A060025 -id:A060025 - cp's OEIS frontend

A060028 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 9.

1, 0, 1, 1, 2, 2, 4, 4, 7, 7, 10, 11, 16, 16, 22, 23, 29, 29, 36, 34, 41, 37, 40, 32, 32, 14, 6, -22, -44, -90, -130, -203, -270, -378, -487, -642, -803, -1027, -1260, -1568, -1899, -2320, -2774, -3342, -3955, -4706, -5526, -6507, -7579, -8854, -10243, -11872, -13656

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+8 into 8 parts and the number of partitions of n+8 into 9 parts. - Wesley Ivan Hurt, Apr 16 2019

Ray Chandler, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, -1, 0, 2, 1, 1, 1, 0, -1, -1, -1, -2, -1, -1, 1, 1, 2, 1, 1, 1, 0, -1, -1, -1, -2, 0, 1, 0, 0, 1, 0, 1, 0, 0, -1, -1, 1).

Cf. A026814, A026815.

Cf. For other values of N: A060022 (N=3), A060023 (N=4), A060024 (N=5), A060025 (N=6), A060026 (N=7), A060027 (N=8), this sequence (N=9), A060029 (N=10).

With[{den=Times@@Table[(1-x^n),{n,9}]},CoefficientList[Series[(1-x-x^9)/ den,{x,0,60}],x]] (* Harvey P. Dale, May 22 2012 *)

a(n) = A026814(n+8) - A026815(n+8). - Wesley Ivan Hurt, Apr 16 2019

A060022 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 3.

Original entry on oeis.org

1, 0, 1, 0, 0, -1, -1, -3, -3, -5, -6, -8, -9, -12, -13, -16, -18, -21, -23, -27, -29, -33, -36, -40, -43, -48, -51, -56, -60, -65, -69, -75, -79, -85, -90, -96, -101, -108, -113, -120, -126, -133, -139, -147, -153, -161, -168, -176, -183, -192, -199, -208, -216, -225, -233, -243, -251, -261, -270, -280

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference between the number of partitions of n+2 into 2 parts and the number of partitions of n+2 into 3 parts. - Wesley Ivan Hurt, Apr 16 2019

Colin Barker, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A004526, A069905.

Cf. For other values of N: this sequence (N=3), A060023 (N=4), A060024 (N=5), A060025 (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

Vec((1 - x - x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)) + O(x^40)) \\ Colin Barker, Apr 17 2019

a(n) = A004526(n+2) - A069905(n+2). - Wesley Ivan Hurt, Apr 16 2019
From Colin Barker, Apr 17 2019: (Start)
G.f.: (1 - x - x^3) / ((1 - x)^3*(1 + x)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n>5.
(End)

A060023 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 4.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, -1, -1, -3, -4, -7, -8, -13, -15, -20, -24, -31, -35, -44, -50, -60, -68, -80, -89, -104, -115, -131, -145, -164, -179, -201, -219, -243, -264, -291, -314, -345, -371, -404, -434, -471, -503, -544, -580, -624, -664, -712, -755, -808, -855, -911, -963, -1024, -1079, -1145

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+3 into 3 parts and the number of partitions of n+3 into 4 parts. - Wesley Ivan Hurt, Apr 16 2019

Colin Barker, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -2, 0, 0, 1, 1, -1).

Cf. A026810, A069905.

Cf. For other values of N: A060022 (N=3), this sequence (N=4), A060024 (N=5), A060025 (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

I:=[1,0,1,1,1,0,1,-1,-1,-3]; [n le 10 select I[n] else Self(n-1)+Self(n-2)-2*Self(n-5)+Self(n-8)+Self(n-9)-Self(n-10): n in [1..60]]; // Vincenzo Librandi, Jun 23 2015

CoefficientList[Series[(1-x-x^4)/Times@@(1-x^Range[4]),{x,0,60}],x] (* or *) LinearRecurrence[{1,1,0,0,-2,0,0,1,1,-1},{1,0,1,1,1,0,1,-1,-1,-3},70] (* Harvey P. Dale, Jan 14 2015 *)

Vec((1 - x - x^4) / ((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^40)) \\ Colin Barker, Apr 17 2019

a(n) = A069905(n+3) - A026810(n+3). - Wesley Ivan Hurt, Apr 16 2019
From Colin Barker, Apr 17 2019: (Start)
G.f.: (1 - x - x^4) / ((1 - x)^4*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n>9.
(End)

A060024 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 5.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 1, 2, 0, 0, -3, -3, -8, -10, -16, -20, -29, -35, -47, -56, -72, -85, -105, -122, -148, -171, -202, -231, -270, -306, -353, -397, -453, -507, -573, -637, -715, -791, -881, -970, -1075, -1178, -1298, -1417, -1554, -1691, -1846, -2001, -2177, -2353, -2550, -2748, -2969

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+4 into 4 parts and the number of partitions of n+4 into 5 parts. - Wesley Ivan Hurt, Apr 16 2019

Colin Barker, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1).

Cf. A026810, A026811.

