A371764 -id:A371764 - cp's OEIS frontend

A372148 a(n) = A371764(n, 2).

1, 14, 86, 374, 1382, 4694, 15206, 47894, 148262, 453974, 1380326, 4177814, 12607142, 37968854, 114201446, 343194134, 1030762022, 3094645334, 9288654566, 27875400854, 83645076902, 250972979414, 752994435686, 2259134301974

Offset: 1

Detlef Meya, Apr 20 2024

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A371764.

a := n -> 2*(4*3^n - 9*2^n + 7) - `if`(n=1, 1, 0);
seq(a(n), n = 1..24);  # Peter Luschny, Apr 20 2024

A372148[n_] := 2*(4*3^n - 9*2^n + 7) - Boole[n == 1]; Array[A372148,50] (* or *)
LinearRecurrence[{6, -11, 6}, {1, 14, 86, 374}, 50] (* Paolo Xausa, May 25 2024 *)

a(n) = 2*(4*3^n - 9*2^n + 7) - [n = 1]. - Hugo Pfoertner, Apr 20 2024

G.f.: x*(1 + 8*x + 13*x^2 + 6*x^3)/((1 - x)*(1 - 2*x)*(1 - 3*x)). - Stefano Spezia, Apr 21 2024

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

Original entry on oeis.org

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

Peter Luschny, Apr 15 2024

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.

			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

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

Cf. A371761, A344920, A099594.

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

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);

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.
```

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

cp's OEIS Frontend

A372148 a(n) = A371764(n, 2).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Julia

Maple

Mathematica

Python

Formula