A006578 -id:A006578 - cp's OEIS frontend

A102214 Expansion of (1 + 4x + 4x^2)/((1+x)*(1-x)^3).

1, 6, 16, 30, 49, 72, 100, 132, 169, 210, 256, 306, 361, 420, 484, 552, 625, 702, 784, 870, 961, 1056, 1156, 1260, 1369, 1482, 1600, 1722, 1849, 1980, 2116, 2256, 2401, 2550, 2704, 2862, 3025, 3192, 3364, 3540, 3721, 3906, 4096, 4290, 4489, 4692, 4900

Offset: 0

Creighton Dement, Feb 17 2005

A floretion-generated sequence.
a(n) gives the number of triples (x,y,x+y) with positive integers satisfying x < y and x + y <= 3*n. - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
Number of different partitions of numbers x + y = z such that {x,y,z} are integers {1,2,3,...,3n} and z > y > x. - Artur Jasinski, Feb 09 2010
Second bisection preceded by zero is A152743. - Bruno Berselli, Oct 25 2011
a(n) has no final digit 3, 7, 8. - Paul Curtz, Mar 04 2020
One odd followed by three evens.
From Paul Curtz, Mar 06 2020: (Start)
b(n) = 0, 1, 6, 16,  30, 49, ... = 0, a(n).
(  25, 12,  4, 0,  1,  6,  16, 30, ...
  -13, -8, -4  1,  5, 10,  14, 19, ...
    5,  4,  5, 4,  5,  4,   5,  4, ... .)
b(-n) = 0, 4, 12, 25, 42, 64, 90, 121, ... .
A154589(n) are in the main diagonal of b(n) and b(-n). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A016778, A038764, A001859, A006578, A069905, A077043, A274221, A330707.

Cf. A000326, A016789, A152743, A001651, A032766, A047237, A047251, A154589.

[(6*n*(3*n+4)+(-1)^n+7)/8: n in [0..60]]; // Vincenzo Librandi, Oct 26 2011

aa = {}; Do[i = 0; Do[Do[Do[If[x + y == z, i = i + 1], {x, y + 1, 3 n}], {y, 1, 3 n}], {z, 1, 3 n}]; AppendTo[aa, i], {n, 1, 20}]; aa (* Artur Jasinski, Feb 09 2010 *)

a(n)=(6*n*(3*n+4)+(-1)^n+7)/8 \\ Charles R Greathouse IV, Apr 16 2020

G.f.: -(4*x^2 + 4*x + 1)/((x+1)*(x-1)^3) = (1+2*x)^2/((1+x)*(1-x)^3).
a(2n) = A016778(n) = (3n+1)^2.
a(n) + a(n+1) = A038764(n+1).
a(n) = floor( (3*n+2)/2 ) * ceiling( (3*n+2)/2 ). - Marcus Schmidt (marcus-schmidt(AT)gmx.net), Jan 13 2006
a(n) = (6*n*(3*n+4) + (-1)^n+7)/8. - Bruno Berselli, Oct 25 2011
a(n) = A198392(n) + A198392(n-1). - Bruno Berselli, Nov 06 2011
From Paul Curtz, Mar 04 2020: (Start)
a(n) = A006578(n) + A001859(n) + A077043(n+1).
a(n) = A274221(2+2*n).
a(20+n) - a(n) = 30*(32+3*n).
a(1+2*n) = 3*(1+n)*(2+3*n).
a(n) = A047237(n) * A047251(n).
a(n) = A001651(n+1) * A032766(n).(End)
E.g.f.: ((4 + 21*x + 9*x^2)*cosh(x) + 3*(1 + 7*x + 3*x^2)*sinh(x))/4. - Stefano Spezia, Mar 04 2020

Definition rewritten by Bruno Berselli, Oct 25 2011

A330707 a(n) = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4.

