A317614 -id:A317614 - cp's OEIS frontend

A325655 Triangle read by rows: T(n, k) = (1/4)(2(-1 + (-1)^n)k - 2k^2n + n(2 - (-1)^k - (-1)^n + 2*n^2)), with 0 <= k < n.

1, 4, 4, 15, 14, 7, 32, 32, 24, 16, 65, 64, 53, 42, 21, 108, 108, 96, 84, 60, 36, 175, 174, 159, 144, 115, 86, 43, 256, 256, 240, 224, 192, 160, 112, 64, 369, 368, 349, 330, 293, 256, 201, 146, 73, 500, 500, 480, 460, 420, 380, 320, 260, 180, 100, 671, 670, 647, 624, 579, 534, 467, 400, 311, 222, 111

Offset: 1

Stefano Spezia, May 13 2019

T(n, k) is the k-subdiagonal 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     4
3  |   15    14     7
4  |   32    32    24    16
5  |   65    64    53    42    21
...
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) = 6 + 8 = 14, and T(3, 2) = 7.

Cf. A317614, A322277, A323723 (k = 1), A325656 (row sums), A325657 (diagonal).

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

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

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

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

T(n, k) = (1/4)*(2*(- 1 + (- 1)^n)*k - 2*k^2*n + n*(2 + (- 1)^(1+k) + (- 1)^(1 + n) + 2*n^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 + 3*y^2 - 2*y^3 + 2*x*(- 1 + y^2) + x^4*(- 1 + 3*y^2) + x^2*(- 6 + 6*y + 2*y^2 - 6*y^3) + x^3*(- 2 + 4*y + 2*y^2 - 4*y^3))/((- 1 + x)^4*(1 + x)^2*(- 1 + y)^3*(1 + y)).
E.g.f.: (1/4)*exp(- x - y)*(- exp(2*x)*x + exp(2*y)*(x + 2*y) + 2*exp(2*(x + y))*(3*x^2 + x^3 - y - x*(- 2 + y + y^2))).
T(n, k) = (1/2)*n*(n^2 - k^2) if n and k are both even; T(n, k) = (1/2)*n*(n^2 - k^2 + 1) if n is even and k is odd; T(n, k) = (1/2)*(n*(n^2 - k^2 + 1) - 2*k) if n is odd and k is even; T(n, k) = (1/2)*(n*(n^2 - k^2 + 2) - 2*k) if n and k are both odd.
Diagonal: T(n, n-1) = A325657(n).
1st column: T(n, 0) = A317614(n).

A340804 Triangle read by rows: T(n, k) = 1 + k(n - 1) + (2k - n - 1)*(k mod 2) with 0 < k <= n.

Original entry on oeis.org

1, 1, 3, 1, 5, 9, 1, 7, 11, 13, 1, 9, 13, 17, 25, 1, 11, 15, 21, 29, 31, 1, 13, 17, 25, 33, 37, 49, 1, 15, 19, 29, 37, 43, 55, 57, 1, 17, 21, 33, 41, 49, 61, 65, 81, 1, 19, 23, 37, 45, 55, 67, 73, 89, 91, 1, 21, 25, 41, 49, 61, 73, 81, 97, 101, 121, 1, 23, 27, 45, 53, 67, 79, 89, 105, 111, 131, 133

Offset: 1

Stefano Spezia, Jan 22 2021

T(n, k) is the k-th diagonal element 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.

It includes exclusively all the odd numbers (A005408). Except the term 1, all the other odd numbers appear a finite number of times.

			1
1,  3
1,  5,  9,
1,  7, 11, 13
1,  9, 13, 17, 25
1, 11, 15, 21, 29, 31
1, 13, 17, 25, 33, 37, 49
...

Stefano Spezia, Table of n, a(n) for n = 1..11325 (first 150 rows of the triangle, flattened).

Cf. A005408, A317614 (row sums).

Cf. A000012 (1st column), A006010 (sum of the first n rows), A060747 (2nd column), A074147 (antidiagonals of M matrices), A241016 (row sums of M matrices), A317617 (column sums of M matrices), A322277 (permanent of M matrices), A323723 (subdiagonal sum of M matrices), A323724 (superdiagonal sum of M matrices).