Cf. For other values of N: A060022 (N=3), A060023 (N=4), this sequence (N=5), A060025 (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

CoefficientList[Series[(1-x-x^5)/(Times@@(1-x^Range[5])),{x,0,60}],x] (* or *) LinearRecurrence[{1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1},{1,0,1,1,2,1,2,1,2,0,0,-3,-3,-8,-10},60] (* Harvey P. Dale, Dec 21 2015 *)

Vec((1 - x + x^2)*(1 - x^2 - x^3) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)) + O(x^40)) \\ Colin Barker, Apr 17 2019

a(0)=1, a(1)=0, a(2)=1, a(3)=1, a(4)=2, a(5)=1, a(6)=2, a(7)=1, a(8)=2, a(9)=0, a(10)=0, a(11)=-3, a(12)=-3, a(13)=-8, a(14)=-10, a(n) = a(n-1)+ a(n-2)-a(n-5)-a(n-6)-a(n-7)+a(n-8)+a(n-9)+a(n-10)-a(n-13)- a(n-14)+ a(n-15). - Harvey P. Dale, Dec 21 2015
a(n) = A026810(n+4) - A026811(n+4). - Wesley Ivan Hurt, Apr 16 2019
G.f.: (1 - x + x^2)*(1 - x^2 - x^3) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Apr 17 2019

A060026 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 7.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 3, 5, 5, 7, 6, 9, 6, 8, 5, 5, -1, -2, -13, -18, -33, -45, -68, -86, -121, -151, -198, -244, -310, -373, -464, -553, -671, -793, -948, -1107, -1309, -1517, -1771, -2039, -2360, -2696, -3098, -3519, -4011, -4534, -5137, -5774, -6508, -7283, -8163, -9099, -10153, -11269

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+6 into 6 parts and the number of partitions of n+6 into 7 parts. - Wesley Ivan Hurt, Apr 16 2019

Ray Chandler, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, -1, 0, 1, 1, 2, 0, 0, 0, -2, -1, -1, 0, 1, 1, 0, 1, 0, 0, -1, -1, 1).

Cf. A026812, A026813.

Cf. For other values of N: A060022 (N=3), A060023 (N=4), A060024 (N=5), A060025 (N=6), this sequence (N=7), A060027 (N=8), A060028 (N=9), A060029 (N=10).

With[{nn=7},CoefficientList[Series[(1-x-x^nn)/Times@@(1-x^Range[nn]),{x,0,60}],x]] (* Harvey P. Dale, May 15 2016 *)

a(n) = A026812(n+6) - A026813(n+6). - Wesley Ivan Hurt, Apr 16 2019

A060027 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 8.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 6, 6, 9, 9, 13, 12, 16, 15, 18, 15, 18, 12, 12, 2, -3, -20, -31, -59, -81, -122, -160, -222, -280, -369, -457, -581, -708, -878, -1055, -1286, -1528, -1833, -2158, -2559, -2985, -3504, -4059, -4721, -5433, -6271, -7172, -8224, -9355, -10660, -12067

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+7 into 7 parts and the number of partitions of n+7 into 8 parts. - Wesley Ivan Hurt, Apr 16 2019

Ray Chandler, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, -1, 0, 1, 2, 1, 0, 1, -1, -1, -2, -1, -1, 1, 0, 1, 2, 1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 1, -1).

Cf. A026813, A026814.

Cf. For other values of N: A060022 (N=3), A060023 (N=4), A060024 (N=5), A060025 (N=6), A060026 (N=7), this sequence (N=8), A060028 (N=9), A060029 (N=10).

With[{nn=8},CoefficientList[Series[(1-x-x^nn)/Times@@(1-x^Range[nn]),{x,0,60}],x]] (* Harvey P. Dale, May 15 2016 *)

a(n) = A026813(n+7) - A026814(n+7). - Wesley Ivan Hurt, Apr 16 2019

A060029 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 10.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 11, 12, 18, 19, 26, 29, 37, 40, 51, 53, 65, 68, 79, 80, 92, 87, 94, 84, 82, 58, 45, -1, -36, -109, -180, -297, -413, -594, -780, -1042, -1325, -1704, -2112, -2647, -3228, -3961, -4772, -5769, -6867, -8206, -9682, -11446, -13402, -15710

Offset: 0

N. J. A. Sloane, Mar 17 2001

Difference of the number of partitions of n+9 into 9 parts and the number of partitions of n+9 into 10 parts. - Wesley Ivan Hurt, Apr 16 2019

Ray Chandler, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, 0, 0, -1, 1, 1, 1, 2, 0, 0, -1, -1, -1, -1, -3, 0, 0, 1, 1, 2, 2, 1, 1, 0, 0, -3, -1, -1, -1, -1, 0, 0, 2, 1, 1, 1, -1, 0, 0, 0, -1, 0, -1, 0, 0, 1, 1, -1).

Cf. A026815, A026816.

Cf. For other values of N: A060022 (N=3), A060023 (N=4), A060024 (N=5), A060025 (N=6), A060026 (N=7), A060027 (N=8), A060028 (N=9), this sequence (N=10).

CoefficientList[Series[(1-x-x^10)/Times@@(1-x^Range[10]),{x,0,60}],x] (* Harvey P. Dale, May 15 2016 *)

a(n) = A026815(n+9) - A026816(n+9). - Wesley Ivan Hurt, Apr 16 2019

cp's OEIS Frontend

A060028 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 9.

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A060022 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 3.

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A060023 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 4.

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A060024 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 5.

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A060026 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 7.

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A060027 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 8.

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A060029 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 10.

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