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 A201552 Square array read by diagonals: T(n,k) = number of arrays of n integers in -k..k with sum equal to 0. 17
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Equivalently, the number of compositions of n*(k + 1) into n parts with maximum part size 2*k+1. - Andrew Howroyd, Oct 14 2017 LINKS R. H. Hardin, Table of n, a(n) for n = 1..9999 Wikipedia, Dirichlet kernel. 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 EXAMPLE 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,... 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 *) PROG (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 CROSSREFS Columns 1-10: A002426, A005191, A025012, A025014, A201549, A201550, A201551, A322538, A322539, A322540. Rows 3-10: A003215, A063496(n+1), A083669, A201553, A201554, A322535, A322536, A322537. Cf. A286928. Sequence in context: A199898 A320904 A193844 * A216182 A143524 A134249 Adjacent sequences: A201549 A201550 A201551 * A201553 A201554 A201555 KEYWORD nonn,tabl AUTHOR R. H. Hardin, Dec 02 2011 STATUS approved

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Last modified April 22 14:29 EDT 2024. Contains 371904 sequences. (Running on oeis4.)