A317614 -id:A317614 - cp's OEIS frontend

A374668 a(n) is the permanent of the n-th order Hankel matrix M whose generic element is given by M(i,j) = A317614(i+j-1) with i,j = 1, ..., n.

1, 1, 31, 10254, 12238276, 41596930860, 309346186680924, 4522151204857137264, 116038936382978521700928, 4918677318766771695942334272, 323424014903141386787887115413440, 31725978444319999354629697685162941056, 4460612938377274751881312432310360618154240

Offset: 0

Stefano Spezia, Jul 15 2024

nonn

The Hankel transform of A317614 has the following polynomial as g.f. 1 + x - x^2 - 624*x^3 - 9216*x^4 + 138240*x^5 - 331776*x^6: the matrices are singular for n > 6.

			a(7) = 4522151204857137264:
  [  1,   4,  15,  32,  65, 108,  175]
  [  4,  15,  32,  65, 108, 175,  256]
  [ 15,  32,  65, 108, 175, 256,  369]
  [ 32,  65, 108, 175, 256, 369,  500]
  [ 65, 108, 175, 256, 369, 500,  671]
  [108, 175, 256, 369, 500, 671,  864]
  [175, 256, 369, 500, 671, 864, 1105]
which is the singular matrix M of minimal order.

N. J. A. Sloane, Transforms.

Cf. A317614.

Cf. A000583 (trace of M), A006010 (sum of the 1st row or column of M), A035287 (super- or subdiagonal sum of M), A346174, A374708 (k-th super- or subdiagonal sum of M).

A317614[n_]:=(1/2)*(n^3 + n*Mod[n,2]); a[n_]:=Permanent[Table[A317614[i+j-1], {i, n}, {j, n}]]; Join[{1}, Array[a, 12]]

A346174 Inverse binomial transform of A317614.

Original entry on oeis.org

0, 1, 6, 30, 120, 420, 1344, 4032, 11520, 31680, 84480, 219648, 559104, 1397760, 3440640, 8355840, 20054016, 47628288, 112066560, 261488640, 605552640, 1392771072, 3183476736, 7235174400, 16357785600, 36805017600, 82443239424, 183911841792, 408692981760, 904963031040

Offset: 0

Stefano Spezia, Jul 08 2021

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Binomial Transform
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).

Cf. A000079, A007531, A128789, A257872 (-8*log(2)), A317614.

LinearRecurrence[{8,-24,32,-16},{0,1,6,30,120,420},30]

O.g.f.: x*(1 - 2*x + 6*x^2 - 8*x^3 + 4*x^4)/(1 - 2*x)^4.
E.g.f.: x*(1 + exp(2*x)*(3 + 6*x + 2*x^2))/4.
a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4) for n > 5.
a(n) = 2^(n-4)*n*(n + 1)*(n + 2) with a(0) = 0 and a(1) = 1.
a(n) = A000079(n-4)*A007531(n+2) for n > 1.
a(n) ~ A128789(n)/16.
Sum_{n>0} 1/a(n) = 8*log(2) - 13/3 = 1.21184411114622914200452363833...

A381374 Little Hankel transform of A317614: a(n) = A317614(n+1)^2 - A317614(n)*A317614(n+2).

Original entry on oeis.org

1, 1, 97, 49, 769, 289, 2977, 961, 8161, 2401, 18241, 5041, 35617, 9409, 63169, 16129, 104257, 25921, 162721, 39601, 242881, 58081, 349537, 82369, 487969, 113569, 663937, 152881, 883681, 201601, 1153921, 261121, 1481857, 332929, 1875169, 418609, 2342017, 519841, 2891041

Offset: 1

Stefano Spezia, Feb 21 2025

Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A317614.

Cf. A056221, A056222, A239607, A374668.

