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%I #27 Feb 13 2024 08:17:05
%S 1,1,4,1,10,10,1,18,45,20,1,28,126,140,35,1,40,280,560,350,56,1,54,
%T 540,1680,1890,756,84,1,70,945,4200,7350,5292,1470,120,1,88,1540,9240,
%U 23100,25872,12936,2640,165,1,108
%N Triangle read by rows. T{n, k] = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.
%C Coefficient triangle of certain polynomials N(3; m,x).
%F The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=3) Laguerre triangle L(3; n+m, m) = A062137(n+m, m), n >= 0, is N(3; m, x)/(1-x)^(2*(m+2)), with the row polynomials N(3; m, x) := Sum_{k=0..m} a(m, k)*x^k.
%F N(3; m, x) := ((1-x)^(2*(m+2)))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+4))); a(m, k) = [x^k]N(3; m, x).
%F N(3; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+3-j)!/((m+3)!*(m-j)!))*(x^(m-j))*(1-x)^j).
%F N(3; m, x)= x^m*(2*m+3)! * 2F1(-m, -m; -2*m-3; (x-1)/x)/((m+3)!*m!). [_Jean-François Alcover_, Sep 18 2013]
%e From _Zerinvary Lajos_, Jan 02 2006: (Start)
%e As a square array:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 4, 10, 18, 28, 40, 54, 70, 88, ...
%e 10, 45, 126, 280, 540, 945, 1540, ...
%e 20, 140, 560, 1680, 4200, 9240, ...
%e 35, 350, 1890, 7350, 23100, ...
%e 56, 756, 5292, 25872, ...
%e ... (End)
%t NN[3, m_, x_] := x^m*(2*m+3)!*Hypergeometric2F1[-m, -m, -2*m-3, (x-1)/x]/((m+3)!*m!); Table[CoefficientList[NN[3, m, x], x], {m, 0, 9}] // Flatten (* _Jean-François Alcover_, Sep 18 2013 *)
%t P[c_, n_, z_] := Sum[Binomial[n, k] Pochhammer[n - k + c, k] z^k / k!, {k, 0, n}];
%t CL[c_] := Table[CoefficientList[P[c, n, z], z], {n, 0, 5}] // TableForm
%t CL[4] (* _Peter Luschny_, Feb 12 2024 *)
%Y Family of polynomials: A008459 (c=1), A132813 (c=2), A062196 (c=3), this sequence (c=4), A062264 (c=5), A062190 (c=6).
%Y Cf. A000292.
%K nonn,tabl
%O 0,3
%A _Wolfdieter Lang_, Jun 19 2001
%E New name by _Peter Luschny_, Feb 12 2024
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