A083392 -id:A083392 - cp's OEIS frontend

A338456 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).

1, 1, 4, 45, 968, 34265, 1799748, 131572357, 12770710096, 1589142683313, 246658484353100

Offset: 0

Stefano Spezia, Oct 28 2020

			a(2) = 4 because the hafnian of
0  1  1  2
1  0  1  1
1  1  0  1
2  1  1  0
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 4.

Wikipedia, Hafnian
Wikipedia, Symmetric matrix
Wikipedia, Toeplitz Matrix

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)).
Cf. A202038, A336114, A336286, A336400.

k[i_]:=Floor[i/2]; M[i_, j_, n_]:=Part[Part[ToeplitzMatrix[Array[k, n]], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 5, 0]

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; }
a(n) = {my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]);); s/(n!*2^n);} \\ Michel Marcus, Nov 11 2020

a(5) from Michel Marcus, Nov 11 2020

a(6)-a(10) from Pontus von Brömssen, Oct 14 2023

A122432 Riordan array (1/(1+x)^3,x).

Original entry on oeis.org

1, -3, 1, 6, -3, 1, -10, 6, -3, 1, 15, -10, 6, -3, 1, -21, 15, -10, 6, -3, 1, 28, -21, 15, -10, 6, -3, 1, -36, 28, -21, 15, -10, 6, -3, 1, 45, -36, 28, -21, 15, -10, 6, -3, 1, -55, 45, -36, 28, -21, 15, -10

Offset: 0

Paul Barry, Sep 04 2006

Sequence array for (-1)^n*C(n+2,2). Inverse of A122431. Row sums are -A083392(n+1). Antidiagonal sums are (-1)^n*A002623(n).
Call the unsigned version of this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/
having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A127893. - Peter Bala, Jul 22 2014
From Wolfdieter Lang, Apr 05 2020: (Start)
Triangle T(n, k) has the k=0 column (-1)^n*A000217(n+1) = (-1)^n*binomial(n+2, 2), then repeated and down-shifted.
The unsigned triangle, i.e., Tup(n, k) := (-1)^(n-k)*T(n-1,k-1) = binomial(n-k+2, 2) with n >= 1, k = 1..n, gives the number of triangles of length k (in some units), for k = 1..n, in the matchstick arrangement (or tower of cards, with n cards as basis) with an enclosing triangle of length n, but only triangles with orientation (up) like the enclosing triangle are counted. The total number of matchsticks (cards) is 3*A000217(n). (See the comment by Andrew Howroyd in A085691). Recurrence: Tup(n, k) = 0 for n < k, Tup(1, 1) = 1, and Tup(n, k) = Tup(n-1, k) + n - k + 1, for n >= 2, k = 1..n. Row sums give A000292(n). (End)

			The triangle T(n, k) begins:
n\k  0   1   2   3   4   5   6  7  8  9 ...
-------------------------------------------
0:   1
1  :-3   1
2:   6  -3   1
3: -10   6  -3   1
4:  15 -10   6  -3   1
5; -21  15 -10   6  -3   1
6:  28 -21  15 -10   6  -3   1
7: -36  28 -21  15 -10   6  -3  1
8:  45 -36  28 -21  15 -10   6 -3  1
9: -55  45 -36  28 -21  15 -10  6 -3  1
... reformattet by - _Wolfdieter Lang_, Apr 05 2020

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000217, A000292, A083392, A085691, A122431, A127893.

/* As triangle */ [[(-1)^(n-k)*Binomial(n-k+2, 2): k in [1..n]]: n in [1..10]]; // G. C. Greubel, Oct 29 2017

Table[(-1)^(n - k)*Binomial[n - k + 2, 2], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 29 2017 *)

for(n=0,10, for(k=0,n, print1((-1)^(n-k)*binomial(n-k+2,2), ", "))) \\ G. C. Greubel, Oct 29 2017

Number triangle T(n, k) = [k<=n]*(-1)^(n-k)*binomial(n-k+2, 2).

Recurrence: T(n, k) = - T(n-1, k) + (-1)^(n-k)*(n-k+1), for n >= 0, and k = 0..n. - Wolfdieter Lang, Apr 06 2020

A332566 a(n) is the permanent of an n X n symmetric Toeplitz matrix M(n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).

Original entry on oeis.org

1, 0, 1, 2, 16, 150, 2333, 45840, 1227816, 40715300, 1701223409, 84902728550, 5108474886424, 357837483830570, 29336856811970045, 2745407159100236484, 294324995624694053072, 35473014438701226021416, 4818705384665419284918401, 727012502373285844943278058, 122057159014198483893887865744

Offset: 0

Stefano Spezia, Feb 16 2020

nonn

			For n = 4 the matrix M(4) is
    0 1 1 2
    1 0 1 1
    1 1 0 1
    2 1 1 0
with permanent a(4) = 16.

