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A156308 Inverse of triangle S(n,m) defined by sequence A156290, n>=1, 1<=m<=n. 10
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 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..55.

D. E. Knuth, Johann Faulhaber and sums of powers, 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)

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}]]

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.

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

EXTENSIONS

Edited by Max Alekseyev, Mar 05 2018

STATUS

approved

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Last modified August 18 00:28 EDT 2018. Contains 313817 sequences. (Running on oeis4.)