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%I #15 Sep 08 2022 08:45:20
%S 1,1,1,1,1,1,1,2,1,1,1,3,3,1,1,1,5,7,4,1,1,1,8,19,13,5,1,1,1,13,51,46,
%T 21,6,1,1,1,21,141,166,89,31,7,1,1,1,34,393,610,393,151,43,8,1,1,1,55,
%U 1107,2269,1761,776,235,57,9,1,1,1,89,3139,8518,7985,4056,1363,344,73,10,1,1
%N A number triangle of sums of binomial products.
%C Columns include A000045, A002426, A026641. Rows include A000012, A000027, A002061(n+1). Row sums are A110542.
%H G. C. Greubel, <a href="/A110541/b110541.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F T(n,k) = Sum_{j=0..n-k} C((k-1)*(n-k)-(k-2)*j, j)*C(j, n-k-j).
%e Rows begin
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 2, 1, 1;
%e 1, 3, 3, 1, 1;
%e 1, 5, 7, 4, 1, 1;
%e 1, 8, 19, 13, 5, 1, 1;
%e 1, 13, 51, 46, 21, 6, 1, 1;
%e 1, 21, 141, 166, 89, 31, 7, 1, 1;
%e As a number square read by antidiagonals, rows begin
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 3, 4, 5, 6, ...
%e 1, 3, 7, 13, 21, 31, ...
%e 1, 5, 19, 46, 89, 151, ...
%e 1, 8, 51, 166, 393, 776, ...
%t T[n_, k_] := Sum[Binomial[(k-1)*(n-k) - (k-2)*j, j]*Binomial[j, n-k-j], {j, 0, n-k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Aug 31 2017 *)
%o (PARI) for(n=0,20, for(k=0,n, print1(sum(j=0,n-k, binomial((k-1)*(n-k) -(k-2)*j, j)*binomial(j, n-k-j)), ", "))) \\ _G. C. Greubel_, Aug 31 2017
%o (Magma) [[(&+[Binomial((k-1)*(n-k)-(k-2)*j, j)*Binomial(j, n-k-j): j in [0..n-k]]): k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Feb 19 2019
%o (Sage) [[sum(binomial((k-1)*(n-k) -(k-2)*j, j)*binomial(j, n-k-j) for j in (0..n-k)) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Feb 19 2019
%o (GAP) Flat(List([0..12], n-> List([0..n], k-> Sum([0..n-k], j-> Binomial((k-1)*(n-k)-(k-2)*j, j)*Binomial(j, n-k-j) )))); # _G. C. Greubel_, Feb 19 2019
%K easy,nonn,tabl
%O 0,8
%A _Paul Barry_, Jul 25 2005
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