Vaclav Kotesovec, Table of n, a(n) for n = 0..35
Wikipedia, Symmetric matrix
Wikipedia, Toeplitz Matrix

Cf. A004526, A083392, A331491, A333119.

nmax:=20; k[i_]:=Floor[i/2];a[n_]:=If[n==0,1,Permanent[ToeplitzMatrix[Array[k, n], Array[k, n]]]]; Table[a[n],{n,0,nmax}]

tm(n) = {my(m = matrix(n, n, i, j, if (i==1, floor(j/2), if (j==1, floor(i/2))))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
a(n) = matpermanent(tm(n));

A266085 Alternating sum of heptagonal numbers.

Original entry on oeis.org

0, -1, 6, -12, 22, -33, 48, -64, 84, -105, 130, -156, 186, -217, 252, -288, 328, -369, 414, -460, 510, -561, 616, -672, 732, -793, 858, -924, 994, -1065, 1140, -1216, 1296, -1377, 1462, -1548, 1638, -1729, 1824, -1920, 2020, -2121, 2226, -2332, 2442, -2553

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Heptagonal Number
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A000566, A002413, A006578, A008728, A035608, A083392, A089594.

Unsigned terms give antidiagonal sums of A204154. - Nathaniel J. Strout, Nov 14 2019

[((10*n^2+4*n-3)*(-1)^n+3)/8: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

R:=PowerSeriesRing(Integers(), 50); [0] cat  Coefficients(R!(-x*(1 - 4*x)/((1 - x)*(1 + x)^3))); // Marius A. Burtea, Nov 13 2019

Table[((10 n^2 + 4 n - 3) (-1)^n + 3)/8, {n, 0, 50}]
CoefficientList[Series[(x - 4 x^2)/(x^4 + 2 x^3 - 2 x - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)
LinearRecurrence[{-2,0,2,1},{0,-1,6,-12},60] (* Harvey P. Dale, Jan 26 2023 *)

x='x+O('x^100); concat(0, Vec(-x*(1-4*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

G.f.: -x*(1 - 4*x)/((1 - x)*(1 + x)^3).
a(n) = ((10*n^2 + 4*n - 3)*(-1)^n + 3)/8.
a(n) = Sum_{k = 0..n} (-1)^k*A000566(k).
Lim_{n -> infinity} a(n + 1)/a(n) = -1.
a(n) = (-1)^n*A008728(5*n-5) for n>0. - Bruno Berselli, Dec 21 2015
E.g.f.: (1/8)*exp(-x)*(-3 + 3*exp(2*x) - 14*x + 10*x^2). - Stefano Spezia, Nov 13 2019

A331491 a(n) is the permanent of a 2n X 2n antisymmetric Toeplitz matrix M(2n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).

Original entry on oeis.org

1, -1, 8, -965, 301864, -276973609, 529706205072, -1976989515848629, 12817424808315680000, -136266429300554940901097, 2240244443768853657066332152, -54675928167021488863788002983045, 1910142516402733768189592370043464696, -92787876901046051283841308281722409846473

Offset: 0

Stefano Spezia, Jan 18 2020

sign

Conjecture: for n > 0, det(M(2n)) = n^2 = A000290(n) with det(M(0)) = 1.

			For n = 2 the matrix M(4) is
   0  1  1  2
  -1  0  1  1
  -1 -1  0  1
  -2 -1 -1  0
with permanent a(2) = 8.

Vaclav Kotesovec, Table of n, a(n) for n = 0..18
Wikipedia, Skew-symmetric matrix
Wikipedia, Toeplitz Matrix

Cf. A000290, A004526, A083392.

nmax:=13; k[i_]:=Floor[i/2]; a[n_]:=If[n==0,1,Permanent[ToeplitzMatrix[-Array[k, n], Array[k, n]]]]; Table[a[2n],{n,0,nmax}]

tm(n) = {my(m = matrix(n, n, i, j, if (i==1, floor(j/2), if (j==1, -floor(i/2))))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m; }
a(n) = matpermanent(tm(2*n));

A266086 Alternating sum of 9-gonal (or nonagonal) numbers.