Table[1+k(n-1)+(2k-n-1)Mod[k,2],{n,12},{k,n}]//Flatten

T(n, k) = 1 + k*(n - 1) + (2*k - n - 1)*(k % 2); \\ Michel Marcus, Jan 25 2021

O.g.f.: (1 + y - 3*y^2 + y^3 + x*(-1 - y + 5*y^2 + y^3))/((-1 + x)^2*(-1 + y)^2*(1+y)^2).

E.g.f.: exp(x - y)*(1 + x + 2*y + exp(2*y)*(1 + x*(-1 + 2*y)))/2.

A304487 a(n) = (3 + 2n - 3n^2 + 4n^3 - 3((-1 + n) mod 2))/6.

Original entry on oeis.org

1, 4, 15, 36, 73, 128, 207, 312, 449, 620, 831, 1084, 1385, 1736, 2143, 2608, 3137, 3732, 4399, 5140, 5961, 6864, 7855, 8936, 10113, 11388, 12767, 14252, 15849, 17560, 19391, 21344, 23425, 25636, 27983, 30468, 33097, 35872, 38799, 41880, 45121, 48524, 52095

Offset: 1

Stefano Spezia, Aug 17 2018

a(n) is the trace of an n X n matrix A in which the entries are 1 through n^2, spiraling inward starting with 1 in the (1,1)-entry (proved).

The first three terms of a(n) coincide with those of A317614.

			For n = 1 the matrix A is
   1
with trace Tr(A) = a(1) = 1.
For n = 2 the matrix A is
   1, 2
   4, 3
with Tr(A) = a(2) = 4.
For n = 3 the matrix A is
   1, 2, 3
   8, 9, 4
   7, 6, 5
with Tr(A) = a(3) = 15.
For n = 4 the matrix A is
   1,  2,  3, 4
  12, 13, 14, 5
  11, 16, 15, 6
  10,  9,  8, 7
with Tr(A) = a(4) = 36.

Stefano Spezia, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A317614, A228958, A045991, A085046.

Cf. A126224 (determinant of the matrix A), A317298 (first differences).

a_n:=List([1..43], n->(3 + 2*n - 3*n^2 + 4*n^3 - 3*RemInt(-1 + n, 2))/6);

List([1..43],n->(3+2*n-3*n^2+4*n^3-3*((-1+n) mod 2))/6); # Muniru A Asiru, Sep 17 2018

I:=[1,4,15,36,73]; [n le 5 select I[n] else 3*Self(n-1)-2*Self(n-2)-2*Self(n-3)+3*Self(n-4)-Self(n-5): n in [1..43]]; // Vincenzo Librandi, Aug 26 2018

seq((3+2*n-3*n^2+4*n^3-3*modp((-1+n),2))/6,n=1..43); # Muniru A Asiru, Sep 17 2018

Table[1/6 (3 + 2 n - 3 n^2 + 4 n^3 - 3 Mod[-1 + n, 2]), {n, 1, 43}] (* or *)
CoefficientList[ Series[x*(1 + x + 5 x^2 + x^3)/((-1 + x)^4 (1 + x)), {x, 0, 43}], x] (* or *)
LinearRecurrence[{3, -2, -2, 3, -1}, {1, 4, 15, 36, 73}, 43]

a(n):=(3 + 2*n - 3*n^2 + 4*n^3 - 3*mod(-1 + n, 2))/6$ makelist(a(n), n, 1, 43);

Vec(x*(1 + x + 5*x^2 + x^3)/((-1 + x)^4*(1 + x)) + O(x^44))

a(n) = (3 + 2*n - 3*n^2 + 4*n^3 - 3*((-1 + n)%2))/6

a(n) = A045991(n) - Sum_{k=2..n-1} A085046(k) for n > 2 (proved).
G.f.: x*(1 + x + 5 x^2 + x^3)/((-1 + x)^4 (1 + x)).
a(n) + a(n + 1) = A228958(2*n + 1).
From Colin Barker, Aug 17 2018: (Start)
a(n) = (2*n - 3*n^2 + 4*n^3) / 6 for n even.
a(n) = (3 + 2*n - 3*n^2 + 4*n^3) / 6 for n odd.
a(n) = 3*a(n - 1) - 2*a(n - 2) - 2*a(n - 3) + 3*a(n - 4) - a(n - 5) for n > 5.
(End)
E.g.f.: (1/12)*exp(-x)*(-3 + exp(2*x)*(3 + 6*x + 18*x^2 + 8*x^3)). - Stefano Spezia, Feb 10 2019

A322844 a(n) = (1/12)n^2(3(1 + n^2) - 2(2 + n^2)*(n mod 2)).

