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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,n-1} = [0,...,0,1; 0,...,0,1,1; ...; 0,1,...,1; 1,...,1], an n X n unit-primitive matrix (see [Jeffery]). Let M_n=[A_{N,n-1}]^4. Then t(n,k)=[M_n]_(1,k), that is, the n-th 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,n-1} be the n X n unit-primitive 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_(r-1)(x) - U_(r-2)(x)  (r>1). Define the column vectors V_(k-1) = (U_(k-1)(cos(Pi/N)), U_(k-1)(cos(3*Pi/N)), ..., U_(k-1)(cos((2*n-1)*Pi/N)))^T, where T denotes matrix transpose. Let S_N = [V_0, V_1, ..., V_(n-1)] be the n X n matrix formed by taking V_(k-1) as column k-1. 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,...,n-1}. Then t(n,k) = [X_N]_(k-1,k-1), 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_(k-1)(cos((2*m-1)*Pi/N)))^2]. Finally, this sequence is related to A057059, since X_N = [sum_{m=1,...,n} A057059(n,m)*A_{N,m-1}] is also an integral linear combination of unit-primitive matrices from the N-th set. - L. Edson Jeffery, Jan 20 2012

Row sums: n*(n+1)*(2*n+1)/6. - L. Edson Jeffery, Jan 25 2013

n-th row = partial sums of n-th 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, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, 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, Unit-primitive matrices

FORMULA

t(n,m) = m*(2*n - m + 1)/2.

t(n,m) = A000217(n) - A000217(n-m). - L. Edson Jeffery, Jan 16 2013

Let v = d*h with h odd be an integer factorization, then v = t(d+(h-1)/2, h) if h+1 <= 2*d, and v = t(d+(h-1)/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+(5-1)/2, 5) in column 5 since 5+1 <= 2*3, t(5+(3-1)/2, 3), t(1+(15-1)/2, 2*1) in column 2 since 15+1 > 2*1, and t(15+(1-1)/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 !! (n-1)

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

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Last modified October 22 02:06 EDT 2017. Contains 293756 sequences.