Original entry on oeis.org

0, 1, 3, 7, 13, 20, 28, 38, 50, 63, 77, 93, 111, 130, 150, 172, 196, 221, 247, 275, 305, 336, 368, 402, 438, 475, 513, 553, 595, 638, 682, 728, 776, 825, 875, 927, 981, 1036, 1092, 1150, 1210, 1271, 1333, 1397, 1463

Offset: 0

Paul Curtz, Dec 27 2019

Essentially four odds followed by four evens.
Last digit is neither 4 nor 9.
Essentially twice or twin sequences in the hexagonal spiral from A002265.
                  21  21  21  22  22  22  22
                21  14  14  14  14  15  15  23
              20  13   8   8   8   9   9  15  23
            20  13   8   4   4   4   4   9  15  23
          20  13   7   3   1   1   1   5   9  16  23
        20  13   7   3   1   0   0   2   5  10  16  24
          19  12   7   3   0   0   2   5  10  16  24
            19  12   7   3   2   2   5  10  16  24
              19  12   6   6   6   6  10  17  24
                19  12  11  11  11  11  17  25
                  18  18  18  18  17  17  25
.
There are 12 twin sequences. 6 of them (A001859, A006578, A077043, A231559, A024219, A281026) are in the OEIS. a(n) is the seventh.
  0, 1, 3, 7, 13, 20, 28, 38, 50, ...
  1, 2, 4, 6,  7,  8, 10, 12, 13, ...
  1, 2, 2, 1,  1,  2,  2,  1,  1, ... period 4. See A014695.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A001859, A002265, A006578, A014695, A016921, A024219, A033579, A077043, A231559, A281026.

[(3*n^2+n-1+ (-1)^Floor(n/2))/4: n in [0..60]]; // G. C. Greubel, Dec 30 2019

seq((3*n^2+n-1+sqrt(2)*sin((2*n+1)*Pi/4))/4, n = 0..60); # G. C. Greubel, Dec 30 2019

LinearRecurrence[{3,-4,4,-3,1}, {0,1,3,7,13}, 60] (* Amiram Eldar, Dec 27 2019 *)

concat(0, Vec(x*(1 + 2*x^2) / ((1 - x)^3*(1 + x^2)) + O(x^60))) \\ Colin Barker, Dec 27 2019

[(3*n^2+n-1+(-1)^floor(n/2))/4 for n in (0..60)] # G. C. Greubel, Dec 30 2019

a(n) = A231559(-n).
a(1+2*n) + a(2+2*n) = A033579(n+1).
a(40+n) - a(n) = 1210, 1270, 1330, 1390, 1450, ... . See 10*A016921(n).
From Colin Barker, Dec 27 2019: (Start)
G.f.: x*(1 + 2*x^2) / ((1 - x)^3*(1 + x^2)).
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>4.
(End)
E.g.f.: (cos(x) + sin(x) + (-1 + 4*x + 3*x^2)*exp(x))/4. - Stefano Spezia, Dec 27 2019
a(n) = ( 3*n^2 + n - 1 + sqrt(2)*sin((2*n+1)*Pi/4) )/4 = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4. - G. C. Greubel, Dec 30 2019

A104567 Triangle read by rows: T(i,j) = i-j+1 if j is odd; T(i,j) = 2(i-j+1) if j is even (1 <= j <= i).

Original entry on oeis.org

1, 2, 2, 3, 4, 1, 4, 6, 2, 2, 5, 8, 3, 4, 1, 6, 10, 4, 6, 2, 2, 7, 12, 5, 8, 3, 4, 1, 8, 14, 6, 10, 4, 6, 2, 2, 9, 16, 7, 12, 5, 8, 3, 4, 1, 10, 18, 8, 14, 6, 10, 4, 6, 2, 2, 11, 20, 9, 16, 7, 12, 5, 8, 3, 4, 1, 12, 22, 10, 18, 8, 14, 6, 10, 4, 6, 2, 2, 13, 24, 11, 20, 9, 16, 7, 12, 5, 8, 3, 4, 1, 14

Offset: 1

Gary W. Adamson, Mar 16 2005

T(i,j) is the (i,j)-entry (1<=j<=i) of the product R*H of the infinite lower triangular matrices R = [1; 1,1; 1,1,1; 1,1,1,1; ...] and H = [1; 1,2; 1,2,1; 1 2,1,2; ...]. Row sums yield A006578. H*R yields A104566. - Emeric Deutsch, Mar 24 2005

			The first few rows are:
  1;
  2, 2;
  3, 4, 1;
  4, 6, 2, 2;

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001082, A006578, A104566, A104568.

Cf. A006578, A104566.

