A141611 Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).

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

Offset

0,2

Comments

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

T(n, k) = (k+1)*(n-k+1)*binomial(n,k).

Sum_{k=0..n} T(n, k) = A007466(n+1) (row sums).

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

From G. C. Greubel, Sep 22 2024: (Start)

T(2*n, n) = A037966(n+1).

T(2*n-1, n) = 2*A085373(n-1), for n >= 1.

Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n) - 2*[n=2].

Example

Triangle begins as:
   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[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten

Prog

(PARI) T(n, m)=(n - m + 1)*(m + 1)*binomial(n, m) \\ Charles R Greathouse IV, Feb 15 2017
(Magma)
A141611:= func< n, k | (k+1)*(n-k+1)*Binomial(n, k) >;
[A141611(n, k): k in [0..n], n in [0..14]]; // G. C. Greubel, Sep 22 2024
(SageMath)
def A141611(n, k): return (k+1)*(n-k+1)*binomial(n, k)
flatten([[A141611(n, k) for k in range(n+1)] for n in range(15)]) # G. C. Greubel, Sep 22 2024

Crossrefs

Cf. A003506, A007466 (row sums), A037966, A085373.

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

Extensions

Offset corrected by G. C. Greubel, Sep 22 2024

Status

approved