LinearRecurrence[{0,5,0,-10,0,10,0,-5,0,1},{1,97,49,769,289,2977,961,8161,2401,18241},38]

a(n) = (10 + 6*(-1)^n + 4*n*(n + 2)*(3*(n + 1)^2 + (-1)^n*(2*n^2 + 4*n + 5)))/16.
a(n) = 5*a(n-2) - 10*a(n-4) + 10*a(n-6) - 5*a(n-8) + a(n-10) for n > 10.
G.f.: (1 + x + 92*x^2 + 44*x^3 + 294*x^4 + 54*x^5 + 92*x^6 - 4*x^7 + x^8 + x^9)/(1 - x^2)^5.
E.g.f.: ((4 + 3*x + 123*x^2 + 10*x^3 + 5*x^4)*cosh(x) + (1 + 69*x + 21*x^2 + 50*x^3 + x^4)*sinh(x))/4.
a(2*n) = A239607(n).

A322277 Permanent of an n X n square matrix M(n) formed by writing the numbers 1, ..., n^2 successively forward and backward along the rows in zig-zag pattern.

Original entry on oeis.org

1, 11, 490, 60916, 15745548, 7477647372, 5799397213200, 6925325038489152, 11958227405868674880, 28853103567727115409600, 93561657023119005869616000, 398720531811315564754326938880, 2174628314166392755825875267321600, 14941853448103858870808931238617312000

Offset: 1

Stefano Spezia, Dec 01 2018

nonn

M(n) is defined as M[i,j,n] = j + n*(i-1) if i is odd and M[i,j,n] = n*i - j + 1 if i is even.
det(M(1)) = 1, det(M(2)) = -5 and det(M(n)) = 0 for n > 2 (proved).
The trace of the matrix M(n) is A317614(n).

			For n = 1 the matrix M(1) is
  1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
  1, 2
  4, 3
with permanent a(2) = 11.
For n = 3 the matrix M(3) is
  1, 2, 3
  6, 5, 4
  7, 8, 9
with permanent a(3) = 490.

Vaclav Kotesovec, Table of n, a(n) for n = 1..35

Cf. A317614 (trace of matrix M(n)).

Cf. A241016 (row sums of M matrices), A317617 (column sums of M matrices), A074147 (antidiagonals of M matrices).

with(LinearAlgebra):
a := n -> Permanent(Matrix(n, (i, j) -> 1-j+i*n+(-1+2*j-n)*modp(i,2))):
seq(a(n), n = 1 .. 20);

M[i_, j_, n_] := 1 - j + i n + (-1 + 2 j - n) Mod[i, 2]; a[n_] := Permanent[Table[M[i, j, n], {i, n}, {j, n}]]; Array[a, 20]

a(n) = matpermanent(matrix(n, n, i, j, if (i % 2, j + n*(i-1), n*i - j + 1)));
vector(20, n, a(n))

A317617 Triangle T read by rows: T(n, k) = (n^3 + n)/2 + (k - (n + 1)/2)*(n mod 2).

Original entry on oeis.org

1, 5, 5, 14, 15, 16, 34, 34, 34, 34, 63, 64, 65, 66, 67, 111, 111, 111, 111, 111, 111, 172, 173, 174, 175, 176, 177, 178, 260, 260, 260, 260, 260, 260, 260, 260, 365, 366, 367, 368, 369, 370, 371, 372, 373, 505, 505, 505, 505, 505, 505, 505, 505, 505, 505, 666

Offset: 1

Stefano Spezia, Aug 01 2018

T(n, k) is the sum of the terms of the k-th column of an n X n square matrix M formed by writing the numbers 1, ..., n^2 successively forward and backward along the rows in zig-zag pattern (proved). The n X n square matrix M is defined as M[i, j, n] = j + n*(i - 1) if i is odd and M[i, j, n] = n*i - j + 1 if i is even (see the examples below).

The rows of even indices of the triangle T are made of all the same repeating number.

			n\k|   1   2   3   4   5   6
---+------------------------
1  |   1
2  |   5   5
3  |  14  15  16
4  |  34  34  34  34
5  |  63  64  65  66  67
6  | 111 111 111 111 111 111
...
For n = 1 the matrix M is
  1
with column sum 1.
For n = 2 the matrix M is
  1, 2
  4, 3
with column sums 5, 5.
For n = 3 the matrix M is
  1, 2, 3
  6, 5, 4
  7, 8, 9
with column sums 14, 15, 16.

