login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

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's Kernel (http://en.wikipedia.org/wiki/Dirichlet_kernel); 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 30 15:26 EST 2023. Contains 359945 sequences. (Running on oeis4.)