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A062264 Coefficient triangle of certain polynomials N(4; m,x).

%I A062264 #23 Mar 04 2025 08:12:44
%S A062264 1,1,5,1,12,15,1,21,63,35,1,32,168,224,70,1,45,360,840,630,126,1,60,
%T A062264 675,2400,3150,1512,210,1,77,1155,5775,11550,9702,3234,330,1,96,1848,
%U A062264 12320,34650,44352,25872,6336,495,1,117,2808,24024,90090,162162,144144,61776,11583,715
%N A062264 Coefficient triangle of certain polynomials N(4; m,x).
%C A062264 The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=4) Laguerre triangle L(4; n+m,m) = A062140(n+m,m), n >= 0, is N(4; m,x)/(1-x)^(5+2*m), with the row polynomials N(4; m,x) := Sum_{k=0..m} T(m,k)*x^k.
%H A062264 G. C. Greubel, <a href="/A062264/b062264.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A062264 T(m, k) = [x^k] N(4; m, x), with N(4; m, x) = ((1-x)^(2*m+5))*(d^m/dx^m)((x^m)/(m!*(1-x)^(m+5))).
%F A062264 N(4; m, x) = Sum_{j=0..m} (binomial(m, j)*(2*m+4-j)!/((m+4)!*(m-j)!)*(x^(m-j))*(1-x)^j).
%F A062264 From _G. C. Greubel_, Mar 03 2025: (Start)
%F A062264 T(n, k) = binomial(n,k)*binomial(n+4,k).
%F A062264 Sum_{k=0..n} (-1)^k*T(n, k) = (1/4)*( (1+(-1)^n)*(-1)^((n+2)/2)*(n^2 + 5*n - 2)*Catalan((n+2)/2)/(n+1) + 8*(1-(-1)^n)*(-1)^((n+1)/2)*Catalan((n+1)/2) ). (End)
%e A062264 Triangle begins as:
%e A062264   1;
%e A062264   1,   5;
%e A062264   1,  12,   15;
%e A062264   1,  21,   63,    35;
%e A062264   1,  32,  168,   224,     70;
%e A062264   1,  45,  360,   840,    630,    126;
%e A062264   1,  60,  675,  2400,   3150,   1512,    210;
%e A062264   1,  77, 1155,  5775,  11550,   9702,   3234,    330;
%e A062264   1,  96, 1848, 12320,  34650,  44352,  25872,   6336,    495;
%e A062264   1, 117, 2808, 24024,  90090, 162162, 144144,  61776,  11583,   715;
%e A062264   1, 140, 4095, 43680, 210210, 504504, 630630, 411840, 135135, 20020, 1001;
%t A062264 A062264[n_, k_]:= Binomial[n,k]*Binomial[n+4,k];
%t A062264 Table[A062264[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 03 2025 *)
%o A062264 (Magma)
%o A062264 A062264:= func< n,k | Binomial(n,k)*Binomial(n+4,k) >;
%o A062264 [A062264(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 03 2025
%o A062264 (SageMath)
%o A062264 def A062264(n,k): return binomial(n,k)*binomial(n+4,k)
%o A062264 print(flatten([[A062264(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Mar 03 2025
%Y A062264 Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), A062196 (c=3), A062145 (c=4), this sequence (c=5), A062190 (c=6).
%Y A062264 Columns: A028347 (k=2), A104473 (k=3), A104474 (k=4), A104475 (k=5), A027814 (k=6), A103604 (k=7), A104476 (k=8), A104478 (k=9).
%Y A062264 Diagonals: A000332 (k=n), A027810 (k=n-1), A105249 (k=n-2), A105250 (k=n-3), A105251 (k=n-4), A105252 (k=n-5), A105253 (k=n-6), A105254 (k=n-7).
%Y A062264 Sums: A002694 (row).
%K A062264 nonn,tabl
%O A062264 0,3
%A A062264 _Wolfdieter Lang_, Jun 19 2001