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

0,5

Links

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

Formula

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

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

Example

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;

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

Prog

(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

Crossrefs

Cf. A000009.

Sequence in context: A321121 A093735 A298918 * A162516 A336693 A193793

Adjacent sequences: A156221 A156222 A156223 * A156225 A156226 A156227

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 06 2009

Extensions

Edited by G. C. Greubel, Dec 31 2021

Status

approved