A177851 - OEIS

%I #16 Oct 14 2024 01:34:42

%S 1,2,2,3,5,7,4,9,16,25,5,14,30,55,91,6,20,50,105,196,336,7,27,77,182,

%T 378,714,1254,8,35,112,294,672,1386,2640,4719,9,44,156,450,1122,2508,

%U 5148,9867,17875,10,54,210,660,1782,4290,9438,19305,37180,68068

%N Triangle read by rows: T(n, m) = binomial(n + m - 3, m - 1)*(2 * m + n - 2) / m, for n>=1 and 1<=m<=n.

%C T(n, m) is the number of linearly independent homogeneous harmonic polynomials of degree m in n variables.

%D Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 170

%F Row sums are ((3*n-1)*binomial(2*n-2,n)/(n-1)-1) for n>=2.

%e Triangle starts:

%e   {1},

%e   {2, 2},

%e   {3, 5, 7},

%e   {4, 9, 16, 25},

%e   {5, 14, 30, 55, 91},

%e   {6, 20, 50, 105, 196, 336},

%e   {7, 27, 77, 182, 378, 714, 1254},

%e   {8, 35, 112, 294, 672, 1386, 2640, 4719},

%e   {9, 44, 156, 450, 1122, 2508, 5148, 9867, 17875},

%e   {10, 54, 210, 660, 1782, 4290, 9438, 19305, 37180, 68068}.

%p T := (n,m) -> ((2*m + n - 2)/m)*binomial(n + m - 3, m - 1):

%p for n from 1 to 10 do lprint(seq(T(n,k), k=1..n)) od; # _Peter Luschny_, Dec 16 2015

%t Flatten[Table[Table[((2*m + n - 2)/m)*Binomial[n + m - 3, m - 1], {m, 1, n}], {n, 1, 10}]]

%K nonn,tabl,easy

%O 1,2

%A _Roger L. Bagula_, May 14 2010