1,2

Row sums of the anti-diagonals are:

{1, 3, 10, 43, 237, 1633, 13848, 143595, 1813225, 27681727}.

Recursion for the first row sum is:A001053;

a[0] = 1; a[1] = 2;

a[n_] := a[n] = ( n + 1)*a[n - 1] + a[n - 2];

Table[a[n], {n, 0, 10}]

None of the others appear to be in OEIS and I haven't found

algorithms for them.

A(n.k)=(m*n - m*k + 1)A(n - 1, k - 1) + (m*k - (m - 1))A(n - 1, k) + j*A(n - 2, k - 1); t(n,m)=Sum[A(n,k),{k,0,n}];{m,1,10} t_out(n,m)=t(n-m+1,m].

{1},

{2, 1},

{7, 2, 1},

{30, 10, 2, 1},

{157, 64, 13, 2, 1},

{972, 532, 110, 16, 2, 1},

{6961, 5448, 1249, 168, 19, 2, 1},

{56660, 66440, 17816, 2416, 238, 22, 2, 1},

{516901, 941056, 306619, 44160, 4141, 320, 25, 2, 1},

{5225670, 15189776, 6185828, 981184, 92292, 6532, 414, 28, 2, 1}

a0 = {{1, 2, 7, 30, 157, 972, 6961, 56660, 516901, 5225670}, {1, 2, 10, 64, 532, 5448, 66440, 941056, 15189776, 275298080}, {1, 2, 13, 110, 1249, 17816, 306619, 6185828, 143193901, 3741598910}, {1, 2, 16, 168, 2416, 44160, 981184, 25687424, 774547456, 26437363200}, {1, 2, 19, 238, 4141, 92292, 2512589, 80864308, 3004542341, 126595099862}, {1, 2, 22, 320, 6532, 171752, 5535256, 211370240, 9333502096, 467943326240}, {1, 2, 25, 414, 9697, 293808, 10938775, 483362756, 24728071981, 1437611714190}, {1, 2, 28, 520, 13744, 471456, 19911104, 999326848, 58120246016, 3843930851840}, {1, 2, 31, 638, 18781, 719420, 33981769, 1909453844, 124420335781, 9224289932390}, {1, 2, 34, 768, 24916, 1054152, 55065064, 3424575488, 247120085776, 20298092788512}}; Table[Table[a0[[m]][[n - m + 1]], {m, 1, n}], {n, 1, Length[a0]}]; Flatten[%]

Cf. A001053.

Sequence in context: A019426 A128747 A124392 this_sequence A121416 A089329 A097411

Adjacent sequences: A144443 A144444 A144445 this_sequence A144447 A144448 A144449

nonn,uned

Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 05 2008