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!)
A338796 Triangle T read by rows: T(n, k) is the k-th row sum of the symmetric Toeplitz matrix M(n) whose first row consists of a single zero followed by successive positive integers repeated (A004526). 1
0, 1, 1, 2, 2, 2, 4, 3, 3, 4, 6, 5, 4, 5, 6, 9, 7, 6, 6, 7, 9, 12, 10, 8, 8, 8, 10, 12, 16, 13, 11, 10, 10, 11, 13, 16, 20, 17, 14, 13, 12, 13, 14, 17, 20, 25, 21, 18, 16, 15, 15, 16, 18, 21, 25, 30, 26, 22, 20, 18, 18, 18, 20, 22, 26, 30, 36, 31, 27, 24, 22, 21, 21, 22, 24, 27, 31, 36 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
FORMULA
O.g.f.: (2*x^3*y^2 + y^2*(1 + y) + x^2*(y - 3*y^2) - x*(-1 + 2*y + y^2))/((1 - x)^3*(1 + x) *(1 - y)^3*(1 + y)).
E.g.f.: exp(-x-y)*(exp(2*x) + 2*exp(2*(x+y))*(x*(3 + x) - 2*x*y + 2*y^2 - 1))/8.
T(n, k) = (-1)^k + (-1)^(n-k+1) + 4*k^2 + 4*n + 2*n^2 - 4*k*(n + 1))/8.
T(n, 1) = T(n, n) = A002620(n).
T(n, 2) = A033638(n-1).
T(n, 3) = A290743(n-2).
Sum_{k=1..n} T(n, k) = A212964(n+1).
EXAMPLE
n\k| 1 2 3 4 5 6
---+------------
1 | 0
2 | 1 1
3 | 2 2 2
4 | 4 3 3 4
5 | 6 5 4 5 6
6 | 9 7 6 6 7 9
...
For n = 4 the matrix M(4) is
0 1 1 2
1 0 1 1
1 1 0 1
2 1 1 0
and therefore T(4, 1) = 4, T(4, 2) = 3, T(4, 3) = 3 and T(4, 4) = 4.
MATHEMATICA
T[n_, k_]:=((-1)^k+(-1)^(n-k+1)+4k^2+4n+2n^2-4k(n+1))/8; Flatten[Table[T[n, k], {n, 12}, {k, n}]] (* or *)
r[n_]:=Table[SeriesCoefficient[(2x^3y^2+y^2(1+y)+x^2(y-3y^2)-x(-1+2y+y^2))/((1-x)^3(1+x)(1-y)^3(1+y)), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, n}]; Flatten[Array[r, 12]] (* or *)
r[n_]:=Table[SeriesCoefficient[1/8 E^(-x-y)(-1+E^(2 x)+2 E^(2 (x+y))(x (3+x)-2 x y+2 y^2)), {x, 0, i}, {y, 0, j}]i!j!, {i, n, n}, {j, n}]; Flatten[Array[r, 12]]
PROG
(PARI) tm(n) = {my(m = matrix(n, n, i, j, if (i==1, j\2, if (j==1, i\2)))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
T(n, k) = my(m = tm(n)); sum(i=1, n, m[i, k]);
matrix(10, 10, n, k, if (n>=k, T(n, k), 0)) \\ Michel Marcus, Nov 12 2020
CROSSREFS
Cf. A004526.
Cf. A002378 (conjectured determinant of M(2n+1)), A083392 (conjectured determinant of M(n+1)), A332566 (permanent of M(n)), A333119 (k-th super- and subdiagonal sums of the matrix M(n)), A338456 (hafnian of M(n)).
Sequence in context: A335607 A366192 A347628 * A085454 A083403 A114091
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, Nov 12 2020
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 March 29 09:28 EDT 2024. Contains 371268 sequences. (Running on oeis4.)