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A075362 Triangle read by rows with the n-th row containing the first n multiples of n. 13
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

(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

Viewed as a sequence of rows, consider the subsequences (of rows) that contain every positive integer. The lexicographically latest of these subsequences consists of the rows with row numbers in A066680 U {1}; this is the only one that contains its own row numbers only once. - Peter Munn, Dec 04 2019

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

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. A066680, A223544.

Cf. A003991 (square multiplication table).

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 January 29 05:04 EST 2022. Contains 350672 sequences. (Running on oeis4.)