A316273 Triangle T(n,k), 1 <= k <= n, read by rows: T(n,1) = A239605(n); T(n,k) = Sum_{i=1..k-1} (T(n-i,k-1) - T(n-i,k)).

1, 1, 1, 2, 0, 1, 4, 2, 0, 1, 10, 2, 1, 0, 1, 24, 8, 3, 1, 0, 1, 66, 16, 6, 2, 1, 0, 1, 178, 50, 15, 7, 2, 1, 0, 1, 508, 128, 45, 14, 6, 2, 1, 0, 1, 1464, 380, 118, 43, 15, 6, 2, 1, 0, 1, 4320, 1084, 345, 114, 42, 14, 6, 2, 1, 0, 1

Offset

1,4

Links

Table of n, a(n) for n=1..66.

Formula

T(n,k) = A239605(n) if k=1, Sum_{i=1..k-1} (T(n-i,k-1) - T(n-i,k)) if 2 <= k <= n, 0 otherwise.

Let M(n,k) = T(n-1,k); then T(n,k) = Sum_{m>=0} M^m.

Example

Triangle starts:
{
  {   1},
  {   1,    1},
  {   2,    0,   1},
  {   4,    2,   0,   1},
  {  10,    2,   1,   0,  1},
  {  24,    8,   3,   1,  0,  1},
  {  66,   16,   6,   2,  1,  0, 1},
  { 178,   50,  15,   7,  2,  1, 0, 1},
  { 508,  128,  45,  14,  6,  2, 1, 0, 1},
  {1464,  380, 118,  43, 15,  6, 2, 1, 0, 1},
  {4320, 1084, 345, 114, 42, 14, 6, 2, 1, 0, 1}
}

Mathematica

Clear[t, n, k, i, nn, x];
coeff = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
   1, 1, 1, 1, 1, 1};
mp[m_, e_] :=
If[e == 0, IdentityMatrix@Length@m, MatrixPower[m, e]]; nn =
Length[coeff]; cc = Range[nn]*0 + 1; Monitor[
Do[Clear[t]; t[n_, 1] := t[n, 1] = cc[[n]];
  t[n_, k_] :=
   t[n, k] =
    If[n >= k,
     Sum[t[n - i, k - 1], {i, 1, k - 1}] -
      Sum[t[n - i, k], {i, 1, k - 1}], 0];
  A4 = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}];
  A5 = A4[[1 ;; nn - 1]]; A5 = Prepend[A5, ConstantArray[0, nn]];
  cc = Total[
    Table[coeff[[n]]*mp[A5, n - 1][[All, 1]], {n, 1, nn}]]; , {i, 1,
   nn}], i];
cc;
Flatten[Table[Table[A4[[n, k]], {k, 1, n}], {n, 1, 11}]]

Crossrefs

Cf. A239605.

Sequence in context: A266213 A289522 A361397 * A124915 A322084 A158239

Adjacent sequences: A316270 A316271 A316272 * A316274 A316275 A316276

Keyword

nonn,tabl

Author

Mats Granvik, Jun 28 2018

Status

approved