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A363733
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Array read by upwards antidiagonals. The family of polynomials generated by the Möbius matrix (A113704) evaluated over the nonnegative integers.
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2
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1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 6, 3, 1, 0, 3, 10, 12, 4, 1, 0, 2, 22, 30, 20, 5, 1, 0, 4, 34, 93, 68, 30, 6, 1, 0, 2, 78, 246, 276, 130, 42, 7, 1, 0, 4, 130, 768, 1028, 655, 222, 56, 8, 1, 0, 3, 278, 2190, 4180, 3130, 1338, 350, 72, 9, 1
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OFFSET
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0,8
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COMMENTS
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The name expresses the 'row view' of the array. The 'column view' regards the array as the collection of the inverse Möbius transforms of the power sequences k^n = 0^n, 1^n, 2^n, .... (n >= 0). Viewed this way, the array is a generalization of the number of divisors sequence tau (A000005), to which it reduces in the case k = 1.
The array has offset (0, 0). It uses the usual definition of 'k divides n' as described in Apostol, rather than the shortened version, which restricts to values k > 0 as some programs do (but not SageMath). Such a restriction makes sense in the context of rational numbers but not in the case of natural numbers.
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REFERENCES
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Tom M. Apostol, Introduction to Analytic Number Theory, Springer 1976, p. 14.
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LINKS
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FORMULA
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A(n, k) = Sum_{j=0..n} divides(j, n) * k^j, where divides(k, n) <-> [k = n or (k > 0 and n mod k = 0)], and '[ ]' denotes the Iverson bracket.
The columns are the inverse Möbius transforms of the powers x^n, x >= 0.
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EXAMPLE
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Array A(n, k) starts:
[0] 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
[1] 0, 1, 2, 3, 4, 5, 6, 7, 8, ... A001477
[2] 0, 2, 6, 12, 20, 30, 42, 56, 72, ... A002378
[3] 0, 2, 10, 30, 68, 130, 222, 350, 520, ... A034262
[4] 0, 3, 22, 93, 276, 655, 1338, 2457, 4168, ...
[5] 0, 2, 34, 246, 1028, 3130, 7782, 16814, 32776, ...
[6] 0, 4, 78, 768, 4180, 15780, 46914, 118048, 262728, ...
[7] 0, 2, 130, 2190, 16388, 78130, 279942, 823550, 2097160, ...
[8] 0, 4, 278, 6654, 65812, 391280, 1680954, 5767258, 16781384, ...
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Triangle T(n, k) starts:
[0] 1;
[1] 0, 1;
[2] 0, 1, 1;
[3] 0, 2, 2, 1;
[4] 0, 2, 6, 3, 1;
[5] 0, 3, 10, 12, 4, 1;
[6] 0, 2, 22, 30, 20, 5, 1;
[7] 0, 4, 34, 93, 68, 30, 6, 1;
[8] 0, 2, 78, 246, 276, 130, 42, 7, 1;
[9] 0, 4, 130, 768, 1028, 655, 222, 56, 8, 1;
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MAPLE
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divides := (k, n) -> ifelse(k = n or (k > 0 and irem(n, k) = 0), 1, 0):
A := (n, k) -> local j; add(divides(j, n) * k^j, j = 0 ..n):
for n from 0 to 8 do seq(A(n, k), k = 0..8) od;
# If we introduce the 'inverse Möbius transform' InvMoebius acting on s ...
InvMoebius := (s, n) -> local j; add(divides(j, n) * s(j), j = 0 ..n):
# ... the transposed array is given by applying InvMoebius to the powers r^m:
seq(lprint(seq(InvMoebius(m -> r^m, n), n = 0..8)), r = 0..8);
# For instance we see that the number of divisors is the inverse
# Moebius transform of the constant sequence s = 1.
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PROG
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(SageMath)
def A(n, k): return sum(j.divides(n) * k^j for j in (0..n))
for n in srange(9): print([A(n, k) for k in (0..8)])
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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