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A062196 Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).

%I A062196 #43 Feb 22 2025 01:30:55
%S A062196 1,1,3,1,8,6,1,15,30,10,1,24,90,80,15,1,35,210,350,175,21,1,48,420,
%T A062196 1120,1050,336,28,1,63,756,2940,4410,2646,588,36,1,80,1260,6720,14700,
%U A062196 14112,5880,960,45,1,99,1980,13860,41580,58212,38808,11880,1485,55
%N A062196 Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).
%C A062196 Also the coefficient triangle of certain polynomials N(2; m,x) := Sum_{k=0..m} T(m,k)*x^k. The e.g.f. of the m-th (unsigned) column sequence without leading zeros of the generalized (a=2) Laguerre triangle L(2; n+m,m) = A062139(n+m,m), n >= 0, is N(2; m,x)/(1-x)^(3+2*m), with the row polynomials N(2; m,x).
%H A062196 G. C. Greubel, <a href="/A062196/b062196.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A062196 T(m, k) = [x^k] N(2; m, x), where N(2; m, x) = ((1-x)^(3+2*m))*(d^m/dx^m)(x^m/(m!*(1-x)^(m+3))).
%F A062196 N(2; m, x) = Sum_{j=0..m} ((binomial(m, j)*(2*m+2-j)!/((m+2)!*(m-j)!)*(x^(m-j)))*(1-x)^j).
%F A062196 T(n,m) = binomial(n, m)*(binomial(n+1, m) + binomial(n+1, m-1)). - _Vladimir Kruchinin_, Apr 06 2018
%F A062196 From _G. C. Greubel_, Feb 21 2025: (Start)
%F A062196 T(2*n, n) = (n+1)^2*A000108(n)*A000108(n+1).
%F A062196 T(2*n-1, n) = (4*n^2 - 1)*A000108(n-1)*A000108(n), n >= 1.
%F A062196 T(2*n+1, n) = (1/2)*binomial(n+2,2)*A000108(n+1)*A000108(n+2). (End)
%e A062196 Triangle starts:
%e A062196   n\k 0...1.....2......3..... 4.....;
%e A062196   [0] 1;
%e A062196   [1] 1,  3;
%e A062196   [2] 1,  8,    6;
%e A062196   [3] 1, 15,   30,    10;
%e A062196   [4] 1, 24,   90,    80,    15;
%e A062196   [5] 1, 35,  210,   350,   175,    21;
%e A062196   [6] 1, 48,  420,  1120,  1050,   336,    28;
%e A062196   [7] 1, 63,  756,  2940,  4410,  2646,   588,    36;
%e A062196   [8] 1, 80, 1260,  6720, 14700, 14112,  5880,   960,   45;
%e A062196   [9] 1, 99, 1980, 13860, 41580, 58212, 38808, 11880, 1485, 55.
%p A062196 T := (n, k) -> binomial(n, k)*binomial(n + 2, k);
%p A062196 seq(seq(T(n, k), k=0..n), n=0..9); # _Peter Luschny_, Sep 30 2021
%t A062196 A062196[n_, k_]:= Binomial[n, k]*Binomial[n+2, k];
%t A062196 Table[A062196[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 21 2025 *)
%o A062196 (Magma)
%o A062196 A062196:= func<n,k | Binomial(n,k)*Binomial(n+2,k) >;
%o A062196 [A062196(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 21 2025
%o A062196 (SageMath)
%o A062196 def A062196(n,k): return binomial(n,k)*binomial(n+2,k)
%o A062196 print(flatten([[A062196(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Feb 21 2025
%Y A062196 Family of polynomials (see A062145): A008459 (c=1), A132813 (c=2), this sequence (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).
%Y A062196 Sums include: A001791 (row), (-1)^n*A089849(n+1) (alternating sign row).
%Y A062196 Diagonals: A000217 (k=n), A002417 (k=n-1), A001297 (k=n-2), A105946 (k=n-3), A105947 (k=n-4), A105948 (k=n-5), A107319 (k=n-6).
%Y A062196 Columns: A005563 (k=1), A033487 (k=2), A027790 (k=3), A107395 (k=4), A107396 (k=5), A107397 (k=6), A107398 (k=7), A107399 (k=8).
%Y A062196 Cf. A000108, A089849.
%K A062196 nonn,easy,tabl
%O A062196 0,3
%A A062196 _Wolfdieter Lang_, Jun 19 2001
%E A062196 New name by _Peter Luschny_, Sep 30 2021