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A171822 Triangle T(n,k) = binomial(2*n-k, k)*binomial(n+k, 2*k), read by rows. 2

%I

%S 1,1,1,1,9,1,1,30,30,1,1,70,225,70,1,1,135,980,980,135,1,1,231,3150,

%T 7056,3150,231,1,1,364,8316,34650,34650,8316,364,1,1,540,19110,132132,

%U 245025,132132,19110,540,1,1,765,39600,420420,1288287,1288287,420420,39600,765,1

%N Triangle T(n,k) = binomial(2*n-k, k)*binomial(n+k, 2*k), read by rows.

%H G. C. Greubel, <a href="/A171822/b171822.txt">Rows n = 0..100 of the triangle, flattened</a>

%F T(n, k) = binomial(2*n-k, k)*binomial(n+k, 2*k) = A054142(n, k)*A085478(n, k).

%F Sum_{k=0..n} T(n, k) = Hypergeometric 4F3([-n, -n, 1/2 -n, n+1], [1/2, 1, -2*n], 1) = A183160(n). - _G. C. Greubel_, Feb 22 2021

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 9, 1;

%e 1, 30, 30, 1;

%e 1, 70, 225, 70, 1;

%e 1, 135, 980, 980, 135, 1;

%e 1, 231, 3150, 7056, 3150, 231, 1;

%e 1, 364, 8316, 34650, 34650, 8316, 364, 1;

%e 1, 540, 19110, 132132, 245025, 132132, 19110, 540, 1;

%e 1, 765, 39600, 420420, 1288287, 1288287, 420420, 39600, 765, 1;

%e 1, 1045, 75735, 1166880, 5465460, 9018009, 5465460, 1166880, 75735, 1045, 1;

%t Table[Binomial[2*n-k, k]*Binomial[n+k, 2*k], {n,0,10}, {k,0,n}]//Flatten

%o (Sage) flatten([[binomial(2*n-k, k)*binomial(n+k, 2*k) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Feb 22 2021

%o (Magma) [Binomial(2*n-k, k)*Binomial(n+k, 2*k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Feb 22 2021

%Y Cf. A054142, A085478, A183160.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Dec 19 2009

%E Edited by _G. C. Greubel_, Feb 22 2021

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Last modified June 14 21:27 EDT 2021. Contains 345041 sequences. (Running on oeis4.)