T:=proc(i,j) if j>i then 0 elif j mod 2 = 1 then i-j+1 elif j mod 2 = 0 then 2*(i-j+1) else fi end: for i from 1 to 14 do seq(T(i,j),j=1..i) od; # yields sequence in triangular form # Emeric Deutsch, Mar 24 2005

Table[If[OddQ[j],i-j+1,2(i-j+1)],{i,20},{j,i}]//Flatten (* Harvey P. Dale, Sep 03 2018 *)

T(i,j) = i-j+1 if j is odd; T(i,j) = 2(i-j+1) if j is even (1 <= j <= i). - Emeric Deutsch, Mar 24 2005

More terms from Emeric Deutsch, Mar 24 2005

A254707 Expansion of (1 + 2x^2) / ((1 - x^2)^2 (1 - x^3) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 0, 4, 1, 8, 4, 15, 8, 25, 15, 38, 25, 55, 38, 77, 55, 103, 77, 135, 103, 173, 135, 217, 173, 268, 217, 327, 268, 393, 327, 468, 393, 552, 468, 645, 552, 748, 645, 862, 748, 986, 862, 1122, 986, 1270, 1122, 1430, 1270, 1603, 1430, 1790, 1603, 1990, 1790

Offset: 0

Michael Somos, Feb 06 2015

The number of quadruples of integers [x, u, v, w] which satisfy x > u > v > w >=0, n+7 = x+u, u+v != x+w, and x+u+v+w is even.

			G.f. = 1 + 4*x^2 + x^3 + 8*x^4 + 4*x^5 + 15*x^6 + 8*x^7 + 25*x^8 + ...

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

Cf. A006578, A254594, A254708.

a[ n_] := Quotient[ n^3 + If[ OddQ[n], 8 n^2 + 9 n + 18, 17 n^2 + 84 n + 148], 96];
a[ n_] := Module[{m = n}, SeriesCoefficient[ If[ n < 0, m = -9 - n; -2 - x^2, 1 + 2 x^2] / ((1 - x^2)^2 (1 - x^3) (1 - x^4)), {x, 0, m}]];
a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 7, u + v != x + w, x + u + v + w == 2 k}, {x, u, v, w, k}, Integers, 10^9];

{a(n) = (n^3 + if( n%2, 8*n^2 + 9*n + 18, 17*n^2 + 84*n + 148)) \ 96};

{a(n) = polcoeff( if( n<0, n = -9-n; -2 - x^2, 1 + 2*x^2) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};

G.f.: (1 + 2*x^2) / (1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11).
0 = a(n) + a(n+1) - a(n+2) - 2*a(n+3) - 2*a(n+4) + 2*a(n+6) + 2*a(n+7) + a(n+8) - a(n+9) - a(n+10) + 3 for all n in Z.
a(n+3) - a(n) = 0 if n even else A006578((n+5)/2) for all n in Z.
a(n+2) = 2*A254594(n) + A254594(n+2) for all n in Z.
a(n) = -A254708(-9 - n) for all n in Z.

A104566 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product H*R of the infinite lower triangular matrices H = [1; 1,2; 1,2,1; 1 2,1,2; ...] and R = [1; 1,1; 1,1,1; 1,1,1,1; ...].

Original entry on oeis.org

1, 3, 2, 4, 3, 1, 6, 5, 3, 2, 7, 6, 4, 3, 1, 9, 8, 6, 5, 3, 2, 10, 9, 7, 6, 4, 3, 1, 12, 11, 9, 8, 6, 5, 3, 2, 13, 12, 10, 9, 7, 6, 4, 3, 1, 15, 14, 12, 11, 9, 8, 6, 5, 3, 2, 16, 15, 13, 12, 10, 9, 7, 6, 4, 3, 1, 18, 17, 15, 14, 12, 11, 9, 8, 6, 5, 3, 2, 19, 18, 16, 15, 13, 12, 10, 9, 7, 6, 4, 3, 1

Offset: 1

Gary W. Adamson, Mar 15 2005

			The first few rows are
  1;
  3, 2;
  4, 3, 1;
  6, 5, 3, 2;
  ...

Row sums yield A001082.
Columns 1, 3, 5, ... (starting at the diagonal entry) yield A032766.
Columns 2, 4, 6, ... (starting at the diagonal entry) yield A045506.
Row sums = 1, 5, 8, 16, 21, ... (generalized octagonal numbers, A001082). A006578(2n-1) = A001082(2n).

