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A348539 Triangle T(n, m) = binomial(n+2, m)*binomial(n+2, n-m), read by rows.

%I A348539 #27 Jun 30 2025 18:57:07
%S A348539 1,3,3,6,16,6,10,50,50,10,15,120,225,120,15,21,245,735,735,245,21,28,
%T A348539 448,1960,3136,1960,448,28,36,756,4536,10584,10584,4536,756,36,45,
%U A348539 1200,9450,30240,44100,30240,9450,1200,45,55,1815,18150,76230,152460,152460,76230,18150,1815,55
%N A348539 Triangle T(n, m) = binomial(n+2, m)*binomial(n+2, n-m), read by rows.
%F A348539 G.f.: (x^2*y^2 - 2*x*y + x^2 - 2*x + 1)/(2*x^4*y^2*sqrt(x^2*y^2 + (-2*x^2-2*x)*y + x^2 - 2*x + 1)) + (x*y + x - 1)/(2*x^4*y^2).
%F A348539 G.f.: diff(N(x,y),x)*N(x,y)/(x*y^2), where N(x,y) is the g.f. of the Narayana numbers A001263.
%e A348539 Triangle starts:
%e A348539   [0]  1;
%e A348539   [1]  3,   3;
%e A348539   [2]  6,  16,    6;
%e A348539   [3] 10,  50,   50,   10;
%e A348539   [4] 15, 120,  225,  120,   15;
%e A348539   [5] 21, 245,  735,  735,  245,  21;
%e A348539   [6] 28, 448, 1960, 3136, 1960, 448, 28.
%e A348539   ...
%e A348539 Taylor series: 1 + 3*x*(y + 1) + 2*x^2*(3*y^2 + 8*y + 3) + 10*x^3*(y^3 + 5*y^2 + 5*y + 1) + 15*x^4 (y^4 + 8*y^3 + 15*y^2 + 8*y + 1) + O(x^5)
%p A348539 T := (n, k) -> binomial(n+2, k) * binomial(n+2, n-k):
%p A348539 for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # _Peter Luschny_, Oct 22 2021
%t A348539 T[n_, m_] := Binomial[n + 2, m] * Binomial[n + 2, n - m]; Table[T[n, m], {n, 0, 9}, {m, 0, n}] // Flatten (* _Amiram Eldar_, Oct 22 2021 *)
%o A348539 (Maxima) T(n,m):=binomial(n+2,m)*binomial(n+2,n-m);
%Y A348539 Cf. A001263, A000217, A002694 (with offset 0 are row sums).
%K A348539 nonn,tabl
%O A348539 0,2
%A A348539 _Vladimir Kruchinin_, Oct 21 2021