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Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).
6

%I #15 Sep 22 2024 02:19:36

%S 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,

%T 225,72,7,8,98,378,700,700,378,98,8,9,128,588,1344,1750,1344,588,128,

%U 9,10,162,864,2352,3780,3780,2352,864,162,10,11,200,1215,3840,7350,9072,7350,3840,1215,200,11

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

%C 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

%H Indranil Ghosh, <a href="/A141611/b141611.txt">Table of n, a(n) for n = 1..5151 (Rows 0..100 of triangle, flattened)</a>

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

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

%F 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

%F From _G. C. Greubel_, Sep 22 2024: (Start)

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

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

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

%e Triangle begins as:

%e 1;

%e 2, 2;

%e 3, 8, 3;

%e 4, 18, 18, 4;

%e 5, 32, 54, 32, 5;

%e 6, 50, 120, 120, 50, 6;

%e 7, 72, 225, 320, 225, 72, 7;

%e 8, 98, 378, 700, 700, 378, 98, 8;

%e 9, 128, 588, 1344, 1750, 1344, 588, 128, 9;

%e 10, 162, 864, 2352, 3780, 3780, 2352, 864, 162, 10;

%e 11, 200, 1215, 3840, 7350, 9072, 7350, 3840, 1215, 200, 11;

%e ...

%e From _Peter Bala_, Mar 06 2017: (Start)

%e Factorization as a square array

%e /1 \ /1 2 3 4...\ /1 2 3 4...\

%e |2 2 | | 2 6 12...| |2 8 12 32...|

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

%e |4 12 12 4 | | 4...| |4 32 120 320...|

%e |... | | | |... |

%e (End)

%t T[n_, m_]:= (n-m+1)*(m+1)*Binomial[n,m];

%t Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten

%o (PARI) T(n,m)=(n - m + 1)*(m + 1)*binomial(n, m) \\ _Charles R Greathouse IV_, Feb 15 2017

%o (Magma)

%o A141611:= func< n,k | (k+1)*(n-k+1)*Binomial(n,k) >;

%o [A141611(n,k): k in [0..n], n in [0..14]]; // _G. C. Greubel_, Sep 22 2024

%o (SageMath)

%o def A141611(n,k): return (k+1)*(n-k+1)*binomial(n,k)

%o flatten([[A141611(n,k) for k in range(n+1)] for n in range(15)]) # _G. C. Greubel_, Sep 22 2024

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

%K nonn,tabl,easy

%O 0,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Aug 22 2008

%E Offset corrected by _G. C. Greubel_, Sep 22 2024