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A113582 Triangle T(n,m) read by rows: T(n,m) = (n - m)*(n - m + 1)*m*(m + 1)/4 + 1. 4
1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 7, 10, 7, 1, 1, 11, 19, 19, 11, 1, 1, 16, 31, 37, 31, 16, 1, 1, 22, 46, 61, 61, 46, 22, 1, 1, 29, 64, 91, 101, 91, 64, 29, 1, 1, 37, 85, 127, 151, 151, 127, 85, 37, 1, 1, 46, 109, 169, 211, 226, 211, 169, 109, 46, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

From Paul Barry, Jan 07 2009: (Start)

This triangle follows a general construction method as follows: Let a(n) be an integer sequence with a(0)=1, a(1)=1. Then T(n,k,r) := [k<=n](1+r*a(k)*a(n-k)) defines a symmetrical triangle.

Row sums are n + 1 + r*Sum_{k=0..n} a(k)*a(n-k) and central coefficients are 1+r*a(n)^2.

Here a(n) = C(n+1,2) and r=1.

Row sums are A154322 and central coefficients are A154323. (End)

LINKS

G. C. Greubel, Rows n=0..100 of triangle, flattened

FORMULA

T(n,m) = (n - m)*(n - m + 1)*m*(m + 1)/4 + 1.

EXAMPLE

{1},

{1, 1},

{1, 2, 1},

{1, 4, 4, 1},

{1, 7, 10, 7, 1},

{1, 11, 19, 19, 11, 1},

{1, 16, 31, 37, 31, 16, 1},

{1, 22, 46, 61, 61, 46, 22, 1},

{1, 29, 64, 91, 101, 91, 64, 29, 1},

{1, 37, 85, 127, 151, 151, 127, 85, 37, 1},

{1, 46, 109, 169, 211, 226, 211, 169, 109, 46, 1}

MATHEMATICA

t[n_, m_] = (n - m)*(n - m + 1)*m*(m + 1)/4 + 1; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]//Flatten

PROG

(MAGMA) /* As triangle: */ [[(n-m)*(n-m+1)*m*(m+1)/4+1: m in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 12 2016

(PARI) for(n=0, 15, for(k=0, n, print1((n-k)*(n-k+1)*k*(k+1)/4 + 1, ", "))) \\ G. C. Greubel, Aug 31 2018

CROSSREFS

Sequence in context: A128562 A034368 A296157 * A347147 A295213 A118245

Adjacent sequences:  A113579 A113580 A113581 * A113583 A113584 A113585

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Aug 25 2008

STATUS

approved

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Last modified December 7 19:20 EST 2021. Contains 349585 sequences. (Running on oeis4.)