Stefano Spezia, First 150 rows of the triangle, flattened

Cf. A006003, A000027, A000035, A037270 (row sums).
A317614(n): the trace of the n X n square matrix M.
A074147(n): the elements of the antidiagonal of the n X n square matrix M.
A241016(n): the triangle of the row sums of the n X n square matrix M.
A246697(n): the right diagonal of the triangle T.

A317617 := function(n)
local i, j, t;
for i in [1 .. n] do
   for j in [1 .. i] do
      t := (i^3 + i)/2 + (j - (i + 1)/2)*(i mod 2);
      Print(t, "\t");
   od;
   Print("\n");
od;
end;
A317617(11); # yields sequence in triangular form

Flat(List([1..11],n->List([1..n],k->(n^3+n)/2+(k-(n+1)/2)*(n mod 2)))); # Muniru A Asiru, Aug 24 2018

[[(n^3 + n)/2 + (k - (n + 1)/2)*(n mod 2): k in [1..n]]: n in [1..11]];

a:=(n,k)->(n^3+n)/2+(k-(n+1)/2)*modp(n,2): seq(seq(a(n,k),k=1..n),n=1..11); # Muniru A Asiru, Aug 24 2018

f[n_] := Table[SeriesCoefficient[(x*(x*(5 - 7*y) + x^4*(1 - 2*y) - x^3*(-3 + y) - 3*x^2*(-1 + y) + y))/((-1 + x)^4*(1 + x)^2*(-1 + y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 1, n}]; Flatten[Array[f, 11]]
T[i_, j_, n_] := If[OddQ@ i, j + n*(i - 1), n*i - j + 1]; f[n_] := Plus @@@ Transpose[ Table[T[i, j, n], {i, n}, {j, n}]]; Array[f, 11] // Flatten  (* Robert G. Wilson v, Aug 01 2018 *)
f[n_] := Table[SeriesCoefficient[1/4 E^(-x + y) (1 - x - 2 y + E^(2 x) (-1 + 3 x + 6 x^2 + 2 x^3 + 2 y)), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 1, n}]; Flatten[Array[f, 11]] (* Stefano Spezia, Jan 10 2019 *)

sjoin(v, j) := apply(sconcat, rest(join(makelist(j, length(v)), v)))$ display_triangle(n) := for i from 1 thru n do disp(sjoin(makelist((i^3+i)/2+(j-(i+1)/2)*mod(i, 2), j, 1, i), " ")); display_triangle(10);

M(i,j,n) = if (i % 2, j + n*(i-1), n*i - j + 1);
T(n, k) = sum(i=1, n, M(i,k,n));
tabl(nn) = for(n=1, nn, for(k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Aug 09 2018

# by formula
for (n in 1:11){
   t <- c(n, "")
   for(j in 1:n){
      t <- c(t, (n^3+n)/2+(j-(n+1)/2)*(n%%2), "")
   }
   cat(t, "\n")
} # yields sequence in triangular form
(MATLAB and FreeMat)
for(i=1:11);
   for(j=1:i);
      t=(i^3 + i)/2 + (j - (i + 1)/2)*mod(i,2);
      fprintf('%0.f\t', t);
   end
   fprintf('\n');
end % yields sequence in triangular form

T(n, k) = A006003(n) + (k - (A000027(n) + 1)/2)*A000035(n).
G.f.: x*(x*(5 - 7*y) + x^4*(1 - 2*y) - x^3*(- 3 + y) - 3*x^2*(- 1 + y) + y)/((-1 + x)^4*(1 + x)^2*(-1 + y)^2).
E.g.f.: (1/4)*exp(-x + y)*(1 - x - 2*y + exp(2*x)*(-1 + 3*x + 6*x^2 + 2*x^3 + 2*y)). - Stefano Spezia, Jan 10 2019

A317297 a(n) = (n - 1)(4n^2 - 8*n + 5).