Original entry on oeis.org

0, 0, 5, 6, 68, 50, 333, 196, 1040, 540, 2525, 1210, 5220, 2366, 9653, 4200, 16448, 6936, 26325, 10830, 40100, 16170, 58685, 23276, 83088, 32500, 114413, 44226, 153860, 58870, 202725, 76880, 262400, 98736, 334373, 124950, 420228, 156066, 521645, 192660, 640400, 235340

Offset: 0

Stefano Spezia, Dec 28 2018

Conjectures: (Start)
For n > 1, a(n) is the absolute value of the trace of the 2nd exterior power of an n X n square matrix M(n) 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). Equivalently, a(n) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of the matrix M(n), or the absolute value of the sum of all principal minors of M(n) of size 2.
For k > 2, the trace of the k-th exterior power of the matrix M(n) is equal to zero.
(End)

Stefano Spezia, Table of n, a(n) for n = 0..10000
Wikipedia, Characteristic polynomial
Wikipedia, Exterior algebra
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

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

Cf. A002415, A037270, A074147 (antidiagonals of M matrices), A241016 (row sums of M matrices), A317617 (column sums of M matrices), A322277 (permanent of matrix M(n)), A323723 (subdiagonal sum of M matrices), A323724 (superdiagonal sum of M matrices), A325516 (k-superdiagonal sum of M matrices), A325655 (k-subdiagonal sum of M matrices).

Flat(List([0..50], n->(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*(n mod 2))));

[IsEven(n) select (1/4)*n^2*(1 + n^2) else (1/12)*(- 1 + n)*n^2*(1 + n): n in [0..50]];

a:=n->(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*modp(n,2)): seq(a(n), n=0..50);

a[n_]:=(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*Mod[n,2]); Array[a,50,0]
LinearRecurrence[{0,5,0,-10,0,10,0,-5,0,1},{0,0,5,6,68,50,333,196,1040,540},50] (* Harvey P. Dale, Aug 23 2025 *)

a(n):=(1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*mod(n,2))$ makelist(a(n), n, 0, 50);

a(n) = (1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*(n % 2));

a(n) = abs(polcoeff(charpoly(matrix(n, n, i, j, if (i %2, j + n*(i-1), n*i - j + 1))), n-2)); \\ Michel Marcus, Feb 06 2019

[int(n**2*(3*(1 + n**2) - 2*(2 + n**2)*pow(n, 1, 2))/12) for n in range(0,50)]

O.g.f.: -x^2*(5 + 6*x + 43*x^2 + 20*x^3 + 43*x^4 + 6*x^5 + 5*x^6)/((-1 + x)^5*(1 + x)^5).
E.g.f.: (1/(12*x^2))*exp(-x)*(24 - 60*exp(x) + 21*x + 9*x^2 + 2*x^3 + x^4 + exp(2*x)*(36 - 33*x + 15*x^2 - 4*x^3 + 2*x^4)).
a(n) = (1/4)*n^2*(1 + n^2) for n even.
a(n) = (1/2)*A037270(n) for n even.
a(n) = (1/12)*(-1 + n)*n^2*(1 + n) for n odd.
a(n) = A002415(n) for n odd.
a(2*n+1) = 5*a(2*n-1) - 10*a(2*n-3) + 10*a(2*n-5) - 5*a(2*n-7) + a(2*n-9), for n > 4.
a(2*n) = 5*a(2*n-2) - 10*a(2*n-4) + 10*a(2*n-6) - 5*a(2*n-8) + a(2*n-10), for n > 4.
O.g.f. for a(2*n+1): -x*(2*(3 + 10*x + 3*x^2))/(-1 + x)^5.
O.g.f. for a(2*n): x*(-5 - 43*x - 43*x^2 - 5*x^3)/(-1 + x)^5.
E.g.f. for a(2*n+1): (1/12)*(6*x*cosh(sqrt(x)) + sqrt(x)*(6 + x)*sinh(sqrt(x))).
E.g.f. for a(2*n): (1/4)*(x*(8 + x)*cosh(sqrt(x)) + 2*sqrt(x)*(1 + 3*x)*sinh(sqrt(x))).
Sum_{k>=1} 1/a(2*k) = (1/6)*(12 + Pi^2 - 6*Pi*coth(Pi/2)) = 0.21955691692893092525407699347398665248691900...
Sum_{k>=1} 1/a(2*k+1) = 3*(5 - Pi^2/2) = 0.1955933983659620717482635001857732970...
Sum_{k>=2} 1/a(k) = 17 - (4*Pi^2)/3 - Pi*coth(Pi/2) = 0.415150315294892997002340493659759949516369894...

