A156742 - OEIS

%I #18 Sep 04 2024 21:58:32

%S 1,1,1,1,231,1,1,10626,10626,1,1,230230,10590580,230230,1,1,3108105,

%T 3097744650,3097744650,3108105,1,1,30045015,404255676825,

%U 8758872997875,404255676825,30045015,1,1,225792840,29367745734600,8590065627370500,8590065627370500,29367745734600,225792840,1

%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 = 9, read by rows.

%H G. C. Greubel, <a href="/A156742/b156742.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 = 9.

%e Triangle begins as:

%e   1;

%e   1,        1;

%e   1,      231,            1;

%e   1,    10626,        10626,             1;

%e   1,   230230,     10590580,        230230,            1;

%e   1,  3108105,   3097744650,    3097744650,      3108105,        1;

%e   1, 30045015, 404255676825, 8758872997875, 404255676825, 30045015, 1;

%t T[n_, k_, m_]:= Round[Product[Binomial[2*(n+j), 2*(k+j)]/Binomial[2*(n-k+j), 2*j], {j,0,m}]];

%t Table[T[n, k, 9], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 19 2021 *)

%o (Magma)

%o A156742:= func< n,k | Round( (&*[Binomial(2*(n+j), 2*(k+j))/Binomial(2*(n-k+j), 2*j): j in [0..9]]) ) >;

%o [A156742(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 19 2021

%o (Sage)

%o def A156742(n, k): return round( product( binomial(2*(n+j), 2*(k+j))/binomial(2*(n-k+j), 2*j) for j in (0..9)) )

%o flatten([[A156742(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 19 2021

%Y Cf. A086645 (m=0), A156739 (m=6), A156740 (m=7), A156741 (m=8), this sequence (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 19 2021