Original entry on oeis.org

0, -1, 8, -16, 30, -45, 66, -88, 116, -145, 180, -216, 258, -301, 350, -400, 456, -513, 576, -640, 710, -781, 858, -936, 1020, -1105, 1196, -1288, 1386, -1485, 1590, -1696, 1808, -1921, 2040, -2160, 2286, -2413, 2546, -2680, 2820, -2961, 3108, -3256, 3410

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Nonagonal Number
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A001106, A006578, A007584, A035608, A083392, A089594.

[(14*(-1)^n*n^2 + 4*(-1)^n*n - 5*(-1)^n + 5)/8: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

Table[((14 n^2 + 4 n - 5) (-1)^n + 5)/8, {n, 0, 44}]
CoefficientList[Series[(x - 6 x^2)/(x^4 + 2 x^3 - 2 x - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)

x='x+O('x^100); concat(0, Vec(-x*(1-6*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

G.f.: -x*(1 - 6*x)/((1 - x)*(1 + x)^3).
a(n) = ((14*n^2 + 4*n - 5)*(-1)^n + 5)/8.
a(n) = Sum_{k = 0..n} (-1)^k*A001106(k).
Lim_{n -> infinity} a(n + 1)/a(n) = -1.

A266087 Alternating sum of 11-gonal (or hendecagonal) numbers.

Original entry on oeis.org

0, -1, 10, -20, 38, -57, 84, -112, 148, -185, 230, -276, 330, -385, 448, -512, 584, -657, 738, -820, 910, -1001, 1100, -1200, 1308, -1417, 1534, -1652, 1778, -1905, 2040, -2176, 2320, -2465, 2618, -2772, 2934, -3097, 3268, -3440, 3620, -3801, 3990, -4180

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A006578, A007586, A035608, A051682, A083392, A089594.

[(18*(-1)^n*n^2 + 4*(-1)^n*n - 7*(-1)^n + 7)/8: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

Table[((18 n^2 + 4 n - 7) (-1)^n + 7)/8, {n, 0, 43}]
CoefficientList[Series[(x - 8 x^2)/(x^4 + 2 x^3 - 2 x - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)
Accumulate[Times@@@Partition[Riffle[PolygonalNumber[11,Range[0,50]],{1,-1},{2,-1,2}],2]] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{-2,0,2,1},{0,-1,10,-20},50] (* Harvey P. Dale, Aug 27 2019 *)

x='x+O('x^100); concat(0, Vec(-x*(1-8*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

G.f.: -x*(1 - 8*x)/((1 - x)*(1 + x)^3).
a(n) = ((18*n^2 + 4*n - 7)*(-1)^n + 7)/8.
a(n) = Sum_{k = 0..n} (-1)^k*A051682(k).
Lim_{n -> infinity} a(n + 1)/a(n) = -1.
E.g.f.: (1/4)*(9*x^2 - 11*x)*cosh(x) - (1/4)*(9*x^2 - 11*x - 7)*sinh(x). - G. C. Greubel, Jan 27 2016

A266088 Alternating sum of 12-gonal (or dodecagonal) numbers.

Original entry on oeis.org

0, -1, 11, -22, 42, -63, 93, -124, 164, -205, 255, -306, 366, -427, 497, -568, 648, -729, 819, -910, 1010, -1111, 1221, -1332, 1452, -1573, 1703, -1834, 1974, -2115, 2265, -2416, 2576, -2737, 2907, -3078, 3258, -3439, 3629, -3820, 4020, -4221, 4431, -4642

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

More generally, the ordinary generating function for the alternating sum of k-gonal numbers is -x*(1 - (k - 3)*x)/((1 - x)*(1 + x)^3).

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A006578, A007587, A035608, A051624, A083392, A089594.

[1+(-1)^n*(5*n^2+n-2)/2: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

Table[1 + (-1)^n (5 n^2 + n - 2)/2, {n, 0, 43}]
CoefficientList[Series[-x (1 - 9 x)/((1 - x) (1 + x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)

x='x+O('x^100); concat(0, Vec(-x*(1-9*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

G.f.: -x*(1 - 9*x)/((1 - x)*(1 + x)^3).
a(n) = 1 + (-1)^n*(5*n^2 + n - 2)/2.
a(n) = Sum_{k = 0..n} (-1)^k*A051624(k).
Lim_{n -> infinity} a(n + 1)/a(n) = -1.

A267831 Expansion of (1 + 5x - 7x^2 - 3x^3)/((1 - x)(1 + x^2)^2).

