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A141611 A symmetrical triangle of coefficients read by rows: t(n,m)=(n - m + 1)*(m + 1)*binomial[n, m]. 5
1, 2, 2, 3, 8, 3, 4, 18, 18, 4, 5, 32, 54, 32, 5, 6, 50, 120, 120, 50, 6, 7, 72, 225, 320, 225, 72, 7, 8, 98, 378, 700, 700, 378, 98, 8, 9, 128, 588, 1344, 1750, 1344, 588, 128, 9, 10, 162, 864, 2352, 3780, 3780, 2352, 864, 162, 10, 11, 200, 1215, 3840, 7350, 9072, 7350 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row sums: [1, 4, 14, 44, 128, 352, 928, 2368, 5888, 14336, 34304, ...] = A007466.

Read as a square array, this array factorizes as M*transpose(M), where M = (k*binomial(n,k))n,k>=1 = A003506(n,k). - Peter Bala, Mar 06 2017

LINKS

Indranil Ghosh, Table of n, a(n) for n = 1..5151 (Rows 0..100 of triangle, flattened)

FORMULA

O.g.f. (1 - (1 + t)*x + 2*t*x^2)/(1 - (1 + t)*x)^3 = 1 + (2 + 2*t)*x + (3 + 8*t + 3*t^2)*x^2 + (4 + 18*t + 18*t^2 + 4*t^3)*x^3 + .... - Peter Bala, Mar 06 2017

EXAMPLE

{1},

{2, 2},

{3, 8, 3},

{4, 18, 18, 4},

{5, 32, 54, 32, 5},

{6, 50, 120, 120, 50, 6},

{7, 72, 225, 320, 225, 72, 7},

{8, 98, 378, 700, 700, 378, 98, 8},

{9, 128, 588, 1344, 1750, 1344, 588, 128, 9},

{10, 162, 864, 2352, 3780, 3780, 2352, 864, 162, 10},

{11, 200, 1215, 3840, 7350, 9072, 7350, 3840, 1215, 200, 11}

...

From Peter Bala, Mar 06 2017: (Start)

Factorization as a square array

  /1         \ /1  2  3  4...\ /1  2   3   4...\

  |2  2      | |   2  6 12...| |2  8  12  32...|

  |3  6  3   |*|      3 12...|=|3 18  54 120...|

  |4 12 12 4 | |         4...| |4 32 120 320...|

  |...       | |             | |...            |

(End)

MATHEMATICA

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

PROG

(PARI) t(n, m)=(n - m + 1)*(m + 1)*binomial(n, m) \\ Charles R Greathouse IV, Feb 15 2017

CROSSREFS

A007466 (row sums). Cf. A003506.

Sequence in context: A295943 A296804 A296952 * A234357 A145596 A186753

Adjacent sequences:  A141608 A141609 A141610 * A141612 A141613 A141614

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula and Gary W. Adamson, Aug 22 2008

STATUS

approved

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Last modified April 9 17:32 EDT 2020. Contains 333361 sequences. (Running on oeis4.)