A349107 a(n) is the permanent of the n X n matrix A(n) that is defined as A[i,j,n] = n - abs((n + 1)/2 - i) - abs((n + 1)/2 - j).

Original entry on oeis.org

1, 1, 2, 22, 292, 9084, 314736, 19224816, 1267665984, 127896194880, 13696865136000, 2061743814864000, 325942368613966080, 68443327006163424000, 14983681934750599526400, 4184458128589740299827200, 1211736134642288777186918400, 434251427188367439407838412800, 160701529762439051943130553548800

Offset: 0

Stefano Spezia, Nov 08 2021

nonn

A(n) is an n X n matrix whose elements start from 1 at the corners and get higher, the more they are at the center (see the examples).

det(A(1)) = 1 and det(A(n)) = 0 for n > 1.

			For n = 5 the matrix A(5) is
   1, 2, 3, 2, 1
   2, 3, 4, 3, 2
   3, 4, 5, 4, 3
   2, 3, 4, 3, 2
   1, 2, 3, 2, 1
with permanent a(5) = 9084.
For n = 6 the matrix A(6) is
   1, 2, 3, 3, 2, 1
   2, 3, 4, 4, 3, 2
   3, 4, 5, 5, 4, 3
   3, 4, 5, 5, 4, 3
   2, 3, 4, 4, 3, 2
   1, 2, 3, 3, 2, 1
with permanent a(6) = 314736.

Vaclav Kotesovec, Table of n, a(n) for n = 0..36

Cf. A000982 (trace of matrix A(n)), A317614 (elements sum of matrix A(n)), A349108.

A[i_, j_, n_] := n - Abs[(n + 1)/2 - i] - Abs[(n + 1)/2 - j]; a[n_]:=Permanent[Table[A[i,j,n],{i,n},{j,n}]]; Array[a,18]

a(n) = matpermanent(matrix(n, n, i, j, n - abs((n + 1)/2 - i) - abs((n + 1)/2 - j))); \\ Michel Marcus, Nov 08 2021

a(2*n) = A349108(2*n).

A350050 a(n) = (2n^4 - 6(-1)^nn^2 - 2n^2 + 3*(-1)^n - 3)/96.

Original entry on oeis.org

0, 0, 0, 2, 4, 14, 24, 52, 80, 140, 200, 310, 420, 602, 784, 1064, 1344, 1752, 2160, 2730, 3300, 4070, 4840, 5852, 6864, 8164, 9464, 11102, 12740, 14770, 16800, 19280, 21760, 24752, 27744, 31314, 34884, 39102, 43320, 48260, 53200, 58940, 64680, 71302, 77924, 85514

Offset: 0

Stefano Spezia, Dec 11 2021

Definitions: (Start)
The k-th exterior power of a vector space V of dimension n is a vector subspace spanned by elements, called k-vectors, that are the exterior product of k vectors v_i in V.
Given a square matrix A that describes the vectors v_i in terms of a basis of V, the k-th exterior power of the matrix A is the matrix that represents the k-vectors in terms of the basis of V. (End)
Conjectures: (Start)
For n > 1, a(n) is the absolute value of the trace of the 2nd exterior power of an n X n square matrix A(n) defined as A[i,j,n] = n - abs((n + 1)/2 - j) - abs((n + 1)/2 - i) (see A349107). Equivalently, a(n) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of the matrix A(n), or the absolute value of the sum of all principal minors of A(n) of size 2.
For k > 2, the trace of the k-th exterior power of the matrix A(n) is equal to zero. (End)
The same conjectures hold for an n X n square matrix A(n) defined as A[i,j,n] = (n mod 2) + abs((n + 1)/2 - j) + abs((n + 1)/2 - i) (see A349108).

