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%I #15 Sep 08 2022 08:45:28
%S 1,2,1,7,2,1,24,8,2,1,86,28,9,2,1,314,103,32,10,2,1,1163,382,121,36,
%T 11,2,1,4352,1432,456,140,40,12,2,1,16414,5408,1732,536,160,44,13,2,1,
%U 62292,20546,6608,2064,622,181,48,14,2,1,237590,78436,25314,7960,2429,714,203,52,15,2,1
%N A Fine number related number triangle.
%C First column is A014300. Second column is A114590. Row sums are A001700. Array is given by (f(x)/(x*sqrt(1-4x)), f(x)) where f(x) is g.f. of Fine numbers A000957.
%H G. C. Greubel, <a href="/A124392/b124392.txt">Rows n = 0..100 of triangle, flattened</a>
%F Riordan array ( 1/(x*sqrt(1-4*x)) * (1-sqrt(1-4*x))/(3-sqrt(1-4*x), (1-sqrt(1-4*x))/(3-sqrt(1-4*x)) ).
%F Number triangle T(n, k) = Sum_{j=0..n-k} C(n-j, k)*C(2*j, n-k).
%e Triangle begins
%e 1;
%e 2, 1;
%e 7, 2, 1;
%e 24, 8, 2, 1;
%e 86, 28, 9, 2, 1;
%e 314, 103, 32, 10, 2, 1;
%e 1163, 382, 121, 36, 11, 2, 1;
%e 4352, 1432, 456, 140, 40, 12, 2, 1;
%e 16414, 5408, 1732, 536, 160, 44, 13, 2, 1;
%p seq(seq( add(binomial(n-j, k)*binomial(2*j, n-k), j=0..n-k), k=0..n), n=0..10); # _G. C. Greubel_, Dec 25 2019
%t Table[Sum[Binomial[n-j, k]*Binomial[2*j, n-k], {j,0,n-k}], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 25 2019 *)
%o (PARI) T(n,k) = sum(j=0, n-k, binomial(n-j, k)*binomial(2*j, n-k)); \\ _G. C. Greubel_, Dec 25 2019
%o (Magma) [(&+[Binomial(n-j, k)*Binomial(2*j, n-k): j in [0..n-k]]): k in [0..n], n in [0.10]]; // _G. C. Greubel_, Dec 25 2019
%o (Sage) [[sum(binomial(n-j, k)*binomial(2*j, n-k) for j in (0..n-k)) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 25 2019
%o (GAP) Flat(List([0..10], n-> List([0..n], k-> Binomial(n-j, k)*Binomial(2*j, n-k) ))); # _G. C. Greubel_, Dec 25 2019
%Y Cf. A000957, A001700, A014300, A114590.
%K easy,nonn,tabl
%O 0,2
%A _Paul Barry_, Oct 30 2006
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