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

%I A141611 #19 Aug 16 2025 07:05:12
%S A141611 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 A141611 225,72,7,8,98,378,700,700,378,98,8,9,128,588,1344,1750,1344,588,128,
%U A141611 9,10,162,864,2352,3780,3780,2352,864,162,10,11,200,1215,3840,7350,9072,7350,3840,1215,200,11
%N A141611 Triangle read by rows: T(n, k) = (n-k+1)*(k+1)*binomial(n, k).
%C A141611 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 A141611 Indranil Ghosh, <a href="/A141611/b141611.txt">Table of n, a(n) for n = 1..5151 (Rows 0..100 of triangle, flattened)</a>
%F A141611 T(n, k) = (k+1)*(n-k+1)*binomial(n,k).
%F A141611 Sum_{k=0..n} T(n, k) = A007466(n+1) (row sums).
%F A141611 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 A141611 From _G. C. Greubel_, Sep 22 2024: (Start)
%F A141611 T(2*n, n) = A037966(n+1).
%F A141611 T(2*n-1, n) = 2*A085373(n-1), for n >= 1.
%F A141611 Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n) - 2*[n=2]. (End)
%e A141611 Triangle begins as:
%e A141611    1;
%e A141611    2,   2;
%e A141611    3,   8,    3;
%e A141611    4,  18,   18,    4;
%e A141611    5,  32,   54,   32,    5;
%e A141611    6,  50,  120,  120,   50,    6;
%e A141611    7,  72,  225,  320,  225,   72,    7;
%e A141611    8,  98,  378,  700,  700,  378,   98,    8;
%e A141611    9, 128,  588, 1344, 1750, 1344,  588,  128,    9;
%e A141611   10, 162,  864, 2352, 3780, 3780, 2352,  864,  162,  10;
%e A141611   11, 200, 1215, 3840, 7350, 9072, 7350, 3840, 1215, 200, 11;
%e A141611   ...
%e A141611 From _Peter Bala_, Mar 06 2017: (Start)
%e A141611 Factorization as a square array
%e A141611   /1         \ /1  2  3  4...\ /1  2   3   4...\
%e A141611   |2  2      | |   2  6 12...| |2  8  12  32...|
%e A141611   |3  6  3   |*|      3 12...|=|3 18  54 120...|
%e A141611   |4 12 12 4 | |         4...| |4 32 120 320...|
%e A141611   |...       | |             | |...            |
%e A141611 (End)
%t A141611 T[n_, m_]:= (n-m+1)*(m+1)*Binomial[n,m];
%t A141611 Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten
%o A141611 (PARI) T(n,m)=(n - m + 1)*(m + 1)*binomial(n, m) \\ _Charles R Greathouse IV_, Feb 15 2017
%o A141611 (Magma)
%o A141611 A141611:= func< n,k | (k+1)*(n-k+1)*Binomial(n,k) >;
%o A141611 [A141611(n,k): k in [0..n], n in [0..14]]; // _G. C. Greubel_, Sep 22 2024
%o A141611 (SageMath)
%o A141611 def A141611(n,k): return (k+1)*(n-k+1)*binomial(n,k)
%o A141611 flatten([[A141611(n,k) for k in range(n+1)] for n in range(15)]) # _G. C. Greubel_, Sep 22 2024
%Y A141611  Cf. A003506, A007466 (row sums), A037966, A085373.
%K A141611 nonn,tabl,easy
%O A141611 0,2
%A A141611 _Roger L. Bagula_ and _Gary W. Adamson_, Aug 22 2008
%E A141611 Offset corrected by _G. C. Greubel_, Sep 22 2024