Original entry on oeis.org

0, 5, 34, 111, 260, 505, 870, 1379, 2056, 2925, 4010, 5335, 6924, 8801, 10990, 13515, 16400, 19669, 23346, 27455, 32020, 37065, 42614, 48691, 55320, 62525, 70330, 78759, 87836, 97585, 108030, 119195, 131104, 143781, 157250, 171535, 186660, 202649, 219526, 237315, 256040, 275725, 296394, 318071

Offset: 1

Omar E. Pol, Sep 01 2018

Conjecture: For n > 1, a(n) is the maximum eigenvalue of a 2*(n-1) X 2*(n-1) square matrix M defined as M[i,j,n] = j + n*(i-1) if i is odd and M[i,j,n] = n*i - j + 1 if i is even (see A317614). - Stefano Spezia, Dec 27 2018

Connections can be made to A022144 and A010014. Namely, a formula for A022144 is (2*n+1)^2 - (2*n-1)^2. A formula for A010014 is (2*n+1)^3 - (2*n-1)^3. The general form can be represented by (2*n+1)^d - (2*n-1)^d, where d designates the number of dimensions. When d is 4, a(n) = ((2*(n-1)+1)^4 - (2*(n-1)-1)^4)/16, namely the general form shifted by 1 and divided by 16 is a(n). - Yigit Oktar, Aug 16 2024

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First bisection of A006003.
Nonzero terms give the row sums of A007607.
Conjecture: 0 together with a bisection of A246697.
Cf. A219086 (partial sums).
Cf. A008595, A033430, A135453, A317614.
Cf. A010014, A022144 (see comments)

Table[(n - 1) (4 n^2 - 8 n + 5), {n, 1, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 5, 34, 111}, 50] (* or *) CoefficientList[Series[x (5 + 14 x + 5 x^2)/(1 - x)^4, {x, 0, 50}], x] (* Stefano Spezia, Sep 01 2018 *)

PARI
```
a(n) = (n - 1)*(4*n^2 - 8*n + 5)
```

concat(0, Vec(x^2*(5 + 14*x + 5*x^2)/(1 - x)^4 + O(x^50))) \\ Colin Barker, Sep 01 2018

a(n) = 4*n^3 - 12*n^2 + 13*n - 5 = A033430(n) - A135453(n) + A008595(n) - 5.
G.f.: x^2*(5 + 14*x + 5*x^2)/(1 - x)^4. - Colin Barker, Sep 01 2018
a(n) = 4*a(n - 1) - 6*a(n - 2) + 4*a(n - 3) - a(n - 4) for n > 4. - Stefano Spezia, Sep 01 2018
E.g.f.: exp(x)*(5*x + 12*x^2 + 4*x^3). - Stefano Spezia, Jan 15 2019
a(n) = ((2*(n-1)+1)^4 - (2*(n-1)-1)^4)/16. - Yigit Oktar, Aug 16 2024

A323723 a(n) = (-2 - (-1)^n(-2 + n) + n + 2n^3)/4.

Original entry on oeis.org

0, 0, 4, 14, 32, 64, 108, 174, 256, 368, 500, 670, 864, 1104, 1372, 1694, 2048, 2464, 2916, 3438, 4000, 4640, 5324, 6094, 6912, 7824, 8788, 9854, 10976, 12208, 13500, 14910, 16384, 17984, 19652, 21454, 23328, 25344, 27436, 29678, 32000, 34480, 37044, 39774

Offset: 0

Stefano Spezia, Jan 25 2019

For n > 1, a(n) is the subdiagonal sum of the matrix M(n) whose permanent is A322277(n).

All the terms of this sequence are even numbers (A005843).

Stefano Spezia, Table of n, a(n) for n = 0..10000
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A000982, A004526, A005843, A033431, A033999, A317614, A322277, A323724, A325655.

