This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A363733 #24 Mar 23 2025 11:17:38
%S A363733 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,
%T A363733 93,68,30,6,1,0,2,78,246,276,130,42,7,1,0,4,130,768,1028,655,222,56,8,
%U A363733 1,0,3,278,2190,4180,3130,1338,350,72,9,1
%N A363733 Array read by upwards antidiagonals. The family of polynomials generated by the divisibility matrix (A113704) evaluated over the nonnegative integers.
%C A363733 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.
%C A363733 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.
%D A363733 Tom M. Apostol, Introduction to Analytic Number Theory, Springer 1976, p. 14.
%F A363733 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.
%F A363733 The columns are the inverse Möbius transforms of the powers x^n, x >= 0.
%e A363733 Array A(n, k) starts:
%e A363733 [0] 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
%e A363733 [1] 0, 1, 2, 3, 4, 5, 6, 7, 8, ... A001477
%e A363733 [2] 0, 2, 6, 12, 20, 30, 42, 56, 72, ... A002378
%e A363733 [3] 0, 2, 10, 30, 68, 130, 222, 350, 520, ... A034262
%e A363733 [4] 0, 3, 22, 93, 276, 655, 1338, 2457, 4168, ...
%e A363733 [5] 0, 2, 34, 246, 1028, 3130, 7782, 16814, 32776, ... A131471
%e A363733 [6] 0, 4, 78, 768, 4180, 15780, 46914, 118048, 262728, ...
%e A363733 [7] 0, 2, 130, 2190, 16388, 78130, 279942, 823550, 2097160, ... A190578
%e A363733 [8] 0, 4, 278, 6654, 65812, 391280, 1680954, 5767258, 16781384, ...
%e A363733 A000005,A055895,A363913, ... A066108 (diagonal)
%e A363733 .
%e A363733 Triangle T(n, k) starts:
%e A363733 [0] 1;
%e A363733 [1] 0, 1;
%e A363733 [2] 0, 1, 1;
%e A363733 [3] 0, 2, 2, 1;
%e A363733 [4] 0, 2, 6, 3, 1;
%e A363733 [5] 0, 3, 10, 12, 4, 1;
%e A363733 [6] 0, 2, 22, 30, 20, 5, 1;
%e A363733 [7] 0, 4, 34, 93, 68, 30, 6, 1;
%e A363733 [8] 0, 2, 78, 246, 276, 130, 42, 7, 1;
%e A363733 [9] 0, 4, 130, 768, 1028, 655, 222, 56, 8, 1;
%p A363733 divides := (k, n) -> ifelse(k = n or (k > 0 and irem(n, k) = 0), 1, 0):
%p A363733 A := (n, k) -> local j; add(divides(j, n) * k^j, j = 0 ..n):
%p A363733 for n from 0 to 8 do seq(A(n, k), k = 0..8) od;
%p A363733 # If we introduce the 'inverse Möbius transform' InvMoebius acting on s ...
%p A363733 InvMoebius := (s, n) -> local j; add(divides(j, n) * s(j), j = 0 ..n):
%p A363733 # ... the transposed array is given by applying InvMoebius to the powers r^m:
%p A363733 seq(lprint(seq(InvMoebius(m -> r^m, n), n = 0..8)), r = 0..8);
%p A363733 # For instance we see that the number of divisors is the inverse
%p A363733 # Moebius transform of the constant sequence s = 1.
%o A363733 (SageMath)
%o A363733 def A(n, k): return sum(j.divides(n) * k^j for j in (0..n))
%o A363733 for n in srange(9): print([A(n, k) for k in (0..8)])
%Y A363733 Cf. A113704 (in compact form A113705), A000005 (column 1), A055895 (column 2), A363913 (column 3), A001477 (row 1), A002378 (row 2), A034262 (row 3), A131471 (row 5), A190578 (row 7), A363912 (row sums), A066108 (main diagonal of array).
%Y A363733 Cf. A363734, A363735, A363421.
%K A363733 nonn,tabl
%O A363733 0,8
%A A363733 _Peter Luschny_, Jun 27 2023