A156224 Triangle T(n, k) = binomial(n, k)*(A000009(n) + A000009(n-k) + A000009(k)) - 2, read by rows.

1, 1, 1, 1, 4, 1, 3, 10, 10, 3, 3, 18, 22, 18, 3, 5, 28, 58, 58, 28, 5, 7, 46, 103, 158, 103, 46, 7, 9, 68, 187, 313, 313, 187, 68, 9, 11, 94, 306, 614, 698, 614, 306, 94, 11, 15, 133, 502, 1174, 1636, 1636, 1174, 502, 133, 15, 19, 188, 763, 2038, 3358, 4030, 3358, 2038, 763, 188, 19

Offset: 0

Author

Roger L. Bagula, Feb 06 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
   1;
   1,   1;
   1,   4,   1;
   3,  10,  10,    3;
   3,  18,  22,   18,    3;
   5,  28,  58,   58,   28,    5;
   7,  46, 103,  158,  103,   46,    7;
   9,  68, 187,  313,  313,  187,   68,    9;
  11,  94, 306,  614,  698,  614,  306,   94,  11;
  15, 133, 502, 1174, 1636, 1636, 1174,  502, 133,  15;
  19, 188, 763, 2038, 3358, 4030, 3358, 2038, 763, 188, 19;
		

Links

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

Crossrefs

Cf. A000009.

Programs

Mathematica

T[n_, k_]:= Binomial[n, k]*(PartitionsQ[n] +PartitionsQ[n-k] +PartitionsQ[k]) -2;
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

Sage

# Uses Peter Luschny's program for A000009
def EulerTransform(a):
    @cached_function
    def b(n):
        if n == 0: return 1
        s = sum(sum(d * a(d) for d in divisors(j)) * b(n-j) for j in (1..n))
        return s//n
    return b
a = BinaryRecurrenceSequence(0, 1)
P = EulerTransform(a)
def T(n,k): return binomial(n,k)*(P(n) + P(n-k) + P(k)) - 2
flatten([[T(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Dec 31 2021

Formula

T(n, k) = binomial(n, k)*(A000009(n) + A000009(n-k) + A000009(k)) - 2.

T(n, n-k) = T(n, k).

Extensions

Edited by G. C. Greubel, Dec 31 2021