Flat(List([0..50], n -> (-2-(-1)^n*(-2+n)+n+2*n^3)/4));

[(-2-(-1)^n*(-2+n)+n+2*n^3)/4: n in [0..50]];

a:=n->(-2 - (-1)^n*(-2 + n) + n + 2*n^3)/4: seq(a(n), n=0..50);

a[n_]:=(6 + n + n^3 + 12 Floor[1/2 (-3 + n)] + 4 Floor[1/2 (-3 + n)]^2 - 2 (1 + n) Floor[1/2 (-1 + n)])/2; Array[a,50,0]

makelist((-2-(-1)^n*(-2+n)+n+2*n^3)/4, n, 0, 50);

PARI
```
a(n) = (-2-(-1)^n*(-2+n)+n+2*n^3)/4;
```

[(-2-(-1)**n*(-2+n)+n+2*n**3)/4 for n in range(50)]

O.g.f.: 2*x^2*(2 + 3*x + x^3)/((1 - x)^4*(1 + x)^2).
E.g.f.: (1/4)*exp(-x)*(2 + x)*(1 + exp(2*x)*(-1 + 2*x + 2* x^2)).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 5.
a(n) = (6 + n + n^3 + 12*floor((n - 3)/2) + 4*floor((n - 3)/2)^2 - 2*(1 + n)*floor((n - 1)/2))/2.
a(n) = (-2 - A033999(n)*(-2 + n) + n + A033431(n))/4.
a(n) = n^3/2 for even n; a(n) = (n - 1)*(n^2 + n + 2)/2 otherwise. - Bruno Berselli, Feb 06 2019
a(n) = 2*A004526(n*A000982(n)). [Found by Christian Krause's LODA miner] - Stefano Spezia, Dec 12 2021

A323724 a(n) = n(2(n - 2)*n + (-1)^n + 3)/4.

Original entry on oeis.org

0, 0, 2, 6, 20, 40, 78, 126, 200, 288, 410, 550, 732, 936, 1190, 1470, 1808, 2176, 2610, 3078, 3620, 4200, 4862, 5566, 6360, 7200, 8138, 9126, 10220, 11368, 12630, 13950, 15392, 16896, 18530, 20230, 22068, 23976, 26030, 28158, 30440, 32800, 35322, 37926, 40700

Offset: 0

Stefano Spezia, Jan 25 2019

For n > 1, a(n) is the superdiagonal sum of the matrix M(n) whose permanent is A322277(n).

All the terms of this sequence are even numbers (A005843), but do not end with 4.

Stefano Spezia, Table of n, a(n) for n = 0..10000
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A000290, A000982, A004526, A005843, A005997, A317614, A322277, A323723, A325516.

Flat(List([0..50], n->(1/2)*(-1 + n)^2*n - (-1 + n)*Int(n/2) + 2*(Int(n/2))^2));

[(1/2)*(-1 + n)^2*n - (-1 + n)*Floor(n/2) + 2*(Floor(n/2))^2: n in [0..50]];

a:=n->(1/2)*(-1 + n)^2*n - (-1 + n)*floor(n/2) + 2*(floor(n/2))^2: seq(a(n), n=0..50);

a[n_] := 1/2 (-1 + n)^2 n - (-1 + n) Floor[n/2] + 2 Floor[n/2]^2; Array[a, 50, 0];
Table[n (2 (n - 2) n + (-1)^n + 3)/4, {n, 0, 50}] (* Bruno Berselli, Feb 06 2019 *)
LinearRecurrence[{2,1,-4,1,2,-1},{0,0,2,6,20,40},50] (* Harvey P. Dale, Jan 13 2024 *)

makelist((1/2)*(-1 + n)^2*n - (-1 + n)*floor(n/2) + 2*(floor(n/2))^2, n, 0, 50);

a(n) = (1/2)*(-1 + n)^2*n - (-1 + n)*floor(n/2) + 2*(floor(n/2))^2;

T(i,j,n) = if (i %2, j + n*(i-1), n*i - j + 1);
a(n) = sum(k=1, n-1, T(k,k+1,n)); \\ Michel Marcus, Feb 06 2019

[int((1/2)*(-1 + n)**2*n - (-1 + n)*int(n/2) + 2*(int(n/2))**2) for n in range(0,50)]

