0,5

The row sums are:

{1, 2, 7, 32, 186, 1304, 10720, 101064, 1074832, 12728624, 166105008,...}.

What I have done here is add a new symmetrical part

to the "zero start" Sierpinski -Pascal recursion at "down two" or n-2 in my notation:

m*k*(n - k)*A(n - 2, k - 1, m).

It uses the symmetrical k*(n-k) multiplier.

m=1;

A(n,k,m)= (m*(n - k) + 1)*A(n - 1, k - 1, m) +

(m*k + 1)*A(n - 1, k, m) +

m*k*(n - k)*A(n - 2, k - 1, m).

{1},

{1, 1},

{1, 5, 1},

{1, 15, 15, 1},

{1, 37, 110, 37, 1},

{1, 83, 568, 568, 83, 1},

{1, 177, 2415, 5534, 2415, 177, 1},

{1, 367, 9137, 41027, 41027, 9137, 367, 1},

{1, 749, 32104, 255155, 498814, 255155, 32104, 749, 1},

{1, 1515, 107442, 1409814, 4845540, 4845540, 1409814, 107442, 1515, 1},

{1, 3049, 347945, 7172976, 40220118, 70616830, 40220118, 7172976, 347945, 3049, 1}

Clear[A, n, k, m];

A[n_, 0, m_] := 1;

A[n_, n_, m_] := 1;

A[n_, k_, m_] := (m*(n - k) + 1)*A[n - 1, k - 1, m] + (m* k + 1)*A[n - 1, k, m] + m*k*(n - k)*A[n - 2, k - 1, m];

Table[A[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}];

Table[Flatten[Table[Table[A[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]

Sequence in context: A056940 A157523 A141691 this_sequence A156920 A074060 A157637

Adjacent sequences: A157144 A157145 A157146 this_sequence A157148 A157149 A157150

nonn,tabl,uned

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 24 2009