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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
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
KEYWORD
nonn,tabl
AUTHOR
Mats Granvik, Jun 28 2018
STATUS
approved

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Last modified July 15 18:49 EDT 2024. Contains 374333 sequences. (Running on oeis4.)