O.g.f.: 2*x^2*(1 + x + 3*x^2 + x^3)/((1 - x)^4*(1 + x)^2).
E.g.f.: (1/2)*x*(exp(x)*x*(1 + x) + sinh(x)).
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 5.
a(n) = (1/2)*(-1 + n)^2*n - (-1 + n)*floor(n/2) + 2*floor(n/2)^2.
a(n) = (1/2)*(-1 + n)^2*n - (-1 + n)*A004526(n) + 2*A000290(A004526(n)).
a(n) = (n/2)*((n - 1)^2 + 1) for even n; a(n) = (n/2)*(n - 1)^2 otherwise. - Bruno Berselli, Feb 06 2019
a(n) = 2*A004526(n*A000982(n-1)). [Found by Christian Krause's LODA miner] - Stefano Spezia, Dec 12 2021
a(n) = 2*A005997(n-1) for n >= 2. - Hugo Pfoertner, Dec 13 2021

Definition by Bruno Berselli, Feb 06 2019

A006010 Number of paraffins (see Losanitsch reference for precise definition).

Original entry on oeis.org

1, 5, 20, 52, 117, 225, 400, 656, 1025, 1525, 2196, 3060, 4165, 5537, 7232, 9280, 11745, 14661, 18100, 22100, 26741, 32065, 38160, 45072, 52897, 61685, 71540, 82516, 94725, 108225, 123136, 139520, 157505, 177157, 198612, 221940, 247285, 274721, 304400

Offset: 1

N. J. A. Sloane

This is also the square of the sum of the odd numbers plus the square of the sum of the even numbers, up to n. E.g., a(4) = (1+3)^2 + (2+4)^2 = 52. - Scott R. Shannon, Feb 20 2019

The area of a square whose side is a segment connecting the ends of a broken line (snake), the adjacent links of which are perpendicular and equal to the numbers 1, 2, 3, 4, ..., n. For example, a(5) = 9^2 + 6^2 = 117. - Nicolay Avilov, Aug 02 2022

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Nicolay Avilov, Illustration of a(1)-a(6)
Nicolay Avilov, The problem of a broken line in a square (in Russian).
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A005994, A186424 (2nd differences), A317614 (1st differences), A335648 (partial sums).

CoefficientList[Series[-(x^4 + 2 x^3 + 6 x^2 + 2 x + 1)/((x - 1)^5 (x + 1)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 14 2013 *)
LinearRecurrence[{3,-1,-5,5,1,-3,1},{1,5,20,52,117,225,400},40] (* Harvey P. Dale, Dec 13 2018 *)

Vec(-x*(x^4+2*x^3+6*x^2+2*x+1)/((x-1)^5*(x+1)^2) + O(x^100)) \\ Colin Barker, Oct 05 2015

Sum of [ 1, 3, 9, ... ](A005994) + [ 0, 0, 1, 3, 9, ... ] + 2*[ 0, 1, 5, 15, 35, ... ](binomial(n, 4)).
If n is odd then a(n) = (1/8) * (n^4 + 2*n^3 + 2*n^2 + 2*n + 1) = Det(Transpose[M]*M) where M is the 2 X 3 matrix whose rows are [(n-1)/2, (n-1)/2], [(n-1)/2 + 1, 0] and [(n-1)/2 + 1, (n-1)/2 + 1]. If n is even then a(n) = (1/8) * (n^4 + 2*n^3 + 2*n^2) = Det(Transpose[M]*M) where M is the 2 X 3 matrix whose rows are [n/2, 0], [n/2, n/2] and [n/2 + 1, 0]. - Gerald McGarvey, Oct 30 2007
G.f.: -x*(x^4+2*x^3+6*x^2+2*x+1) / ((x-1)^5*(x+1)^2). - Colin Barker, Mar 20 2013
E.g.f.: (x*(7 + 15*x + 8*x^2 + x^3)*cosh(x) + (1 + 5*x + 15*x^2 + 8*x^3 + x^4)*sinh(x))/8. - Stefano Spezia, Jul 08 2020

More terms from David W. Wilson

A325516 Triangle read by rows: T(n, k) = (1/4)n(1 - (-1)^(n - k) + 2*(n - k)^2), with 0 <= k < n.