Winston de Greef, Table of n, a(n) for n = 0..10000
Wikipedia, Characteristic polynomial
Wikipedia, Exterior algebra
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A000982 (trace of matrix A(n)), A317614 (elements sum of matrix A(n)), A349107, A349108.

Cf. A006325, A112742, A338429.

Table[(2*n^4-6*(-1)^n*n^2-2*n^2+3*(-1)^n-3)/96,{n,0,45}]

a(n) = (2*n^4 - 6*(-1)^n*n^2 - 2*n^2 + 3*(-1)^n - 3)/96 \\ Winston de Greef, Jan 28 2024

O.g.f.: 2*x^3*(1 + x^2)/((1 - x)^5*(1 + x)^3).
E.g.f.: (x*(x^3 + 6*x^2 + 3*x + 3)*cosh(x) + (x^4 + 6*x^3 + 9*x^2 - 3*x - 3)*sinh(x))/48.
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n > 7.
a(n) = A338429(n-2)/2 for n > 2.
a(2*n-1) = 2*A006325(n).
a(2*n) = A112742(n).
Sum_{n>2} 1/a(n) = (45 - 2*Pi^2 - 4*sqrt(3)*Pi*tanh(sqrt(3)*Pi/2))/4 = 0.920755957767250147865...

A325517 a(n) = n((2n + 1)(2n^2 + 2n + 3) - 3(-1)^n)/24.

Original entry on oeis.org

0, 1, 6, 24, 64, 145, 282, 504, 832, 1305, 1950, 2816, 3936, 5369, 7154, 9360, 12032, 15249, 19062, 23560, 28800, 34881, 41866, 49864, 58944, 69225, 80782, 93744, 108192, 124265, 142050, 161696, 183296, 207009, 232934, 261240, 292032, 325489, 361722, 400920, 443200

Offset: 0

Stefano Spezia, May 07 2019

For n > 0, a(n) is the n-th row sum of the triangle A325516.

Stefano Spezia, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A131941, A317614, A322277, A325516.

Flat(List([0..50], n->n*((2*n + 1)*(2*n^2 + 2*n + 3) - 3*(-1)^n)/24));

[n*((2*n + 1)*(2*n^2 + 2*n + 3) - 3*(-1)^n)/24: n in [0..50]];

a:=n->n*((2*n + 1)*(2*n^2 + 2*n + 3) - 3*(-1)^n)/24: seq(a(n), n=0..50);

a[n_]:=n*((2*n + 1)*(2*n^2 + 2*n + 3) - 3*(-1)^n)/24; Array[a,50,0]

a(n) = n*((2*n + 1)*(2*n^2 + 2*n + 3) - 3*(-1)^n)/24;

O.g.f.: x*(1 + 3*x + 7*x^2 + 3*x^3 + 2*x^4)/((1 - x)^5*(1 + x)^2).
E.g.f.: (1/24)*exp(-x)*x*(3 + 21*exp(2*x) + 54*exp(2*x)*x + 30*exp(2*x)*x^2 + 4*exp(2*x)*x^3).
a(n) = 3*a(n-1) - a(n-2) - 5*a(n-3) + 5*a(n-4) + a(n-5) - 3*a(n-6) + a(n-7) for n > 6.
a(n) = n^2*(2*n^2 + 3*n + 4)/12 if n is even.
a(n) = n*(n + 1)*(2*n^2 + n + 3)/12 if n is odd.
a(n) = n*A131941(n). - Stefano Spezia, Dec 21 2021

A325656 a(n) = (1/24)n((4n + 3)(2n^2 + 1) - 3(-1)^n).

Original entry on oeis.org

0, 1, 8, 36, 104, 245, 492, 896, 1504, 2385, 3600, 5236, 7368, 10101, 13524, 17760, 22912, 29121, 36504, 45220, 55400, 67221, 80828, 96416, 114144, 134225, 156832, 182196, 210504, 242005, 276900, 315456, 357888, 404481, 455464, 511140, 571752, 637621, 709004, 786240

Offset: 0

Stefano Spezia, May 13 2019

For n > 0, a(n) is the n-th row sum of the triangle A325655.

