A395188 Triangular array T(n, k) read by rows: Row n > 0 gives the coefficient of x^k in the expansion of f_n(x) = Sum_{k=0..n} T(n, k)*x^k, where T(n, n) and T(0, n) = 1 for any n. The n-th falling diagonal has the ordinary generating function: 1/f_n(x).

1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, 0, -1, 1, 1, 1, 0, -1, 1, 1, 1, -1, 1, 0, -1, -1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, -1, 0, 0, 2, -1, 1, -1, 1, 1, 1, 0, -1, -2, 2, 0, 1, 1, 1, 1, -1, 1, -1, 0, 3, 3, 0, 0, -1, 1, 1, 1, 1, 1, 2, 3, -3, 3, 1, -1, 1, 1, 1, -1

Offset

0,32

Comments

The columns up to k = 5 become periodic if we omit the first term. However column k = 6 is not periodic anymore and is also the last column known as a linear recurrence with constant coefficients.

The first difference between this triangle and A395191 occurs at T(7, 2), this is directly caused by row n=5 which is the first non-palindromic row in both triangles.

Links

Table of n, a(n) for n=0..79.

Formula

Sum_{k=0..n} T(n, k)*x^k = 1/Sum{m=0..oo} T(n+m, m)*x^m, for n > 0.

T(n, k) = Sum_{m=1..k} -T(n-m, k-m)*T(n-k, m), for n > k.

T(n, 2) = T(n-1, 2) - T(n-2, 2) + T(n-3, 2), for n > 5.

T(n, 3) = -T(n-2, 3) - T(n-3, 2) - T(n-5, 2), for n > 8.

T(n, 4) = -T(n-2, 4) + T(n-3, 4) - T(n-4, 4) + T(n-5, 4) - T(n-6, 4) + T(n-7, 4) + T(n-9, 4), for n > 13.

T(n, 5) = T(n-1, 5) - T(n-3, 5) + T(n-5, 5) - T(n-6, 5) + T(n-8, 5) - T(n-9, 5) + T(n-11, 5) - T(n-13, 5) + T(n-14, 5), for n > 19.

T(n, 6) = T(n-2, 6) + T(n-3, 6) - 2*T(n-4, 6) - T(n-5, 6) + 2*T(n-7, 6) - 2*T(n-10, 6) + T(n-12, 6) + 2*T(n-13, 6) - T(n-14, 6) - T(n-15, 6) + T(n-17, 6), for n > 23.

Example

Triangle T(n, k) starts:
  [0] 1;
  [1] 1,   1;
  [2] 1,  -1,   1;
  [3] 1,   1,   1,   1;
  [4] 1,  -1,   0,  -1,   1;
  [5] 1,   1,   0,  -1,   1,   1;
  [6] 1,  -1,   1,   0,  -1,  -1,   1;
  [7] 1,   1,   1,   2,   1,   0,   1,   1;
  [8] 1,  -1,   0,   0,   2,  -1,   1,  -1,   1;
  [9] 1,   1,   0,  -1,  -2,   2,   0,   1,   1,   1;
  ...
The falling diagonal beginning in row n = 2 is: 1, 1, 0, -1, -1, 0, 1, ... . It has the generating function: 1/(1-x+x^2). The coefficients of 1-x+x^2 are 1, -1, 1, this is row n = 2.

Prog

(PARI)
squareRow(n, max_k) = if(n==0, vector(max_k, k, 1), my(f(x)=1+sum(k=1, n, squareRow(n-k, k+1)[k+1]*x^k)); Vec(1/f(x)+O(x^max_k)))
T(n, k) = squareRow(n-k, k+1)[k+1]

Crossrefs

Cf. A395191 (same idea but reversed order of terms relation in rows).

Sequence in context: A261887 A037871 A037853 * A255237 A291954 A375670

Adjacent sequences: A395185 A395186 A395187 * A395189 A395190 A395191

Keyword

sign,easy,tabl

Author

Thomas Scheuerle, Apr 15 2026

Status

approved