A156739 - OEIS

%I #20 Sep 04 2024 18:57:59

%S 1,1,1,1,120,1,1,3060,3060,1,1,38760,988380,38760,1,1,319770,

%T 103285710,103285710,319770,1,1,1961256,5226256926,66199254396,

%U 5226256926,1961256,1,1,9657700,157843517260,16494647553670,16494647553670,157843517260,9657700,1

%N Triangle T(n, k, m) = round( Product_{j=0..m} binomial(2*(n+j), 2*(k+j))/binomial( 2*(n-k+j), 2*j) ), where m = 6, read by rows.

%H G. C. Greubel, <a href="/A156739/b156739.txt">Rows n = 0..30 of the triangle, flattened</a>

%F T(n, k, m) = round( Product_{j=0..m} b(n+j, k+j)/b(n-k+j, j) ), where b(n, k) = binomial(2*n, 2*k) and m = 6.

%e Triangle begins as:

%e   1;

%e   1,       1;

%e   1,     120,          1;

%e   1,    3060,       3060,           1;

%e   1,   38760,     988380,       38760,          1;

%e   1,  319770,  103285710,   103285710,     319770,       1;

%e   1, 1961256, 5226256926, 66199254396, 5226256926, 1961256, 1;

%t b[n_, k_]:= Binomial[2*n, 2*k];

%t T[n_, k_, m_]:= Round[Product[b[n+j, k+j]/b[n-k+j, j], {j,0,m}]];

%t Table[T[n, k, 6], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 18 2021 *)

%o (Magma)

%o A156739:= func< n,k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..6]]) ) >;

%o [A156739(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 18 2021

%o (Sage)

%o def A156739(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..6)) )

%o flatten([[A156739(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 18 2021

%Y Cf. A086645 (m=0), this sequence (m=6), A156740 (m=7), A156741 (m=8), A156742 (m=9).

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 14 2009

%E Definition corrected to give integral terms and edited by _G. C. Greubel_, Jun 18 2021