Stefano Spezia, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A173722, A317614, A322277, A325655.

Flat(List([0..50], n->(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3)));

[(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3): n in [0..50]];

a:=n->(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3): seq(a(n), n=0..50);

a[n_]:=(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3); Array[a,50,0]

a(n) = (1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3);

O.g.f.: -x*(1 + 5*x + 13*x*2 + 9*x^3 + 4*x^4)/((-1 + x)^5*(1 + x)^2).
E.g.f.: (1/24)*exp(-x)*x*(3 + 21*exp(2*x) + 78*exp(2*x)*x + 54*exp(2*x)*x^2 + 8*exp(2*x)*x*3).
a(n) = a(n) = 3*a(n-1) - a(n-2) - 5*a(n-3) + 5*a(n-4) + a(n-5) - 3*a(n-6) + a(n-7) for n > 6.
a(n) = (1/12)*n^2*(4*n^2 + 3*n + 2) if n is even.
a(n) = (1/12)*n*(n + 1)*(4*n^2 - n + 3) if n is odd.
a(n) = n*A173722(2*n). - Stefano Spezia, Dec 21 2021

A335648 Partial sums of A006010.

Original entry on oeis.org

0, 1, 6, 26, 78, 195, 420, 820, 1476, 2501, 4026, 6222, 9282, 13447, 18984, 26216, 35496, 47241, 61902, 80002, 102102, 128843, 160908, 199068, 244140, 297037, 358722, 430262, 512778, 607503, 715728, 838864, 978384, 1135889, 1313046, 1511658, 1733598, 1980883, 2255604

Offset: 0

Stefano Spezia, Jun 15 2020

Stefano Spezia, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-4,-4,10,-4,-4,4,-1).
Index entries for partial sums.

Cf. A000290, A001844, A005408, A007204, A053755, A058331, A059722.

Cf. A006010 (1st differences), A186424 (3rd differences), A317614 (2nd differences).

I:=[0, 1, 6, 26, 78, 195, 420, 820]; [n le 8 select I[n] else 4*Self(n-1)-4*Self(n-2)-4*Self(n-3)+10*Self(n-4)-4*Self(n-5)-4*Self(n-6)+4*Self(n-7)-Self(n-8): n in [1..39]];

Table[(1+n)(5-5(-1)^n+8n+12n^2+8n^3+2n^4)/80,{n,0,38}]

a(n) = (1 + n)*(5 - 5*(-1)^n + 8*n + 12*n^2 + 8*n^3 + 2*n^4)/80;

(x*(1+2*x+6*x^2+2*x^3+x^4)/((1-x)^6*(1+x)^2)).series(x, 39).coefficients(x, False)

a(n) = (1 + n)*(5 - 5*(-1)^n + 8*n + 12*n^2 + 8*n^3 + 2*n^4)/80.
O.g.f.: x*(1 + 2*x + 6*x^2 + 2*x^3 + x^4)/((1 - x)^6*(1 + x)^2).
E.g.f.: (cosh(x) - sinh(x))*(-5 + 5*x + (5 + 65*x + 180*x^2 + 130*x^3 + 30*x^4 + 2*x^5)*(cosh(2*x) + sinh(2*x)))/80.
a(n) = 4*a(n-1) - 4*a(n-2) - 4*a(n-3) + 10*a(n-4) - 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - a(n-8) for n > 7.
a(2*n-1) = n*A053755(n)/5 for n > 0.
a(2*n) = n*A005408(n)*A059722(n-1)/5.
a(2*n+1) - a(2*n-1) = A001844(n)^2 = A007204(n) for n > 0.
a(2*n) - a(2*n-2) = 2*A000290(n)*A058331(n) for n > 0.

A374708 Triangle T read by rows: T(n,k) = (n - k)n(4n^2 - 4nk + 2k^2 - 1 + (-1)^k)/4, with 0 <= k < n.

