OFFSET
0
COMMENTS
With the recurrence relation P(n, k) = P(n-1, k-1) + P(n-k, k) from A008284 (partition of n into k parts), this sequence is defined as T(n, k) := P(-n, k) = P(-n-1, k-1) + P(-n-k, k) = T(n+1, k-1) + T(n+k, k). Replacing n with n-k gives: T(n, k) = T(n-k, k) - T(n-k+1, k-1).
The initializing column T(n>0, 0) can be chosen at random. It is conjectured that T(n, 0) = A010815(n-1) minimizes the row-wise sum of absolute values for growing n.
With increasing k, T(n, k) becomes sinusoidal, i.e. T(n, k) can roughly be described as A * n^(k/2) * cos(Pi * (k/2 + sqrt((2/3) * (n - B)))), with A = sqrt(k/(Pi * n)) * 0.758^k/k! and B = (k/Pi - 1)^2 - k/Pi^2. The trigonometric argument is similar to the one in Rademacher's formula for the partition function (A000041).
REFERENCES
H. Rademacher, On the Partition Function p(n). Proceedings of the London Mathematical Society, s2-43 (1938): 241-254.
LINKS
Friedjof Tellkamp, Table of n, a(n) for n = 0..20000
Friedjof Tellkamp, Plots showing the sinusoidal behavior of T(n, k) for some fixed k
H. Rademacher, On the Partition Function p(n), Proceedings of the London Mathematical Society, s2-43 (1938): 241-254.
FORMULA
Row-wise sum = 0 for n>0.
T(n, k>0) = 0 for n <= k*(k-1)/2 = A000217(k-1).
From Friedjof Tellkamp, Feb 11 2025: (Start)
T(n>0, 1) = -A078616(n-1).
G.f.: (-1)^k * x^(1 + k*(k-1)/2) Product_{i>=k+1} (1 - x^i), except for T(0, 0). (End)
EXAMPLE
Upper triangle 1 <= k <= m: number of partitions of m into k positive parts, A008284 (also A072233).
Lower triangle 0 <= k <= n: [1, 1, -1, -1, 0, 1, -1, 1, 0, 0, 0, 1, 0, -1, 0, ...], this sequence.
Column k=0, n >= 1: [1, -1, -1, 0, 0, 1, 0, 1, 0, ...], A010815(n-1).
Each number is the sum of the left below and the k-th below, e.g. 11 = 8 + 3.
m n /k 0 1 2 3 4 5 6 7 8 9 10 11 12 13
13 -13: 0 1 6 14 18 18 14 11 7 5 3 2 1 1
12 -12: 0 1 6 12 15 13 11 7 5 3 2 1 1
11 -11: 0 1 5 10 11 10 7 5 3 2 1 1
10 -10: 0 1 5 8 9 7 5 3 2 1 1
9 -9: 0 1 4 7 6 5 3 2 1 1
8 -8: 0 1 4 5 5 3 2 1 1
7 -7: 0 1 3 4 3 2 1 1
6 -6: 0 1 3 3 2 1 1
5 -5: 0 1 2 2 1 1
4 -4: 0 1 2 1 1
3 -3: 0 1 1 1
2 -2: 0 1 1
1 -1: 0 1
0 0: 1
-1 1: 1 -1
-2 2: -1 0 1
-3 3: -1 1 0 0
-4 4: 0 1 0 -1 0
-5 5: 0 1 -1 0 0 0
-6 6: 1 0 -1 0 0 0 0
-7 7: 0 0 -1 0 1 0 0 0
-8 8: 1 -1 -1 1 0 0 0 0 0
-9 9: 0 -1 0 1 0 0 0 0 0 0
-10 10: 0 -1 0 1 0 0 0 0 0 0 0
-11 11: 0 -1 1 1 0 -1 0 0 0 0 0 0
-12 12: 0 -1 1 1 -1 0 0 0 0 0 0 0 0
-13 13: -1 0 2 0 -1 0 0 0 0 0 0 0 0 0
MATHEMATICA
nmax=20; kmax=Floor[1/2 + Sqrt[2 nmax]]; A010815[n_]:=With[{m = Sqrt[24 n + 1]}, If[IntegerQ[m], KroneckerSymbol[12, m], 0]];
arr=Transpose@Join[{Join[{1}, Table[A010815[n - 1], {n, nmax}]]}, ConstantArray[0, {kmax, nmax + 1}]];
For[ik = 1, ik <= kmax, ik++, For[in = ik, in <= nmax, in++,
arr[[in + 1, ik + 1]]=arr[[in - ik + 1, ik + 1]] - arr[[in - ik + 2, ik]]; ]];
T[n_, k_] := If[k <= kmax, arr[[n + 1, k + 1]], 0];
Flatten@Table[T[n, k], {n, 0, nmax}, {k, 0, n}]
(*alternate program *)
nmax=20; kmax=Floor[1/2 + Sqrt[2 nmax]];
arr=Transpose@Table[CoefficientList[Series[(-1)^k x^(1 + k (k - 1)/2) Product[1 - x^i, {i, k + 1, nmax}], {x, 0, nmax}], x, nmax + 1], {k, 0, kmax}]; arr[[1, 1]]=1;
T[n_, k_] := If[k <= kmax, arr[[n + 1, k + 1]], 0];
Flatten@Table[T[n, k], {n, 0, nmax}, {k, 0, n}]
(* using g.f., Friedjof Tellkamp, Feb 11 2025 *)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Friedjof Tellkamp, Jan 12 2025
STATUS
approved