T:=proc(i,j) if j>i then 0 elif i mod 2 = 1 and j mod 2 = 1 then 3*(i-j)/2+1 elif i mod 2 = 0 and j mod 2 = 0 then 3*(i-j)/2+2 elif i+j mod 2 = 1 then 3*(i-j+1)/2 else fi end: for i from 1 to 14 do seq(T(i,j),j=1..i) od; # yields sequence in triangular form # Emeric Deutsch, Mar 24 2005

For 1 <= j <= i: T(i,j) = 3(i-j+1)/2 if i and j are of opposite parity; T(i,j) = 3(i-j)/2 + 1 if both i and j are odd; T(i,j) = 3(i-j)/2 + 2 if both i and j are even. - Emeric Deutsch, Mar 24 2005

More terms from Emeric Deutsch, Mar 24 2005

A143971 Triangle read by rows, (3n-2) subsequences decrescendo.

Original entry on oeis.org

1, 4, 1, 7, 4, 1, 10, 7, 4, 1, 13, 10, 7, 4, 1, 16, 13, 10, 7, 4, 1, 19, 16, 13, 10, 7, 4, 1, 22, 19, 16, 13, 10, 7, 4, 1, 25, 22, 19, 16, 13, 10, 7, 4, 1, 28, 25, 22, 19, 16, 13, 10, 7, 4, 1, 31, 28, 25, 22, 19, 16, 13, 10, 7, 4, 1

Offset: 1

Gary W. Adamson, Sep 06 2008

Row sums = pentagonal numbers, A000326: (1, 5, 12, 22, 35,...).

The alternating row sums lead to A032766 and the antidiagonal sums to A006578. - Johannes W. Meijer, Sep 05 2013

			Triangle starts
1;
4, 1;
7, 4, 1;
10, 7, 4, 1;
...

Cf. A016777, A000326, A209634.

T := (n, k) -> 3*n-3*k+1: seq(seq(T(n, k), k=1..n), n=1..11); # Johannes W. Meijer, Sep 05 2013

Triangle read by rows, (3n-2) subsequences decrescendo; 1<=k<=n.
(1, 4, 7, 10, 13,...) in every column.
T(n,k) = 3*n - 3*k + 1.

Terms a(17) and a(38) corrected and terms added by Johannes W. Meijer, Sep 05 2013

A209634 Triangle with (1,4,7,10,13,16...,(3*n-2),...) in every column, shifted down twice.

Original entry on oeis.org

1, 4, 7, 1, 10, 4, 13, 7, 1, 16, 10, 4, 19, 13, 7, 1, 22, 16, 10, 4, 25, 19, 13, 7, 1, 28, 22, 16, 10, 4, 31, 25, 19, 13, 7, 1, 34, 28, 22, 16, 10, 4, 37, 31, 25, 19, 13, 7, 1, 40, 34, 28, 22, 16, 10, 4, 43, 37, 31, 25, 19, 13, 7, 1, 46, 40, 34, 28, 22, 16, 10

Offset: 1

Ctibor O. Zizka, Mar 11 2012

OEIS contains a lot of similar sequences, for example A152204, A122196, A173284.
Row sums for this sequence gives A006578.
In general, by given triangle with (A-B,2*A-B,...,A*n-B,...) in every column, shifted down K-times, we have the row sum s(n)= A*(n*n+K*n+nmodK)/(2*K) - B*(n+nmodK)/K. In this sequence K=2,A=3,B=2, in A152204 K=2,A=2,B=1.
No triangle with primes in every column, shifted down by K>=2 in OEIS, no row sums of it in OEIS.
From Johannes W. Meijer, Sep 28 2013: (Start)
Triangle read by rows formed from antidiagonals of triangle A143971.
The alternating row sums equal A004524(n+2) + 2*A004524(n+1).
The antidiagonal sums equal A171452(n+1). (End)

			Triangle:
1
4
7,  1
10, 4
13, 7,  1
16, 10, 4
19, 13, 7,  1
22, 16, 10, 4
25, 19, 13, 7,  1
28, 22, 16, 10, 4
...

Cf. A008315, A011973, A102541, A122196, A122197, A128099, A152198, A152204, A173284, A207538.