Original entry on oeis.org

1, 16, 4, 81, 36, 15, 256, 144, 80, 32, 625, 400, 255, 140, 65, 1296, 900, 624, 396, 240, 108, 2401, 1764, 1295, 896, 609, 364, 175, 4096, 3136, 2400, 1760, 1280, 864, 544, 256, 6561, 5184, 4095, 3132, 2385, 1728, 1215, 756, 369, 10000, 8100, 6560, 5180, 4080, 3100, 2320, 1620, 1040, 500

Offset: 1

Stefano Spezia, Jul 17 2024

T(n, k) is the k-th super- and subdiagonal sum of the Hankel matrix M(n) whose permanent is A374668(n).

			n\k|    0    1    2    3    4    5
---+------------------------------
1  |    1
2  |   16    4
3  |   81   36   15
4  |  256  144   80   32
5  |  625  400  255  140   65
6  | 1296  900  624  396  240  108
      ...
For n = 3 the matrix M is
  [ 1,  4, 15]
  [ 4, 15, 32]
  [15, 32, 65]
and therefore T(3, 0) = 1 + 15 + 65 = 81, T(3, 1) = 4 + 32 = 36, and T(3, 2) = 15.

Cf. A317614 (diagonal), A374668.
Cf. A333119, A330613.
Cf. A000583 (k=0), A035287 (k=1), A123865, A374709 (row sums).

T[n_,k_]:=(n-k)*n*(4*n^2 - 4*n*k+2*k^2-1+(-1)^k)/4; Table[T[n,k],{n,10},{k,0,n-1}]//Flatten

O.g.f.: x*(1 - 4*x^8*y^5 + x*(11 + 2*y) - x^7*y^4*(7 + 16*y) - x^2*(-11 + 6*y - 6*y^2) - x^5*y^2*(2 - 46*y - 3*y^2) - x^6*y^3*(-2 - 27*y + 4*y^2) - x^3*(-1 + 18*y + 38*y^2 - 2*y^3) - x^4*y*(2 + 14*y + 2*y^2 - y^3))/((1 - x)^5*(1 - x*y)^4*(1 + x*y)^2).

T(n,2) = A123865(n-1) for n > 1.

cp's OEIS Frontend

A325655 Triangle read by rows: T(n, k) = (1/4)*(2*(-1 + (-1)^n)*k - 2*k^2*n + n*(2 - (-1)^k - (-1)^n + 2*n^2)), with 0 <= k < n.

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A340804 Triangle read by rows: T(n, k) = 1 + k*(n - 1) + (2*k - n - 1)*(k mod 2) with 0 < k <= n.

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A304487 a(n) = (3 + 2*n - 3*n^2 + 4*n^3 - 3*((-1 + n) mod 2))/6.

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A322844 a(n) = (1/12)*n^2*(3*(1 + n^2) - 2*(2 + n^2)*(n mod 2)).

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A349107 a(n) is the permanent of the n X n matrix A(n) that is defined as A[i,j,n] = n - abs((n + 1)/2 - i) - abs((n + 1)/2 - j).

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A350050 a(n) = (2*n^4 - 6*(-1)^n*n^2 - 2*n^2 + 3*(-1)^n - 3)/96.

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A325655 Triangle read by rows: T(n, k) = (1/4)(2(-1 + (-1)^n)k - 2k^2n + n(2 - (-1)^k - (-1)^n + 2*n^2)), with 0 <= k < n.

A340804 Triangle read by rows: T(n, k) = 1 + k(n - 1) + (2k - n - 1)*(k mod 2) with 0 < k <= n.

A304487 a(n) = (3 + 2n - 3n^2 + 4n^3 - 3((-1 + n) mod 2))/6.

A322844 a(n) = (1/12)n^2(3(1 + n^2) - 2(2 + n^2)*(n mod 2)).

A350050 a(n) = (2n^4 - 6(-1)^nn^2 - 2n^2 + 3*(-1)^n - 3)/96.

A325517 a(n) = n((2n + 1)(2n^2 + 2n + 3) - 3(-1)^n)/24.

A325656 a(n) = (1/24)n((4n + 3)(2n^2 + 1) - 3(-1)^n).

A374708 Triangle T read by rows: T(n,k) = (n - k)n(4n^2 - 4nk + 2k^2 - 1 + (-1)^k)/4, with 0 <= k < n.