A371763 Triangle read by rows: Trace of the Akiyama-Tanigawa algorithm for powers x^2.

0, 1, 1, 5, 6, 4, 13, 18, 15, 9, 29, 42, 39, 28, 16, 61, 90, 87, 68, 45, 25, 125, 186, 183, 148, 105, 66, 36, 253, 378, 375, 308, 225, 150, 91, 49, 509, 762, 759, 628, 465, 318, 203, 120, 64, 1021, 1530, 1527, 1268, 945, 654, 427, 264, 153, 81

Offset: 0

Author

Peter Luschny, Apr 15 2024

Keywords

• nonn
• tabl

Comments

The Akiyama-Tanigawa is a sequence-to-sequence transformation AT := A -> B. If A(n) = 1/(n + 1) then B(n) are the Bernoulli numbers. Tracing the algorithm generates a triangle where the right edge is sequence A and the left edge is its transform B.

Here we consider the sequence A(n) = n^2 that is transformed into sequence B(n) = |A344920(n)|. The case A(n) = n^3 is A371764. Sequence [1, 1, 1, ...] generates A023531 and sequence [0, 1, 2, 3, ...] generates A193738.

In their general form, the AT-transforms of the powers are closely related to the poly-Bernoulli numbers A099594 and generate the rows of the array A371761.

Examples

			Triangle starts:
0:                  0
1:               1,   1
2:             5,   6,   4
3:          13,  18,  15,   9
4:        29,  42,  39,  28,  16
5:      61,  90,  87,  68,  45,  25
6:    125, 186, 183, 148, 105,  66, 36
7:  253, 378, 375, 308, 225, 150, 91, 49
		

Crossrefs

Family of triangles: A023531 (n=0), A193738 (n=1), this triangle (n=2), A371764 (n=3).

Cf. A371761, A344920, A099594.

Programs

Julia

function ATPtriangle(k::Int, len::Int)
    A = Vector{BigInt}(undef, len)
    B = Vector{Vector{BigInt}}(undef, len)
    for n in 0:len-1
        A[n+1] = n^k
        for j = n:-1:1
            A[j] = j * (A[j+1] - A[j])
        end
        B[n+1] = A[1:n+1]
    end
    return B
end
for (n, row) in enumerate(ATPtriangle(2, 9))
    println("$(n-1): ", row)
end

Maple

ATProw := proc(k, n) local m, j, A;
   for m from 0 by 1 to n do
      A[m] := m^k;
      for j from m by -1 to 1 do
         A[j - 1] := j * (A[j] - A[j - 1])
   od od; convert(A, list) end:
ATPtriangle := (p, len) -> local k;
     ListTools:-Flatten([seq(ATProw(p, k), k = 0..len)]):
ATPtriangle(2, 9);

Mathematica

T[n,k] := If[n==k, n^2, (k+1)*(2^(n-k)*(k+2)-3)]; Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Apr 19 2024 *)

Python

# See function ATPowList in A371761.

Formula

T(n, k) = n^2 if n=k, otherwise (k + 1)*(2^(n - k)*(k + 2) - 3). - Detlef Meya, Apr 19 2024