Original entry on oeis.org

1, 6, -3, -16, 1, 22, -3, -32, 1, 38, -3, -48, 1, 54, -3, -64, 1, 70, -3, -80, 1, 86, -3, -96, 1, 102, -3, -112, 1, 118, -3, -128, 1, 134, -3, -144, 1, 150, -3, -160, 1, 166, -3, -176, 1, 182, -3, -192, 1, 198, -3, -208, 1, 214, -3, -224, 1, 230, -3, -240, 1, 246, -3, -256, 1, 262, -3

Offset: 0

Ilya Gutkovskiy, Jan 21 2016

			a(0) = (0 + 1) = 1;
a(1) = (0 + 1) + (2 + 3) = 6;
a(2) = (0 + 1) + (2 + 3) - (4 + 5) = -3;
a(3) = (0 + 1) + (2 + 3) - (4 + 5) - (6 + 7) = -16;
a(4) = (0 + 1) + (2 + 3) - (4 + 5) - (6 + 7) + (8 + 9) = 1;
a(5) = (0 + 1) + (2 + 3) - (4 + 5) - (6 + 7) + (8 + 9) + (10 + 11) = 22, etc.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-2,2,-1,1).

Cf. A000217, A001477, A042948, A004767, A021913, A083392.

&cat [[1,2*(8*n+3),-3,-16*(n+1)]: n in [0..17]]; // Bruno Berselli, Jan 21 2016

I:=[1,6,-3,-16,1]; [n le 5 select I[n] else Self(n-1)-2*Self(n-2)+2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..70]]; // Vincenzo Librandi, Jan 21 2016

Table[Sum[(-1)^(1/2 - Sin[(2 k + 1) (Pi/4)]/Sqrt[2]) (4 k + 1), {k, 0, n}], {n, 0, 70}]
Table[-8 Floor[(n - 3)/4]^2 - 21 Floor[(n - 3)/4] - 8 Floor[(n - 2)/4]^2 + 8 Floor[(n - 1)/4]^2 + 8 Floor[n/4]^2 - 17 Floor[(n - 2)/4] + 13 Floor[(n - 1)/4] + 9 Floor[n/4] - 16, {n, 0, 70}]
Table[-1 + (2 + (1 + 4 n) (1 - (-1)^n)/2) (-1)^((n - 1) n/2), {n, 0, 70}] (* Bruno Berselli, Jan 21 2016 *)
LinearRecurrence[{1, -2, 2, -1, 1}, {1, 6, -3, -16, 1}, 70] (* Vincenzo Librandi, Jan 21 2016 *)

Vec((1 + 5*x - 7*x^2 - 3*x^3)/((1 - x)*(1 + x^2)^2) + O(x^70)) \\ Michel Marcus, Oct 29 2017

[-1+(2+(1+4*n)*(1-(-1)^n)/2)*(-1)^((n-1)*n/2) for n in (0..70)]; # Bruno Berselli, Jan 21 2016

G.f.: (1 + 5*x - 7*x^2 - 3*x^3)/((1 - x)*(1 + x^2)^2).
a(n) = Sum_{k = 0..n} (-1)^(1/2 - sin((2*k + 1)*Pi/4)/sqrt(2))*(4*k + 1).
a(n) = -8*floor((n - 3)/4)^2 - 21*floor((n - 3)/4) - 8*floor((n - 2)/4)^2 + 8*floor((n - 1)/4)^2 + 8*floor(n/4)^2 - 17*floor((n - 2)/4) + 13*floor((n - 1)/4) + 9*floor(n/4) - 16.
a(n) = -1 + (2 + (1 + 4*n)*(1 - (-1)^n)/2)*(-1)^((n-1)*n/2). Therefore: a(4*k) = 1, a(4*k+1) = 2*(8*k+3), a(4*k+2) = -3, a(4*k+3) = -16*(k+1). [Bruno Berselli, Jan 21 2016]

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).

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 12 2020

			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.

Cf. A004526.
Cf. A002620, A033638, A212964, A290743.
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)).

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]]

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

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).

cp's OEIS Frontend

A338456 a(n) is the hafnian of a symmetric Toeplitz matrix M(2n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).

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A122432 Riordan array (1/(1+x)^3,x).

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Magma

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A332566 a(n) is the permanent of an n X n symmetric Toeplitz matrix M(n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).

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A266085 Alternating sum of heptagonal numbers.

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Magma

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A331491 a(n) is the permanent of a 2n X 2n antisymmetric Toeplitz matrix M(2n) whose first row consists of a single zero followed by successive positive integers repeated (A004526).

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A266086 Alternating sum of 9-gonal (or nonagonal) numbers.

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A266087 Alternating sum of 11-gonal (or hendecagonal) numbers.

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A267831 Expansion of (1 + 5x - 7x^2 - 3x^3)/((1 - x)(1 + x^2)^2).