A023531 -id:A023531 - cp's OEIS frontend

A348014 Triangle, read by rows, with row n forming the coefficients in Product_{k=0..n} (1 + k^k*x).

1, 1, 1, 1, 5, 4, 1, 32, 139, 108, 1, 288, 8331, 35692, 27648, 1, 3413, 908331, 26070067, 111565148, 86400000, 1, 50069, 160145259, 42405161203, 1216436611100, 5205269945088, 4031078400000

Offset: 0

Seiichi Manyama, Sep 24 2021

			Triangle begins:
  1;
  1,    1;
  1,    5,      4;
  1,   32,    139,      108;
  1,  288,   8331,    35692,     27648;
  1, 3413, 908331, 26070067, 111565148, 86400000;

Seiichi Manyama, Rows n = 0..37, flattened

Column k=1 gives A001923.
The diagonal of the triangle is A002109.
Cf. A007318, A008955, A023531, A108084, A173007, A173008, A249677.

T(n, k) = if(k==0, 1, if(k==n, prod(j=1, n, j^j), T(n-1, k)+n^n*T(n-1, k-1)));

PARI
```
row(n) = Vecrev(prod(k=1, n, 1+k^k*x));
```

T(0,0) = 1; T(n,k) = T(n-1,k) + n^n * T(n-1,k-1).

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

A375466 Array read by ascending antidiagonals of triangles read by rows: the coefficients of the polynomials n! * m^(n-k) * x^k * A094587(n, k), for m >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 1, 3, 1, 2, 0, 1, 4, 1, 8, 2, 1, 1, 5, 1, 18, 4, 1, 0, 1, 6, 1, 32, 6, 1, 6, 0, 1, 7, 1, 50, 8, 1, 48, 6, 0, 1, 8, 1, 72, 10, 1, 162, 24, 3, 1, 1, 9, 1, 98, 12, 1, 384, 54, 6, 1, 0, 1, 10, 1, 128, 14, 1, 750, 96, 9, 1, 24, 0

Offset: 0

Peter Luschny, Aug 17 2024

			Sequence of polynomials P(n, m) for n = 0, 1, 2, ...:
  [0]   1;
  [1]   1*m   +         x;
  [2]   2*m^2 +     2*m*x +         x^2;
  [3]   6*m^3 +   6*m^2*x +     3*m*x^2 +         x^3;
  [4]  24*m^4 +  24*m^3*x +  12*m^2*x^2 +     4*m*x^3 +        x^4;
  [5] 120*m^5 + 120*m^4*x +  60*m^3*x^2 +  20*m^2*x^3 +    5*m*x^4 +     x^5;
  [6] 720*m^6 + 720*m^5*x + 360*m^4*x^2 + 120*m^3*x^3 + 30*m^2*x^4 + 6*m*x^5 + x^6;
  ...
Array of the coefficients of the polynomials for m = 0, 1, 2, ...:
  [0] 1, 0, 1,  0,  0, 1,    0,   0,  0, 1,     0,    0,   0,  0, 1, ...  A023531
  [1] 1, 1, 1,  2,  2, 1,    6,   6,  3, 1,    24,   24,  12,  4, 1, ...  A094587
  [2] 1, 2, 1,  8,  4, 1,   48,  24,  6, 1,   384,  192,  48,  8, 1, ...
  [3] 1, 3, 1, 18,  6, 1,  162,  54,  9, 1,  1944,  648, 108, 12, 1, ...
  [4] 1, 4, 1, 32,  8, 1,  384,  96, 12, 1,  6144, 1536, 192, 16, 1, ...
  [5] 1, 5, 1, 50, 10, 1,  750, 150, 15, 1, 15000, 3000, 300, 20, 1, ...
  [6] 1, 6, 1, 72, 12, 1, 1296, 216, 18, 1, 31104, 5184, 432, 24, 1, ...
  ...
Seen as triangle:
  1;
  1, 0;
  1, 1, 1;
  1, 2, 1,  0;
  1, 3, 1,  2,  0;
  1, 4, 1,  8,  2, 1;
  1, 5, 1, 18,  4, 1,   0;
  1, 6, 1, 32,  6, 1,   6,  0;
  1, 7, 1, 50,  8, 1,  48,  6, 0;
  1, 8, 1, 72, 10, 1, 162, 24, 3, 1;
  1, 9, 1, 98, 12, 1, 384, 54, 6, 1,  0;

Cf. A094587, A023531.

# Computes the polynomials depending on the parameter m.
P := (n, m) -> ifelse(m = 0, x^n, n! * m^n * hypergeom([-n], [-n], x/m)):
seq(print(simplify(P(n, m))), n = 0..5);
# Computes the array of coefficients:
P := (n, k, m) -> (n!/k!) * m^(n-k) * x^k:
Arow := (m, len) -> local n, k;
seq(seq(coeff(P(n, k, m), x, k), k = 0..n), n = 0..len):
seq(lprint(Arow(n, 4)), n = 0..6);

T(n, m, k) = [x^k] n! * m^n * hypergeom([-n], [-n], x/m), for n > 0.

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A348014 Triangle, read by rows, with row n forming the coefficients in Product_{k=0..n} (1 + k^k*x).

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

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A375466 Array read by ascending antidiagonals of triangles read by rows: the coefficients of the polynomials n! * m^(n-k) * x^k * A094587(n, k), for m >= 0.

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