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A156740 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 = 7, read by rows.

%I A156740 #20 Sep 04 2024 21:59:03
%S A156740 1,1,1,1,153,1,1,4845,4845,1,1,74613,2362745,74613,1,1,735471,
%T A156740 358664691,358664691,735471,1,1,5311735,25533510145,393216056233,
%U A156740 25533510145,5311735,1,1,30421755,1056158828725,160324910200455,160324910200455,1056158828725,30421755,1
%N A156740 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 = 7, read by rows.
%H A156740 G. C. Greubel, <a href="/A156740/b156740.txt">Rows n = 0..30 of the triangle, flattened</a>
%F A156740 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 = 7.
%F A156740 Sum_{k=0..n} T(n, k, 7) = A151614(n).
%e A156740 Triangle begins as:
%e A156740   1;
%e A156740   1,       1;
%e A156740   1,     153,           1;
%e A156740   1,    4845,        4845,            1;
%e A156740   1,   74613,     2362745,        74613,           1;
%e A156740   1,  735471,   358664691,    358664691,      735471,       1;
%e A156740   1, 5311735, 25533510145, 393216056233, 25533510145, 5311735, 1;
%t A156740 b[n_, k_]:= Binomial[2*n, 2*k];
%t A156740 T[n_, k_, m_]:= Round[Product[b[n+j, k+j]/b[n-k+j, j], {j,0,m}]];
%t A156740 Table[T[n, k, 7], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 19 2021 *)
%o A156740 (Magma)
%o A156740 A156740:= func< n,k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..7]]) ) >;
%o A156740 [A156740(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 19 2021
%o A156740 (Sage)
%o A156740 def A156740(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..7)) )
%o A156740 flatten([[A156740(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 19 2021
%Y A156740 Cf. A086645 (m=0), A156739 (m=6), this sequence (m=7), A156741 (m=8), A156742 (m=9).
%Y A156740 Cf. A151614 (row sums).
%K A156740 nonn,tabl
%O A156740 0,5
%A A156740 _Roger L. Bagula_, Feb 14 2009
%E A156740 Definition corrected to give integral terms and edited by _G. C. Greubel_, Jun 19 2021