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A124392 A Fine number related number triangle.

%I A124392 #18 Mar 15 2025 23:29:14
%S A124392 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 A124392 11,2,1,4352,1432,456,140,40,12,2,1,16414,5408,1732,536,160,44,13,2,1,
%U A124392 62292,20546,6608,2064,622,181,48,14,2,1,237590,78436,25314,7960,2429,714,203,52,15,2,1
%N A124392 A Fine number related number triangle.
%C A124392 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 A124392 G. C. Greubel, <a href="/A124392/b124392.txt">Rows n = 0..100 of triangle, flattened</a>
%F A124392 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 A124392 Number triangle T(n, k) = Sum_{j=0..n-k} C(n-j, k)*C(2*j, n-k).
%e A124392 Triangle begins
%e A124392       1;
%e A124392       2,    1;
%e A124392       7,    2,    1;
%e A124392      24,    8,    2,   1;
%e A124392      86,   28,    9,   2,   1;
%e A124392     314,  103,   32,  10,   2,  1;
%e A124392    1163,  382,  121,  36,  11,  2,  1;
%e A124392    4352, 1432,  456, 140,  40, 12,  2, 1;
%e A124392   16414, 5408, 1732, 536, 160, 44, 13, 2, 1;
%p A124392 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 A124392 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 A124392 (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 A124392 (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 A124392 (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 A124392 (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 A124392 Cf. A000957, A001700, A014300, A114590.
%K A124392 easy,nonn,tabl
%O A124392 0,2
%A A124392 _Paul Barry_, Oct 30 2006