

A156308


Inverse of triangle defined by sequence A156290.


9



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
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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 A111125(n,m), appearing in this reference as T_m(n), by S(n,m) = A111125(n,m)  A111125(n1,m), n>=m>=1 (identity on p. 286).
Also, S(n,m)S(n1,m) = A111125(n1,m1), 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

Table of n, a(n) for n=1..55.
D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comp. 61 (1993), no. 203, 277294.


FORMULA

S(n,m) = n/m * C(n + m  1, 2*m  1).
From Peter Bala, May 01 2014: (Start):
The nth row polynomial R(n,x) = 2/x*( T(n,(x + 2)/2)  1 ), where T(n,x) denotes the Chebyshev polynomial of the first kind. The row polynomials form a divisibility sequence, i.e., if n divides m then R(n,x) divides R(m,x) in the ring Z[x].
R(2*n,x) = (x + 4)*U(n1,(x + 2)/2)^2;
R(2*n + 1,x) = ( U(n,(x + 2)/2) + U(n1,(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)


EXAMPLE

S(3,1) = 9, S(3,3) = 1, S(5,3) = 35.


MATHEMATICA

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


CROSSREFS

Column 1 of triangle is A000290.
Column 2 of triangle is A002415.
Column 3 of triangle is A040977.
Column 4 of triangle is A053347.
Column 5 of triangle is A054334.
Cf. A078812, A111125.
Cf. A127677, A208513, A263916.
Sequence in context: A211783 A185780 A051672 * A092162 A073056 A235944
Adjacent sequences: A156305 A156306 A156307 * A156309 A156310 A156311


KEYWORD

easy,nonn,tabl


AUTHOR

Hartmut F. W. Hoft, Feb 07 2009


STATUS

approved



