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A156224
Triangle T(n, k) = binomial(n, k)*(A000009(n) + A000009(n-k) + A000009(k)) - 2, read by rows.
2
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
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
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 06 2009
EXTENSIONS
Edited by G. C. Greubel, Dec 31 2021
STATUS
approved