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A062145 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.

%I A062145 #38 Mar 10 2025 13:38:58
%S A062145 1,1,4,1,10,10,1,18,45,20,1,28,126,140,35,1,40,280,560,350,56,1,54,
%T A062145 540,1680,1890,756,84,1,70,945,4200,7350,5292,1470,120,1,88,1540,9240,
%U A062145 23100,25872,12936,2640,165,1,108,2376,18480,62370,99792,77616,28512,4455,220
%N A062145 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 A062145 Coefficient triangle of certain polynomials N(3; m,x).
%H A062145 G. C. Greubel, <a href="/A062145/b062145.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A062145 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 A062145 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 A062145 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 A062145 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
%F A062145 From _G. C. Greubel_, Mar 07 2025 : (Start)
%F A062145 T(n, k) = binomial(n, k)*binomial(n+3, k).
%F A062145 T(2*n, n) = (1/2)*(n+1)^2*A000108(n)*A000108(n+2).
%F A062145 Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^floor((n+2)/2)*(A047074(n+3) - A047074(n+ 2)). (End)
%e A062145 As a square array:
%e A062145     1,    1,     1,     1,     1,     1,    1,  1, ... A000012;
%e A062145     4,   10,    18,    28,    40,    54,   70, 88, ... A028552;
%e A062145    10,   45,   126,   280,   540,   945, 1540, ....... A105938;
%e A062145    20,  140,   560,  1680,  4200,  9240, ............. A105939;
%e A062145    35,  350,  1890,  7350, 23100, 62370, ............. A027803;
%e A062145    56,  756,  5292, 25872, 99792, .................... A105940;
%e A062145    84, 1470, 12936, 77616, ........................... A105942;
%e A062145   120, 2640, 28512, .................................. A105943;
%e A062145   165, 4455, 57015, .................................. A105944;
%e A062145   ....;
%e A062145 As a triangle:
%e A062145   1;
%e A062145   1,   4;
%e A062145   1,  10,   10;
%e A062145   1,  18,   45,    20;
%e A062145   1,  28,  126,   140,    35;
%e A062145   1,  40,  280,   560,   350,    56;
%e A062145   1,  54,  540,  1680,  1890,   756,    84;
%e A062145   1,  70,  945,  4200,  7350,  5292,  1470,   120;
%e A062145   1,  88, 1540,  9240, 23100, 25872, 12936,  2640,  165;
%e A062145   1, 108, 2376, 18480, 62370, 99792, 77616, 28512, 4455, 220;
%e A062145   ....;
%t A062145 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 A062145 P[c_, n_, z_] := Sum[Binomial[n, k] Pochhammer[n-k+c, k] z^k /k!, {k,0,n}];
%t A062145 CL[c_] := Table[CoefficientList[P[c, n, z], z], {n, 0, 5}] // TableForm
%t A062145 CL[4]  (* _Peter Luschny_, Feb 12 2024 *)
%t A062145 A062145[n_,k_]:= Binomial[n,k]*Binomial[n+3,k];
%t A062145 Table[A062145[n,k], {n,0,12},{k,0,n}]//Flatten (* _G. C. Greubel_, Mar 07 2025 *)
%o A062145 (Magma)
%o A062145 A062145:= func< n,k | Binomial(n,k)*Binomial(n+3,k) >;
%o A062145 [A062145(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 07 2025
%o A062145 (SageMath)
%o A062145 def A062145(n,k): return binomial(n,k)*binomial(n+3,k)
%o A062145 print(flatten([[A062145(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Mar 07 2025
%Y A062145 Family of polynomials: A008459 (c=1), A132813 (c=2), A062196 (c=3), this sequence (c=4), A062264 (c=5), A062190 (c=6).
%Y A062145 Columns: A028552 (k=1), A105938 (k=2), A105939 (k=3), A027803 (k=4), A105940 (k=5), A105942 (k=6), A105943 (k=7), A105944 (k=8).
%Y A062145 Diagonals: A000292 (k=n), A027800 (k=n-1), A107417 (k=n-2), A107418 (k=n-3), A107419 (k=n-4), A107420 (k=n-5), A107421 (k=n-6), A107422 (k=n-7).
%Y A062145 Sums: A002054 (row).
%Y A062145 Cf. A000108, A000292, A047074.
%K A062145 nonn,tabl
%O A062145 0,3
%A A062145 _Wolfdieter Lang_, Jun 19 2001
%E A062145 New name by _Peter Luschny_, Feb 12 2024
%E A062145 More terms from _G. C. Greubel_, Mar 07 2025