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 A156308 Inverse of triangle S(n,m) defined by sequence A156290, n>=1, 1<=m<=n. 10

%I

%S 1,4,1,9,6,1,16,20,8,1,25,50,35,10,1,36,105,112,54,12,1,49,196,294,

%T 210,77,14,1,64,336,672,660,352,104,16,1,81,540,1386,1782,1287,546,

%U 135,18,1,100,825,2640,4290,4004,2275,800,170,20,1

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

%C From _Wolfdieter Lang_, Jun 26 2011: (Start)

%C 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).

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

%C (End)

%C 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

%H D. E. Knuth, <a href="http://arxiv.org/abs/math/9207222//">Johann Faulhaber and sums of powers</a>, Math. Comp. 61 (1993), no. 203, 277-294.

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

%F From _Peter Bala_, May 01 2014: (Start):

%F 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].

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

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

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

%e Triangle starts:

%e n=1: 1

%e n=2: 4, 1

%e n=3: 9, 6, 1

%e n=4: 16, 20, 8, 1

%e ...

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

%Y Same as triangle A208513 with the first column truncated.

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

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

%K easy,nonn,tabl

%O 1,2

%A _Hartmut F. W. Hoft_, Feb 07 2009

%E Edited by _Max Alekseyev_, Mar 05 2018

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Last modified November 21 11:01 EST 2018. Contains 317447 sequences. (Running on oeis4.)