

A141419


A triangular sequence of coefficients of Dynkin diagram weights for the Cartan Groups B_n: t(n,m)=m*(2*n  m + 1)/2. As a rectangle, the accumulation array of A051340.


14



1, 2, 3, 3, 5, 6, 4, 7, 9, 10, 5, 9, 12, 14, 15, 6, 11, 15, 18, 20, 21, 7, 13, 18, 22, 25, 27, 28, 8, 15, 21, 26, 30, 33, 35, 36, 9, 17, 24, 30, 35, 39, 42, 44, 45, 10, 19, 27, 34, 40, 45, 49, 52, 54, 55
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

From Clark Kimberling, Feb 05 2011: (Start)
Row sums are:
{1, 5, 14, 30, 55, 91, 140, 204, 285, 385};
Here all the weights are divided by two where they aren't in Cahn.
(Start) As a rectangle, A141419 is in the accumulation chain
...< A051340 < A141419 < A185874 < A185875 < A185876 < ...
(See A144112 for the definition of accumulation array.)
row 1: A000027
col 1: A000217
diag (1,5,...): A000326 (pentagonal numbers)
diag (2,7,...): A005449 (second pentagonal numbers)
diag (3,9,...): A045943 (triangular matchstick numbers)
diag (4,11,...): A115067
diag (5,13,...): A140090
diag (6,15,...): A140091
diag (7,17,...): A059845
diag (8,19,...): A140672
antidiagonal sums: A000330
(End)
Let N=2*n+1 and k=1,2,...,n. Let A_{N,n1} = [0,...,0,1; 0,...,0,1,1; ...; 0,1,...,1; 1,...,1], an n X n unitprimitive matrix (see [Jeffery]). Let M_n=[A_{N,n1}]^4. Then t(n,k)=[M_n]_(1,k), that is, the nth row of the triangle is given by the first row of M_n.  L. Edson Jeffery, Nov 20 2011
Conjecture. Let N=2*n+1 and k=1,...,n. Let A_{N,0}, A_{N,1}, ..., A_{N,n1} be the n X n unitprimitive matrices (again see [Jeffery]) associated with N, and define the Chebyshev polynomials of the second kind by the recurrence U_0(x) = 1, U_1(x) = 2*x and U_r(x) = 2*x*U_(r1)(x)  U_(r2)(x) (r>1). Define the column vectors V_(k1) = (U_(k1)(cos(Pi/N)), U_(k1)(cos(3*Pi/N)), ..., U_(k1)(cos((2*n1)*Pi/N)))^T, where T denotes matrix transpose. Let S_N = [V_0, V_1, ..., V_(n1)] be the n X n matrix formed by taking V_(k1) as column k1. Let X_N = [S_N]^T*S_N, and let [X_N]_(i,j) denote the entry in row i and column j of X_N, i,j in {0,...,n1}. Then t(n,k) = [X_N]_(k1,k1), and row n of the triangle is given by the main diagonal entries of X_N. Remarks: Hence t(n,k) is the sum of squares t(n,k) = sum[m=1,...,n (U_(k1)(cos((2*m1)*Pi/N)))^2]. Finally, this sequence is related to A057059, since X_N = [sum_{m=1,...,n} A057059(n,m)*A_{N,m1}] is also an integral linear combination of unitprimitive matrices from the Nth set.  L. Edson Jeffery, Jan 20 2012
Row sums: n*(n+1)*(2*n+1)/6.  L. Edson Jeffery, Jan 25 2013
nth row = partial sums of nth row of A004736.  Reinhard Zumkeller, Aug 04 2014
T(n,k) is the number of distinct sums made by at most k elements in {1, 2, ... n}, for 1 <= k <= n, e.g., T(6,2) = the number of distinct sums made by at most 2 elements in {1,2,3,4,5,6}. The sums range from 1, to 5+6=11. So there are 11 distinct sums.  Derek Orr, Nov 26 2014
A number n occurs in this sequence A001227(n) times, the number of odd divisors of n, see A209260.  Hartmut F. W. Hoft, Apr 14 2016


REFERENCES

R. N. Cahn, SemiSimple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0486449998, p. 139.


LINKS

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Johann Cigler, Some elementary observations on Narayana polynomials and related topics, arXiv:1611.05252 [math.CO], 2016. See p. 24.
L. E. Jeffery, Unitprimitive matrices


FORMULA

t(n,m) = m*(2*n  m + 1)/2.
t(n,m) = A000217(n)  A000217(nm).  L. Edson Jeffery, Jan 16 2013
Let v = d*h with h odd be an integer factorization, then v = t(d+(h1)/2, h) if h+1 <= 2*d, and v = t(d+(h1)/2, 2*d) if h+1 > 2*d; see A209260.  Hartmut F. W. Hoft, Apr 14 2016


EXAMPLE

{1},
{2, 3},
{3, 5, 6},
{4, 7, 9, 10},
{5, 9, 12, 14, 15},
{6, 11, 15, 18, 20, 21},
{7, 13, 18, 22, 25, 27, 28},
{8, 15, 21, 26, 30, 33, 35, 36},
{9, 17, 24, 30, 35, 39, 42, 44, 45},
{10, 19, 27, 34, 40, 45, 49, 52, 54, 55}
As a rectangle:
1....2...3...4...5....6....7....8....9...10
3....5...7...9..11...13...15...17...19...21
6....9..12..15..18...21...24...27...30...33
10..14..18..22..26...30...34...38...42...46
15..20..25..30..35...40...45...50...55...60
21..27..33..39..45...51...57...63...69...75
28..35..42..49..56...63...70...77...84...91
36..44..52..60..68...76...84...92..100..108
45..54..63..72..81...90...99..108..117..126
55..65..75..85..95..105..115..125..135..145
Since the odd divisors of 15 are 1, 3, 5 and 15, number 15 appears four times in the triangle at t(3+(51)/2, 5) in column 5 since 5+1 <= 2*3, t(5+(31)/2, 3), t(1+(151)/2, 2*1) in column 2 since 15+1 > 2*1, and t(15+(11)/2, 1).  Hartmut F. W. Hoft, Apr 14 2016


MATHEMATICA

Clear[T, n, m, a] T[n_, m_] = m*(2*n  m + 1)/2; a = Table[Table[T[n, m], {m, 1, n}], {n, 1, 10}]; Flatten[a]


PROG

(Haskell)
a141419 n k = k * (2 * n  k + 1) `div` 2
a141419_row n = a141419_tabl !! (n1)
a141419_tabl = map (scanl1 (+)) a004736_tabl
 Reinhard Zumkeller, Aug 04 2014


CROSSREFS

Cf. A144112, A051340, A141419, A185874, A185875, A185876.
Cf. A057059.
Cf. A004736, A141418.
Cf. A001227, A209260.  Hartmut F. W. Hoft, Apr 14 2016
Sequence in context: A003558 A216066 A234094 * A072451 A023156 A051599
Adjacent sequences: A141416 A141417 A141418 * A141420 A141421 A141422


KEYWORD

nonn,tabl,easy


AUTHOR

Roger L. Bagula, Aug 05 2008


STATUS

approved



