A157523 -id:A157523 - cp's OEIS frontend

A157522 Triangle T(n, k) = f(n, k) + f(n, n-k) - 1, where f(n, k) = k if k <= floor(n/4), floor(n/2) - k if floor(n/4) < k <= floor(n/2), k - floor(n/2) if floor(n/2) < k <= floor(3*n/4), otherwise n-k, read by rows.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 3, 1, 1, 3, 2, 2, 3, 1, 1, 3, 3, 1, 3, 3, 1, 1, 3, 4, 2, 2, 4, 3, 1, 1, 3, 5, 3, 1, 3, 5, 3, 1, 1, 3, 5, 4, 2, 2, 4, 5, 3, 1, 1, 3, 5, 5, 3, 1, 3, 5, 5, 3, 1, 1, 3, 5, 6, 4, 2, 2, 4, 6, 5, 3, 1, 1, 3, 5, 7, 5, 3, 1, 3, 5, 7, 5, 3, 1, 1, 3, 5, 7, 6, 4, 2, 2, 4, 6, 7, 5, 3, 1

Offset: 0

Roger L. Bagula, Mar 02 2009

			Triangle begins as:
  1;
  1, 1;
  1, 1, 1;
  1, 2, 2, 1;
  1, 3, 1, 3, 1;
  1, 3, 2, 2, 3, 1;
  1, 3, 3, 1, 3, 3, 1;
  1, 3, 4, 2, 2, 4, 3, 1;
  1, 3, 5, 3, 1, 3, 5, 3, 1;
  1, 3, 5, 4, 2, 2, 4, 5, 3, 1;
  1, 3, 5, 5, 3, 1, 3, 5, 5, 3, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A157523.

f[n_, k_]= 1 +If[k<=Floor[n/4], k, If[Floor[n/4]G. C. Greubel, Jan 22 2022 *)

def f(n, k):
    if (k <= (n//4)): return k+1
    elif ((n//4) < k <= (n//2)): return (n//2)-k+1
    elif ((n//2) < k <= (3*n//4)): return k+1-(n//2)
    else: return n-k+1
def T(n,k): return f(n,k) + f(n,n-k) - 1
flatten([[T(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jan 22 2022

T(n, k) = f(n, k) + f(n, n-k) - 1, where f(n, k) = k if k <= floor(n/4), floor(n/2) - k if floor(n/4) < k <= floor(n/2), k - floor(n/2) if floor(n/2) < k <= floor(3*n/4), otherwise n-k.
From G. C. Greubel, Jan 22 2022: (Start)
T(n, n-k) = T(n, k).
T(2*n, n) = 1.
T(2*n+1, n) = A040000(n).
Sum_{k=0..n} T(n, k) = A302488(n). (End)

Edited by N. J. A. Sloane, Mar 05 2009

A157637 Triangle, T(n, k, m) = 1 if (k=0 or k=n), otherwise (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mA157636(n, k)T(n-2, k-1, m) for m = 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 16, 16, 1, 1, 42, 136, 42, 1, 1, 99, 816, 816, 99, 1, 1, 219, 3951, 10200, 3951, 219, 1, 1, 466, 16632, 94827, 94827, 16632, 466, 1, 1, 968, 63670, 716160, 1601070, 716160, 63670, 968, 1, 1, 1981, 228112, 4657522, 20836740, 20836740, 4657522, 228112, 1981, 1

Offset: 0

Roger L. Bagula, Mar 03 2009

For the case of m = 0 the triangle becomes T(n, k, 0) = A007318(n, k). - G. C. Greubel, Dec 13 2021

			Triangle begins as:
  1;
  1,    1;
  1,    5,      1;
  1,   16,     16,       1;
  1,   42,    136,      42,        1;
  1,   99,    816,     816,       99,        1;
  1,  219,   3951,   10200,     3951,      219,       1;
  1,  466,  16632,   94827,    94827,    16632,     466,      1;
  1,  968,  63670,  716160,  1601070,   716160,   63670,    968,    1;
  1, 1981, 228112, 4657522, 20836740, 20836740, 4657522, 228112, 1981, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318, A157523, A157636.

A157636[n_, k_]:= If[k==0||k==n, 1, n*k*(n-k)/2];
T[n_, k_, m_]:= T[n,k,m]= If[k==0 || k==n, 1, (m*(n-k) +1)*T[n-1,k-1,m] + (m*k + 1)*T[n-1,k,m] + m*A157636[n, k]*T[n-2,k-1,m]];
Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten

@CachedFunction
def A157636(n,k): return 1 if (k==0 or k==n) else n*k*(n-k)/2
def T(n,k,q): return 1 if (k==0 or k==n) else (q*(n-k) +1)*T(n-1, k-1, q) + (q*k + 1)*T(n-1, k, q) + q*A157636(n, k)*T(n-2, k-1, q)
flatten([[T(n,k,1) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Dec 13 2021

T(n, k, m) = 1 if (k=0 or k=n), otherwise (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*A157636(n, k)*T(n-2, k-1, m) for m = 1.

T(n, k) = T(n, n-k). - G. C. Greubel, Dec 13 2021

Edited by G. C. Greubel, Dec 13 2021

cp's OEIS Frontend

A157522 Triangle T(n, k) = f(n, k) + f(n, n-k) - 1, where f(n, k) = k if k <= floor(n/4), floor(n/2) - k if floor(n/4) < k <= floor(n/2), k - floor(n/2) if floor(n/2) < k <= floor(3*n/4), otherwise n-k, read by rows.

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A157637 Triangle, T(n, k, m) = 1 if (k=0 or k=n), otherwise (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mA157636(n, k)T(n-2, k-1, m) for m = 1, read by rows.

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Programs

Mathematica

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Formula

Extensions

cp's OEIS Frontend

A157522 Triangle T(n, k) = f(n, k) + f(n, n-k) - 1, where f(n, k) = k if k <= floor(n/4), floor(n/2) - k if floor(n/4) < k <= floor(n/2), k - floor(n/2) if floor(n/2) < k <= floor(3*n/4), otherwise n-k, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A157637 Triangle, T(n, k, m) = 1 if (k=0 or k=n), otherwise (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*A157636(n, k)*T(n-2, k-1, m) for m = 1, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A157637 Triangle, T(n, k, m) = 1 if (k=0 or k=n), otherwise (m(n-k) + 1)T(n-1, k-1, m) + (mk + 1)T(n-1, k, m) + mA157636(n, k)T(n-2, k-1, m) for m = 1, read by rows.