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A323345 Square array read by ascending antidiagonals: T(n, k) is the number of partitions of n where parts, if sorted in ascending order, form an arithmetic progression (AP) with common difference of k; n >= 1, k >= 0.

%I A323345 #23 Oct 21 2024 06:33:40
%S A323345 1,2,1,2,1,1,3,2,1,1,2,1,1,1,1,4,2,2,1,1,1,2,2,1,1,1,1,1,4,2,2,2,1,1,
%T A323345 1,1,3,1,1,1,1,1,1,1,1,4,3,2,2,2,1,1,1,1,1,2,2,2,1,1,1,1,1,1,1,1,6,2,
%U A323345 2,2,2,2,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,1,1,1,1
%N A323345 Square array read by ascending antidiagonals: T(n, k) is the number of partitions of n where parts, if sorted in ascending order, form an arithmetic progression (AP) with common difference of k; n >= 1, k >= 0.
%C A323345 T(n, k) is the number of positive integers in the sequence defined, for all i >= 1, by x_1 = n and x_i = (i-1)*(x_(i-1)-k)/i; or defined equivalently by x_i=n/i-(k/2)*(i-1). An x_i positive and integer characterizes the AP-partition with smallest part x_i and number of parts i.
%C A323345 T(n, k) is the number of i, positive integers, such that n - P(k+2,i) is both nonnegative and divisible by i, where P(r,i) denotes the i-th r-gonal number (see A057145).
%F A323345 T(n, 0) = A000005(n), the number of divisors of n.
%F A323345 T(n, 1) = A001227(n), the number of odd divisors of n.
%F A323345 T(n, 2) = A038548(n), the number of divisors of n that are at most sqrt(n).
%F A323345 T(n, 3) = A117277(n).
%F A323345 The g.f. for column d is Sum_{k>=1} x^(k*(d*k-d+2)/2)/(1-x^k) [information taken from A117277]. - _Joerg Arndt_, May 05 2020
%e A323345 There are 4 partitions of 150 such that the parts form an arithmetic progression with common difference of 9:
%e A323345 150 = 150
%e A323345 150 = 41 + 50 + 59
%e A323345 150 = 24 + 33 + 42 + 51
%e A323345 150 = 12 + 21 + 30 + 39 + 48
%e A323345 Then, T(150,9) = 4.
%e A323345 Array begins:
%e A323345      k 0 1 2 3 4 5 6 7 8 9
%e A323345    n +--------------------
%e A323345    1 | 1 1 1 1 1 1 1 1 1 1
%e A323345    2 | 2 1 1 1 1 1 1 1 1 1
%e A323345    3 | 2 2 1 1 1 1 1 1 1 1
%e A323345    4 | 3 1 2 1 1 1 1 1 1 1
%e A323345    5 | 2 2 1 2 1 1 1 1 1 1
%e A323345    6 | 4 2 2 1 2 1 1 1 1 1
%e A323345    7 | 2 2 1 2 1 2 1 1 1 1
%e A323345    8 | 4 1 2 1 2 1 2 1 1 1
%e A323345    9 | 3 3 2 2 1 2 1 2 1 1
%e A323345   10 | 4 2 2 1 2 1 2 1 2 1
%t A323345 T[n_, k_] :=
%t A323345 Module[{c = 0, i = 1, x = n},
%t A323345   While[x >= 1, If[IntegerQ[x], c++]; i++; x = (i-1)*(x-k)/i]; c]
%t A323345 A004736[n_] := Binomial[Floor[3/2 + Sqrt[2*n]], 2] - n + 1
%t A323345 A002260[n_] := n - Binomial[Floor[1/2 + Sqrt[2*n]], 2]
%t A323345 a[n_] := T[A004736[n], A002260[n] - 1]
%t A323345 Table[a[n], {n, 1, 91}]
%t A323345 (* Second program: *)
%t A323345 nmax = 14;
%t A323345 col[k_] := col[k] = CoefficientList[Sum[x^(n(k n - k + 2)/2 - 1)/(1 - x^n), {n, 1, nmax}] + O[x]^nmax, x];
%t A323345 T[n_, k_] := col[k][[n]];
%t A323345 Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Nov 30 2020 *)
%o A323345 (PARI)
%o A323345 T(n,k)=c=0;i=1;x=n;while(x>=1,if(frac(x)==0,c++);i++;x=n/i-(k/2)*(i-1));c
%o A323345 for(s=1,13,for(k=0,s-1,n=s-k;print1(T(n,k),", ")))
%Y A323345 Cf. A000005, A001227, A038548, A117277, A334461, A334541, A334948.
%Y A323345 Cf. A057145.
%K A323345 nonn,tabl
%O A323345 1,2
%A A323345 _Luc Rousseau_, Jan 11 2019