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A075362 Triangle read by rows with the n-th row containing the first n multiples of n. 9
1, 2, 4, 3, 6, 9, 4, 8, 12, 16, 5, 10, 15, 20, 25, 6, 12, 18, 24, 30, 36, 7, 14, 21, 28, 35, 42, 49, 8, 16, 24, 32, 40, 48, 56, 64, 9, 18, 27, 36, 45, 54, 63, 72, 81, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 12, 24, 36, 48, 60, 72, 84 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

T(n,k) = A050873(n,k)*A051173(n,k), 1<=k<=n. [Reinhard Zumkeller, Apr 25 2011]

(Conjecture) Let N=2*n and k=1,...,n. Let A_{N,0}, A_{N,1}, ..., A_{N,n-1} be the n X n unit-primitive matrices (see [Jeffery]) associated with N. 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 B^T denotes the transpose of matrix B. Let S_N=[V_0,V_1,...,V_(n-1)] be the n X n matrix formed by taking the components of vector V_(k-1) as the entries in column k-1 (V_(k-1) gives the ordered spectrum of A_{N,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 also T(n,k)=[X_N]_(k-1,k-1); that is, row n of the triangle is given by the main diagonal entries of X_N. 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]=[V_(k-1)]^T*V_(k-1). - L. Edson Jeffery, Jan 20 2012

Conjecture that antidiagonal sums are A023855. - L. Edson Jeffery, Jan 20 2012

LINKS

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

L. E. Jeffery, Unit-primitive matrices

FORMULA

T(n,k) = n*k, 1 <= k <= n. [Reinhard Zumkeller, Mar 07 2010]

T(n,k) = sum_{i=1..k} i*binomial(k,i)*binomial(n+1-k,n-i), 1<=k<=n. [Mircea Merca, Apr 11 2012]

T(n,k) = A002260(n,k)*A002024(n,k) = (A215630(n,k)-A215631(n,k))/2, 1<=k<=n. - Reinhard Zumkeller, Nov 11 2012

a(n) = A223544(n) - 1; a(n) = i*(t+1), where i = n - t*(t+1)/2, t = floor((-1 + sqrt(8*n-7))/2). - Boris Putievskiy, Jul 24 2013

EXAMPLE

Triangle begins:

1

2 4

3 6 9

4 8 12 16

5 10 15 20 25

6 12 18 24 30 36

MAPLE

T(n, k):=piecewise(k<=n, sum(i*binomial(k, i)*binomial(n+1-k, n-i), i=1..k), k>n, 0) [Mircea Merca, Apr 11 2012]

PROG

(Haskell)

a075362 n k = a075362_tabl !! (n-1) !! (k-1)

a075362_row n = a075362_tabl !! (n-1)

a075362_tabl = zipWith (zipWith (*)) a002260_tabl a002024_tabl

-- Reinhard Zumkeller, Nov 11 2012, Oct 04 2012

CROSSREFS

A002411 gives the sum of the n-th row. A141419 is similarly derived.

Cf. A223544.

Sequence in context: A077583 A153125 A139413 * A110749 A077529 A143516

Adjacent sequences:  A075359 A075360 A075361 * A075363 A075364 A075365

KEYWORD

nonn,tabl,easy

AUTHOR

Amarnath Murthy, Sep 20 2002

EXTENSIONS

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

STATUS

approved

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Last modified April 16 06:51 EDT 2014. Contains 240552 sequences.