OFFSET
1,2
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
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.
D. E. Knuth, Johann Faulhaber and sums of powers, arXiv:math/9207222 [math.CA], 1992; Math. Comp. 61 (1993), no. 203, 277-294.
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
EXAMPLE
Triangle starts:
n=1: 1;
n=2: 4, 1;
n=3: 9, 6, 1;
n=4: 16, 20, 8, 1;
...
MATHEMATICA
S[m_] := Flatten[Table[k/j Binomial[k + j - 1, 2 j - 1], {k, 1, m}, {j, 1, k}]]
PROG
(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
(Magma) [(n/k)*Binomial(n+k-1, 2*k-1): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 01 2022
CROSSREFS
Same as triangle A208513 with the first column truncated.
KEYWORD
AUTHOR
Hartmut F. W. Hoft, Feb 07 2009
EXTENSIONS
Edited by Max Alekseyev, Mar 05 2018
STATUS
approved