 7, 8, 9, 1, 3104, 6075, 1504, 125, 6, 7, 8, 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

Offset

1,3

Comments

Taking only the triangle where 1<=n<=k and reading by rows yields A002024.

Links

Jason Bard, Table of n, a(n) for n = 1..5050

Formula

A(m,m+1) = m^3 for all m >= 1.

A(m,m+2) = m^4 + m^3 - m^2 for all m >= 1.

A(m,m+3) = m^5 + 2m^4 - 2m^2 for all m >= 1.

A(m,m+4) = m^6 + 3m^5 + 2m^4 - 2m^3 - 3m^2 for all m >= 3.

A(m,m+5) = m^7 + 4m^6 + 5m^5 - 5m^3 - 4m^2 for all m >= 4.

...

A(m,m+k) ~ O(m^(k+2)) for all m >= k-1 may be derived similarly.

Example

Top left corner of the array:
   1  1  1  1  1   1    1    1     1      1      1
   2  2  8 20 56 152  416 1136  3104   8480  23168
   3  3  3 27 99 387 1539 6075 24003  94851 374787
   4  4  4  4 64 304 1504 7504 37504 187264 935104
   5  5  5  5  5 125  725 4325 25925 155525 933125
   6  6  6  6  6   6  216 1476 10296  72036 504216
   7  7  7  7  7   7    7  343  2695  21511 172039
   8  8  8  8  8   8    8    8   512   4544  40832
   9  9  9  9  9   9    9    9     9    729   7209
  10 10 10 10 10  10   10   10    10     10   1000
  ...

Mathematica

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]]

Crossrefs

Cf. A000012 (row 1), A080040 (row 2).

Cf. A000578, A002024, A002260.

Sequence in context: A232177 A111377 A014046 * A243919 A231775 A372840

Adjacent sequences: A384144 A384145 A384146 * A384148 A384149 A384150

Keyword

nonn,tabl

Author

Jason Bard, May 25 2025

Status

approved