A394704 Triangle read by rows: coefficients of reciprocal polynomials refining A013499.

1, 1, 1, 6, 1, 1, 26, 26, 1, 1, 100, 310, 100, 1, 1, 372, 2752, 2752, 372, 1, 1, 1379, 21259, 48034, 21259, 1379, 1, 1, 5140, 152644, 665758, 665758, 152644, 5140, 1, 1, 19296, 1052076, 8069856, 15271974, 8069856, 1052076, 19296, 1, 1, 72920, 7082120, 89918720, 290346728, 290346728, 89918720, 7082120, 72920, 1

Offset

0,4

Comments

Define an integer-valued polynomial p_n(x) = Product_{i=1..n} ((n+1)*x+i) * (2*x+1) / n!. Then the n-th row describes the numerator of the generating function Sum_{k >= 0} p_n(k) t^k as a rational function.

Links

Table of n, a(n) for n=0..53.

Example

The first few rows are:
  1,    1;
  1,    6,     1;
  1,   26,    26,     1;
  1,  100,   310,   100,     1;
  1,  372,  2752,  2752,   372,    1;
  1, 1379, 21259, 48034, 21259, 1379, 1;

Prog

(SageMath)
x = polygen(QQ, 'x')
t = x.parent()[['t']].gen()
def row(n):
    pn = prod((n+1)*x+i for i in range(1, n+1)) * (2*x+1) / factorial(n)
    poly = (1 - t)**(n + 2) * sum(pn(k) * t**k for k in range(n + 2)).O(n + 2)
    return poly.coefficients()

Crossrefs

Row sums are A013499(n+1) if n >= 1.

Similar to A108267, A395490.

Sequence in context: A035348 A140945 A141688 * A166960 A155908 A105373

Adjacent sequences: A394701 A394702 A394703 * A394705 A394706 A394707

Keyword

nonn,tabf

Author

F. Chapoton, Apr 26 2026

Status

approved