A156308 Inverse of triangle S(n,m) defined by sequence A156290, n >= 1, 1 <= m <= n.

1, 4, 1, 9, 6, 1, 16, 20, 8, 1, 25, 50, 35, 10, 1, 36, 105, 112, 54, 12, 1, 49, 196, 294, 210, 77, 14, 1, 64, 336, 672, 660, 352, 104, 16, 1, 81, 540, 1386, 1782, 1287, 546, 135, 18, 1, 100, 825, 2640, 4290, 4004, 2275, 800, 170, 20, 1

Offset: 1

Author

Hartmut F. W. Hoft, Feb 07 2009

Keywords

• easy
• nonn
• tabl

Comments

From Wolfdieter Lang, Jun 26 2011: (Start)

This triangle S(n,m) appears as U_m(n) in the Knuth reference on p. 285. It is related to the Riordan triangle T_m(n) = A111125(n,m) by S(n,m) = A111125(n,m) - A111125(n-1,m), n >= m >= 1 (identity on p. 286).

Also, S(n,m)-S(n-1,m) = A111125(n-1,m-1), n >= 2, m >= 1 (identity on p. 286). (End)

These polynomials may be expressed in terms of the Faber polynomials of A263916 and are embedded in A127677 and A208513. - Tom Copeland, Nov 06 2015

Examples

			Triangle starts:
  n=1:  1;
  n=2:  4,  1;
  n=3:  9,  6,  1;
  n=4: 16, 20,  8,  1;
  ...
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened
• Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
• Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.
• Johann Cigler, A bivariate version of the spread polynomials, arXiv:2508.04751 [math.CO], 2025.
• D. E. Knuth, Johann Faulhaber and sums of powers, arXiv:math/9207222 [math.CA], 1992; Math. Comp. 61 (1993), no. 203, 277-294.

Crossrefs

Same as triangle A208513 with the first column truncated.

Columns: A000290 (m=1), A002415 (m=2), A040977 (m=3), A053347 (m=4), A054334 (m=5).

Cf. A078812, A111125, A127677, A208513, A263916.

Programs

Magma

[(n/k)*Binomial(n+k-1, 2*k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 01 2022

Mathematica

S[m_] := Flatten[Table[k/j Binomial[k + j - 1, 2 j - 1], {k, 1, m}, {j, 1, k}]]

Sage

flatten([[(n/k)*binomial(n+k-1, 2*k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 01 2022

Formula

S(n, m) = (n/m) * binomial(n + m - 1, 2*m - 1).

From Peter Bala, May 01 2014: (Start)

The n-th row o.g.f. is polynomial R(n,x) = 2/x*( T(n,(x + 2)/2) - 1 ), where T(n,x) is Chebyshev polynomial of the first kind. They form a divisibility sequence: if n divides m then R(n,x) divides R(m,x) in the ring Z[x].

R(2*n,x) = (x + 4)*U(n-1,(x + 2)/2)^2;

R(2*n + 1,x) = ( U(n,(x + 2)/2) + U(n-1,(x + 2)/2) )^2.

O.g.f.: Sum_{n >= 0} R(n,x)*z^n = z*(1 + z)/( (1 - z)*(1 - (x + 2)*z + z^2) ). (End)

The polynomial R(n,x) defined above satisfies (x + 1/x - 2) * R(n, x + 1/x - 2) = x^n + 1/x^n - 2. - Alexander Burstein, May 23 2021

Extensions

Edited by Max Alekseyev, Mar 05 2018