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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, 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 (list; table; graph; refs; listen; history; text; internal format)
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: A302996 A266213 A289522 * A124915 A322084 A158239

Adjacent sequences:  A316270 A316271 A316272 * A316274 A316275 A316276

KEYWORD

nonn,tabl

AUTHOR

Mats Granvik, Jun 28 2018

STATUS

approved

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Last modified July 18 07:05 EDT 2019. Contains 325134 sequences. (Running on oeis4.)