A135852 - OEIS

%I #26 Feb 16 2022 11:03:33

%S 1,3,2,8,4,3,20,6,9,4,48,8,18,16,5,112,10,30,40,25,6,256,12,45,80,75,

%T 36,7,576,14,63,140,175,126,49,8,1280,16,84,224,350,336,196,64,9,2816,

%U 18,108,336,630,756,588,288,81,10

%N A007318 * A103516 as a lower triangular matrix.

%C Binomial transform of triangle A103516.

%H G. C. Greubel, <a href="/A135852/b135852.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = (A007318 * A103516)(n, k).

%F T(n, 0) = A001792(n).

%F Sum_{k=0..n} T(n, k) = A099035(n+1).

%F T(n, k) = (k+1)*binomial(n, k), with T(n, 0) = (n+2)*2^(n-1), T(n, n) = n+1. - _G. C. Greubel_, Dec 07 2016

%e First few rows of the triangle are:

%e     1;

%e     3,  2;

%e     8,  4,  3;

%e    20,  6,  9,  4;

%e    48,  8, 18, 16,  5;

%e   112, 10, 30, 40, 25,  6;

%e   256, 12, 45, 80, 75, 36,  7;

%e   ...

%t T[n_, k_]:= If[n==0, 1, If[k==0, (n+2)*2^(n-1), (k+1)*Binomial[n, k]]];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 07 2016 *)

%o (Sage)

%o def A135852(n,k):

%o     if (n==0): return 1

%o     elif (k==0): return (n+2)*2^(n-1)

%o     else: return (k+1)*binomial(n, k)

%o flatten([[A135852(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 07 2022

%Y Cf. A007318, A103516.

%Y Cf. A001792 (1st column), A099035 (row sums).

%Y Cf. A135853 (= A103516 * A007318).

%K nonn,tabl

%O 0,2

%A _Gary W. Adamson_, Dec 01 2007

%E Offset changed to 0 by _G. C. Greubel_, Feb 07 2022