A201552 Square array read by diagonals: T(n,k) = number of arrays of n integers in -k..k with sum equal to 0.
1, 1, 3, 1, 5, 7, 1, 7, 19, 19, 1, 9, 37, 85, 51, 1, 11, 61, 231, 381, 141, 1, 13, 91, 489, 1451, 1751, 393, 1, 15, 127, 891, 3951, 9331, 8135, 1107, 1, 17, 169, 1469, 8801, 32661, 60691, 38165, 3139, 1, 19, 217, 2255, 17151, 88913, 273127, 398567, 180325, 8953, 1
Offset: 1
Examples
Some solutions for n=7, k=3: ..1...-2....1...-1....1...-3....0....0....1....2....3...-3....0....2....1....0 .-1....2...-2....2....2....2...-1....0....2....2...-2...-1...-2...-1....2...-1 .-3...-1....1...-3....2....1....0....1....3....0....2....0...-1....2...-2...-1 ..0....3....3....3...-2...-2....3....3...-3...-3....0...-1...-1...-1....0....3 ..2...-1...-1...-1...-3....0...-3...-2....1...-1...-1....1....1....0....3...-1 ..2...-1...-3....0....2....3....0....1...-2....1....1....1....3...-2...-3...-3 .-1....0....1....0...-2...-1....1...-3...-2...-1...-3....3....0....0...-1....3 Table starts: . 1, 1, 1, 1, 1, 1,... . 3, 5, 7, 9, 11, 13,... . 7, 19, 37, 61, 91, 127,... . 19, 85, 231, 489, 891, 1469,... . 51, 381, 1451, 3951, 8801, 17151,... . 141, 1751, 9331, 32661, 88913, 204763,... . 393, 8135, 60691, 273127, 908755, 2473325,... .1107, 38165, 398567, 2306025, 9377467, 30162301,... .3139, 180325, 2636263, 19610233, 97464799, 370487485,...
Links
- R. H. Hardin, Table of n, a(n) for n = 1..9999
- Michelle Rudolph-Lilith and Lyle E. Muller, On a link between Dirichlet kernels and central multinomial coefficients, Discrete Mathematics 338.9 (2015): 1567-1572.
- Wikipedia, Dirichlet kernel.
Crossrefs
Programs
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Maple
seq(print(seq(add((-1)^i*binomial(n, i)*binomial((k+1)*n-(2*k+1)*i-1, n-1), i = 0..floor((1/2)*n)), k = 1..10)), n = 1..10); # Peter Bala, Oct 16 2024
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Mathematica
comps[r_, m_, k_] := Sum[(-1)^i*Binomial[r - 1 - i*m, k - 1]*Binomial[k, i], {i, 0, Floor[(r - k)/m]}]; T[n_, k_] := comps[n*(k + 1), 2*k + 1, n]; Table[T[n - k + 1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 31 2017, after Andrew Howroyd *) -
PARI
comps(r, m, k)=sum(i=0, floor((r-k)/m), (-1)^i*binomial(r-1-i*m, k-1)*binomial(k, i)); T(n,k) = comps(n*(k+1), 2*k+1, n); \\ Andrew Howroyd, Oct 14 2017
Formula
Empirical: T(n,k) = Sum_{i=0..floor(k*n/(2*k+1))} (-1)^i*binomial(n,i)* binomial((k+1)*n-(2*k+1)*i-1,n-1).
The above empirical formula is true and can be derived from the formula for the number of compositions with given number of parts and maximum part size. - Andrew Howroyd, Oct 14 2017
Empirical for rows:
T(1,k) = 1
T(2,k) = 2*k + 1
T(3,k) = 3*k^2 + 3*k + 1
T(4,k) = (16/3)*k^3 + 8*k^2 + (14/3)*k + 1
T(5,k) = (115/12)*k^4 + (115/6)*k^3 + (185/12)*k^2 + (35/6)*k + 1
T(6,k) = (88/5)*k^5 + 44*k^4 + 46*k^3 + 25*k^2 + (37/5)*k + 1
T(7,k) = (5887/180)*k^6 + (5887/60)*k^5 + (2275/18)*k^4 + (357/4)*k^3 + (6643/180)*k^2 + (259/30)*k + 1
T(m,k) = (1/Pi)*integral_{x=0..Pi} (sin((k+1/2)x)/sin(x/2))^m dx; for the proof see Dirichlet Kernel link; so f(m,n) = (1/Pi)*integral_{x=0..Pi} (Sum_{k=-n..n} exp(I*k*x))^m dx = sum(integral(exp(I(k_1+...+k_m).x),x=0..Pi)/Pi,{k_1,...,k_m=-n..n}) = sum(delta_0(k1+...+k_m),{k_1,...,k_m=-n..n}) = number of arrays of m integers in -n..n with sum zero. - Yalcin Aktar, Dec 03 2011
T(n, k) = the constant term in the expansion of (x^(-k) + ... + x^(-1) + 1 + x + ... + x^k)^n = the coefficient of x^(k*n) (i.e., the central coefficient) in the expansion of (1 + x + ... + x^(2*k))^n = the coefficient of x^(k*n) in the expansion of ( (1 - x^(2*k+1))/(1 - x) )^n. Expanding the binomials and collecting terms gives the empirical formula above. - Peter Bala, Oct 16 2024
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