Original entry on oeis.org

1, 4, 2, 15, 6, 3, 32, 20, 8, 4, 65, 40, 25, 10, 5, 108, 78, 48, 30, 12, 6, 175, 126, 91, 56, 35, 14, 7, 256, 200, 144, 104, 64, 40, 16, 8, 369, 288, 225, 162, 117, 72, 45, 18, 9, 500, 410, 320, 250, 180, 130, 80, 50, 20, 10, 671, 550, 451, 352, 275, 198, 143, 88, 55, 22, 11

Offset: 1

Stefano Spezia, May 07 2019

T(n, k) is the k-superdiagonal sum of the matrix M(n) whose permanent is A322277(n).

			The triangle T(n, k) begins:
---+------------------------------
n\k|    0     1     2     3     4
---+------------------------------
1  |    1
2  |    4     2
3  |   15     6     3
4  |   32    20     8     4
5  |   65    40    25    10     5
...
For n = 3 the matrix M(3) is
1, 2, 3
6, 5, 4
7, 8, 9
and therefore T(3, 0) = 1 + 5 + 9 = 15, T(3, 1) = 2 + 4 = 6, and T(3, 2) = 3.

Cf. A000027, A317614, A322277, A325517 (row sums).

Flat(List([1..11], n->List([0..n-1], k->(1/4)*n*(1 - (-1)^(n - k) + 2*(n - k)^2))));

[[(1/4)*n*(1 - (-1)^(n - k) + 2*(n - k)^2): k in [0..n-1]]: n in [1..11]];

a:=(n, k)->(1/4)*n*(1 - (-1)^(n - k) + 2*(n - k)^2): seq(seq(a(n, k), k=0..n-1), n=1..11);

T[n_,k_] := (1/4) n (1 - (-1)^(n - k) + 2 (n - k)^2); Flatten[Table[T[n, k], {n, 1, 11}, {k, 0, n - 1}]]

T(n, k) = (1/4)*n*(1 - (-1)^(n - k) + 2*(n - k)^2);
tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print);
tabl(11) \\ yields sequence in triangular form

O.g.f.: x*(1 - 2*y + y^2 + 2*y^3 + x^4*(1 + y^2) + 2*x^3*(1 + y - 3*y^2 + y^3) + 2*x^2*(3 - 5*y - y^2 + y^3) + x*(2 - 2*y - 6*y^2 + 6*y^3))/((1 - x)^4*(1 + x)^2*(1 - y)^3*(1 + y)).
E.g.f.: (1/4)*exp(-x - y)*(x + exp(2*(x + y))*x*(3 + 2*x^2 + x*(6 - 4*y) - 2*y + 2*y^2)).
T(n, k) = n*(n - k)^2/2 if n and k are both even or odd; T(n, k) = n*(n - k)^2/2 + n/2 otherwise.
1st column: T(n, 1) = A317614(n).
Diagonal: T(n, n-1) = n.

cp's OEIS Frontend

A374668 a(n) is the permanent of the n-th order Hankel matrix M whose generic element is given by M(i,j) = A317614(i+j-1) with i,j = 1, ..., n.

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A346174 Inverse binomial transform of A317614.

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A381374 Little Hankel transform of A317614: a(n) = A317614(n+1)^2 - A317614(n)*A317614(n+2).

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A322277 Permanent of an n X n square matrix M(n) formed by writing the numbers 1, ..., n^2 successively forward and backward along the rows in zig-zag pattern.

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Maple

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A317617 Triangle T read by rows: T(n, k) = (n^3 + n)/2 + (k - (n + 1)/2)*(n mod 2).

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Magma

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Maxima

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A317297 a(n) = (n - 1)*(4*n^2 - 8*n + 5).

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A323723 a(n) = (-2 - (-1)^n*(-2 + n) + n + 2*n^3)/4.

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A317297 a(n) = (n - 1)(4n^2 - 8*n + 5).

A323723 a(n) = (-2 - (-1)^n(-2 + n) + n + 2n^3)/4.

A323724 a(n) = n(2(n - 2)*n + (-1)^n + 3)/4.

A325516 Triangle read by rows: T(n, k) = (1/4)n(1 - (-1)^(n - k) + 2*(n - k)^2), with 0 <= k < n.