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A062190 Coefficient triangle of certain polynomials N(5; m,x).

%I A062190 #31 Mar 02 2025 08:00:45
%S A062190 1,1,6,1,14,21,1,24,84,56,1,36,216,336,126,1,50,450,1200,1050,252,1,
%T A062190 66,825,3300,4950,2772,462,1,84,1386,7700,17325,16632,6468,792,1,104,
%U A062190 2184,16016,50050,72072,48048,13728,1287,1,126,3276,30576,126126,252252,252252,123552,27027,2002
%N A062190 Coefficient triangle of certain polynomials N(5; m,x).
%C A062190 The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=5) Laguerre triangle L(5; n+m,m)= A062138(n+m,m), n >= 0, is N(5; m,x)/(1-x)^(2*(m+3)), with the row polynomials N(5; m,x) := Sum_{k=0..m} a(m,k)*x^k.
%H A062190 G. C. Greubel, <a href="/A062190/b062190.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A062190 T(m, k) = [x^k]N(5; m, x), with N(5; m, x) = ((1-x)^(2*(m+3)))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+6))).
%F A062190 N(5; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j).
%F A062190 N(5; m, x)= x^m*(2*m+5)! * 2F1(-m, -m; -2*m-5; (x-1)/x)/((m+5)!*m!). - _Jean-François Alcover_, Sep 18 2013
%F A062190 T(n, k) = binomial(n, k)*binomial(n+5, k). - _G. C. Greubel_, Feb 28 2025
%e A062190 Triangle begins as:
%e A062190   1;
%e A062190   1,   6;
%e A062190   1,  14,   21;
%e A062190   1,  24,   84,    56;
%e A062190   1,  36,  216,   336,    126;
%e A062190   1,  50,  450,  1200,   1050,    252;
%e A062190   1,  66,  825,  3300,   4950,   2772,     462;
%e A062190   1,  84, 1386,  7700,  17325,  16632,    6468,    792;
%e A062190   1, 104, 2184, 16016,  50050,  72072,   48048,  13728,   1287;
%e A062190   1, 126, 3276, 30576, 126126, 252252,  252252, 123552,  27027,  2002;
%e A062190   1, 150, 4725, 54600, 286650, 756756, 1051050, 772200, 289575, 50050, 3003;
%p A062190 A062190 := proc(m,k)
%p A062190     add( (binomial(m, j)*(2*m+5-j)!/((m+5)!*(m-j)!))*(x^(m-j))*(1-x)^j,j=0..m) ;
%p A062190     coeftayl(%,x=0,k) ;
%p A062190 end proc: # _R. J. Mathar_, Nov 29 2015
%t A062190 NN[5, m_, x_] := x^m*(2*m+5)!*Hypergeometric2F1[-m, -m, -2*m-5, (x-1)/x]/((m+5)!*m!); Table[CoefficientList[NN[5, m, x], x], {m, 0, 8}] // Flatten (* _Jean-François Alcover_, Sep 18 2013 *)
%t A062190 A062190[n_,k_]:= Binomial[n,k]*Binomial[n+5,k];
%t A062190 Table[A062190[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 28 2025 *)
%o A062190 (Magma)
%o A062190 A062190:= func< n,k | Binomial(n,k)*Binomial(n+5,k) >;
%o A062190 [A062190(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 28 2025
%o A062190 (SageMath)
%o A062190 def A062190(n,k): return binomial(n,k)*binomial(n+5,k)
%o A062190 print(flatten([[A062190(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Feb 28 2025
%Y A062190 Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), this sequence (c=6).
%Y A062190 Columns k: A028557 (k=1), A104676 (k=2), A104677 (k=3), A104678 (k=4), A104679 (k=5), A104680 (k=6).
%Y A062190 Diagonals: A000389 (k=n), A027818 (k=n-1), A104670 (k=n-2), A104671 (k=n-3), A104672 (k=n-4), A104673 (k=n-5), A104674 (k=n-6).
%Y A062190 Cf. A003516 (row sums), A113894 (main diagonal).
%K A062190 nonn,tabl
%O A062190 0,3
%A A062190 _Wolfdieter Lang_, Jun 19 2001