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A135853 A103516 * A007318 as an infinite lower triangular matrix.

%I A135853 #27 Feb 16 2022 11:04:09
%S A135853 1,4,2,6,6,3,8,12,12,4,10,20,30,20,5,12,30,60,60,30,6,14,42,105,140,
%T A135853 105,42,7,16,56,168,280,280,168,56,8,18,72,252,504,630,504,252,72,9,
%U A135853 20,90,360,840,1260,1260,840,360,90,10
%N A135853 A103516 * A007318 as an infinite lower triangular matrix.
%H A135853 G. C. Greubel, <a href="/A135853/b135853.txt">Table of n, a(n) for the first 50 rows</a>
%F A135853 T(n, k) = (A103516 * A007318)(n, k).
%F A135853 Sum_{k=0..n} T(n, k) = A135854(n).
%F A135853 T(n, k) = (k+1)*binomial(n+1, k+1), with T(n, n) = n+1, T(n, 0) = 2*(n+1). - _G. C. Greubel_, Dec 07 2016
%F A135853 T(n, 0) = A103517(n). - _G. C. Greubel_, Feb 06 2022
%e A135853 First few rows of the triangle are:
%e A135853    1;
%e A135853    4,   2;
%e A135853    6,   6,  3;
%e A135853    8,  12,  12,   4;
%e A135853   10,  20,  30,  20,   5;
%e A135853   12,  30,  60,  60,  30,   6;
%e A135853   14,  42, 105, 140, 105,  42,   7;
%e A135853   ...
%t A135853 T[n_, k_]:= If[k==n, n+1, If[k==0, 2*(n+1), (k+1)*Binomial[n+1, k+1]]];
%t A135853 Table[T[n, k], {n,0,12}, {k,0,n}]//flatten (* _G. C. Greubel_, Dec 07 2016 *)
%o A135853 (Sage)
%o A135853 def A135853(n,k):
%o A135853     if (n==0): return 1
%o A135853     elif (k==0): return 2*(n+1)
%o A135853     else: return (k+1)*binomial(n+1, k+1)
%o A135853 flatten([[A135853(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 06 2022
%Y A135853 Cf. A007318, A103516.
%Y A135853 Cf. A103517 (1st column), A135854 (row sums).
%Y A135853 Cf. A135852 (= A007318 * A103516).
%K A135853 nonn,tabl
%O A135853 0,2
%A A135853 _Gary W. Adamson_, Dec 01 2007