Cf. (Related to triangle sums) A006578, A000217, A002620, A004524, A171452.

T := (n, k) -> 3*n - 6*k + 4: seq(seq(T(n, k), k=1..floor((n+1)/2)), n=1..15); # Johannes W. Meijer, Sep 28 2013

From Johannes W. Meijer, Sep 28 2013: (Start)
T(n, k) = 3*n - 6*k + 4, n >= 1 and 1 <= k <= floor((n+1)/2).
T(n, k) = A143971(n-k+1, k), n >= 1 and 1 <= k <= floor((n+1)/2). (End)

A212506 Number of (w,x,y,z) with all terms in {1,...,n} and w<=2x and y<=2z.

Original entry on oeis.org

0, 1, 16, 64, 196, 441, 900, 1600, 2704, 4225, 6400, 9216, 12996, 17689, 23716, 30976, 40000, 50625, 63504, 78400, 96100, 116281, 139876, 166464, 197136, 231361, 270400, 313600, 362404, 416025, 476100, 541696, 614656, 693889, 781456

Offset: 0

Clark Kimberling, May 19 2012

For a guide to related sequences, see A211795.

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A211795.

a:= n-> (n*(n+1)/2+floor(n^2/4))^2:
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2012

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w <= 2 x && y <= 2 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212506 *)

a(n) = 2a(n-1)+2a(n-2)-6a(n-3)+6a(n-5)-2a(n-6)-2a(n-7)+a(n-8).
From Alois P. Heinz, May 31 2012: (Start)
a(n) = A006578(n)^2.
G.f.: x*(14*x+30*x^2+42*x^3+17*x^4+4*x^5+1) / ((x+1)^3*(1-x)^5). (End)

A212507 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y<=2z.

Original entry on oeis.org

0, 1, 12, 56, 168, 399, 810, 1480, 2496, 3965, 6000, 8736, 12312, 16891, 22638, 29744, 38400, 48825, 61236, 75880, 93000, 112871, 135762, 161976, 191808, 225589, 263640, 306320, 353976, 406995, 465750, 530656, 602112, 680561, 766428

Offset: 0

Clark Kimberling, May 19 2012

For a guide to related sequences, see A211795.

Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A211795.

[(2*n*(9*n^3+6*n^2+1)-(2*n-1)*(-1)^n-1)/32: n in [0..34]]; // Bruno Berselli, May 31 2012

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < 2 x && y <= 2 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212507 *)
CoefficientList[Series[x (1 + 2 x) (1 + 7 x + 7 x^2 + 3 x^3)/((1 + x)^2 (1 - x)^5), {x, 0, 34}], x] (* Bruno Berselli, May 31 2012 *)

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).
G.f.: x*(1+2*x)*(1+7*x+7*x^2+3*x^3)/((1+x)^2*(1-x)^5). [Bruno Berselli, May 31 2012]
a(n) = (2*n*(9*n^3+6*n^2+1)-(2*n-1)*(-1)^n-1)/32. [Bruno Berselli, May 31 2012]
a(n) = A006578(n) * A077043(n). - Alois P. Heinz, May 31 2012

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

cp's OEIS Frontend

A102214 Expansion of (1 + 4*x + 4*x^2)/((1+x)*(1-x)^3).

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A330707 a(n) = ( 3*n^2 + n - 1 + (-1)^floor(n/2) )/4.

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A104567 Triangle read by rows: T(i,j) = i-j+1 if j is odd; T(i,j) = 2(i-j+1) if j is even (1 <= j <= i).

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A254707 Expansion of (1 + 2*x^2) / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

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PARI

PARI

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A104566 Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product H*R of the infinite lower triangular matrices H = [1; 1,2; 1,2,1; 1 2,1,2; ...] and R = [1; 1,1; 1,1,1; 1,1,1,1; ...].

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Maple

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A143971 Triangle read by rows, (3n-2) subsequences decrescendo.

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A209634 Triangle with (1,4,7,10,13,16...,(3*n-2),...) in every column, shifted down twice.

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A102214 Expansion of (1 + 4x + 4x^2)/((1+x)*(1-x)^3).

A254707 Expansion of (1 + 2x^2) / ((1 - x^2)^2 (1 - x^3) * (1 - x^4)) in powers of x.