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A157523 A general recursion triangle: m=1; Bimodal tent function: t(n,m)=1 + If[m <= Floor[n/4], m, If[m > Floor[n/4] && m <= Floor[n/2], Floor[n/2] - m, If[m > Floor[n/2] && m <= Floor[3*n/4], m - Floor[n/2], n - m]]]; f(n,k)=t(n,k)+t(n,n-k)-1; Recursion: A(n,k,m)=(m*(n - k) + 1)*A(n - 1, k - 1, m) + (m*k + 1)*A(n - 1, k, m) + m*f(n, k)* A(n - 2, k - 1, m) 1
1, 1, 1, 1, 5, 1, 1, 15, 15, 1, 1, 37, 95, 37, 1, 1, 82, 463, 463, 82, 1, 1, 173, 1910, 3799, 1910, 173, 1, 1, 356, 7096, 25672, 25672, 7096, 356, 1, 1, 723, 24645, 150994, 260519, 150994, 24645, 723, 1, 1, 1458, 81499, 804875, 2259903, 2259903, 804875, 81499 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The row sums are:

{1, 2, 7, 32, 171, 1092, 7967, 66250, 613245, 6295472, 70670361,...}.

The half tent recursion in this form goes negative which is small and positive, but this bimodal tent that is derivative like in the third term doesn't. I invented the bimodal integer symmetrical tent function to see how the three term recursion reacted to it.

LINKS

Table of n, a(n) for n=0..52.

FORMULA

Bimodal tent function:

t(n,m)=1 + If[m <= Floor[n/4], m, If[m > Floor[n/4] && m <= Floor[n/2], Floor[n/2] - m, If[m > Floor[n/2] && m <= Floor[3*n/4], m - Floor[n/2], n - m]]];

f(n,k)=t(n,k)+t(n,n-k)-1;

Recursion:

A(n,k,m)=(m*(n - k) + 1)*A(n - 1, k - 1, m) + (m*k + 1)*A(n - 1, k, m) + m*f(n, k)* A(n - 2, k - 1, m)

EXAMPLE

{1},

{1, 1},

{1, 5, 1},

{1, 15, 15, 1},

{1, 37, 95, 37, 1},

{1, 82, 463, 463, 82, 1},

{1, 173, 1910, 3799, 1910, 173, 1},

{1, 356, 7096, 25672, 25672, 7096, 356, 1},

{1, 723, 24645, 150994, 260519, 150994, 24645, 723, 1},

{1, 1458, 81499, 804875, 2259903, 2259903, 804875, 81499, 1458, 1},

{1, 2929, 261234, 3994717, 17386622, 27379355, 17386622, 3994717, 261234, 2929, 1}

MATHEMATICA

Clear[A, a0, b0, n, k, m];

t[n_, m_] = 1 + If[m <= Floor[n/4], m, If[m > Floor[n/ 4] && m <= Floor[n/2], Floor[n/2] - m, If[m > Floor[n/2] && m <= Floor[3*n/4], m - Floor[n/2], n - m]]];

f[n_, k_] := t[n, k] + t[n, n - k] - 1;

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*f[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}];

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

CROSSREFS

Sequence in context: A196019 A056940 A168288 * A141691 A157147 A347973

Adjacent sequences:  A157520 A157521 A157522 * A157524 A157525 A157526

KEYWORD

nonn,tabl,uned

AUTHOR

Roger L. Bagula, Mar 02 2009

STATUS

approved

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Last modified December 5 08:36 EST 2021. Contains 349543 sequences. (Running on oeis4.)