David Neil McGrath - cp's OEIS frontend

A260710 Expansion of 1/(1 - x - x^2 - x^4 + x^5 + x^7).

1, 1, 2, 3, 6, 9, 16, 25, 43, 69, 116, 188, 313, 511, 846, 1386, 2288, 3756, 6191, 10174, 16756, 27552, 45357, 74604, 122787, 201996, 332414, 546901, 899946, 1480699, 2436459, 4008858, 6596366, 10853563, 17858788, 29384804, 48350401, 79555943, 130902711

Offset: 0

David Neil McGrath, Jul 30 2015

This sequence counts partially ordered partitions of (n) into parts 1,2,3,4 where the order (position) of adjacent pairs of numbers (1,2);(2,3);(3,4) is unimportant. Alternatively the order of the complementary pairs (1,4);(1,3);(2,4) is important.

			There are 25 partially ordered partitions of 7, i.e., a(7) = 25. These are (43=34),(421=412),(124=214),(241),(142),(4111),(1411),(1141),(1114),(331),(313),(133),(1132=1123),(2131=1231),(1312=1321),(2311=3211),(31111),(13111),(11311),(11131),(11113),(2221=four),(22111=ten),(211111=six),(1111111).

Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1,1,0,1,-1,0,-1).

Cf. A023435, A080239, A023434, A254685, A116732, A004695.

I:=[1,1,2,3,6,9,16]; [n le 7 select I[n] else Self(n-1)+Self(n-2)+Self(n-4)-Self(n-5)-Self(n-7): n in [1..40]]; // Vincenzo Librandi, Aug 04 2015

LinearRecurrence[{1, 1, 0, 1, -1, 0, -1}, {1, 1, 2, 3, 6, 9, 16}, 50] (* Vincenzo Librandi, Aug 04 2015 *)

Vec(1/(1 - x - x^2 - x^4 + x^5 + x^7) + O(x^50)) \\ Michel Marcus, Aug 06 2015

G.f: 1/(1 - x - x^2 - x^4 + x^5 + x^7).
a(n) = a(n-1) + a(n-2) + a(n-4) - a(n-5) - a(n-7).
Construct the matrix array T(n,j) = [A^*j]*[S^*(j-1)] where A=(1,1,0,1,-1,0,-1) and S=(0,1,0,...) (A063524). [* is convolution operation] Define S^*0=I with I=(1,0,...). a(n) = Sum_{j=1..n} T(n,j).

A260917 Expansion of 1/(1 - x - x^2 - x^3 + x^6 + x^7).

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 23, 41, 74, 132, 236, 422, 754, 1348, 2409, 4305, 7694, 13750, 24573, 43915, 78481, 140255, 250652, 447944, 800528, 1430636, 2556712, 4569140, 8165581, 14592837, 26079086, 46606340, 83290915, 148850489, 266013023, 475396009, 849587598, 1518311204, 2713397556, 4849154954, 8666000202

Offset: 0

David Neil McGrath, Aug 04 2015

This sequence counts the partially ordered partitions of (n) into parts 1,2,3,4 where the order (position) of adjacent pairs (1,3);(3,4);(2,4) is unimportant. Alternatively the order of complementary pairs (1,2);(1,4);(2,3) is important.

			a(7)=41; the corresponding partitions (cf. comment) are: (43), (241=421), (124=142), (412), (214), (4111), (1411), (1141), (1114), (331=313=133), (322), (232), (223), (3112=1312=1132), (2113=2131=2311), (1213=1231), (3121=1321), (3211), (1123), (31111=13111=11311=11131=11113), (2221)=four, (22111)=ten, (211111)=six, (1111111).

Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,0,-1,-1).

Cf. A023435, A080239, A260710.

I:=[1,1,2,4,7,13,23]; [n le 7 select I[n] else Self(n-1) + Self(n-2) + Self(n-3) - Self(n-6) - Self(n-7): n in [1..45]]; // Vincenzo Librandi, Aug 07 2015

CoefficientList[Series[1/(1 - x - x^2 - x^3 + x^6 + x^7), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 07 2015 *)
LinearRecurrence[{1,1,1,0,0,-1,-1},{1,1,2,4,7,13,23},50] (* Harvey P. Dale, Aug 21 2021 *)

Vec(1/(1 - x - x^2 - x^3 + x^6 + x^7) + O(x^50)) \\ Michel Marcus, Aug 06 2015

a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-6) - a(n-7).

G.f.: 1/((1 - x)*(1 - x^2 - 2*x^3 - 2*x^4 - 2*x^5 - x^6)).

A258000 Expansion of 1/(1-x-x^2-x^3-x^4+x^5+x^6+x^7-x^9).

Original entry on oeis.org

1, 1, 2, 4, 8, 14, 26, 48, 89, 164, 302, 557, 1028, 1896, 3496, 6448, 11893, 21935, 40455, 74613, 137613, 253807, 468108, 863354, 1592327, 2936808, 5416499, 9989915, 18424893, 33981939, 62674564, 115593785, 213195313, 393206621, 725210344, 1337541166

Offset: 0

David Neil McGrath, May 16 2015

This sequence counts partially ordered partitions of (n) into parts (1,2,3,4) in which only the position (order) of the 1's are important. The 1's behave as placeholders for unordered 2's,3's and 4's.

			a(6)=26; these are (42,24=one),(411),(141),(114),(33),(321,231=one),(123,132=one),(312),(213),(3111=four),(222),(2211),(1122),(2112),(1221),(1212),(2121),(21111=five),(111111).

Index entries for partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,-1,-1,-1,0,1)

I:=[1,1,2,4,8,14,26,48,89]; [n le 9 select I[n] else Self(n-1)+Self(n-2)+Self(n-3)+Self(n-4)-Self(n-5)-Self(n-6)-Self(n-7)+Self(n-9): n in [1..40]]; // Vincenzo Librandi, May 19 2015

LinearRecurrence[{1, 1, 1, 1, -1, -1, -1, 0, 1}, {1, 1, 2, 4, 8, 14, 26, 48, 89}, 50] (* Vincenzo Librandi, May 19 2015 *)

Vec(1/(-x^9+x^7+x^6+x^5-x^4-x^3-x^2-x+1) + O(x^100)) \\ Colin Barker, May 17 2015

a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9)

G.f.: 1/(1-x-x^2-x^3-x^4+x^5+x^6+x^7-x^9).

More terms from Vincenzo Librandi, May 19 2015

A257932 Expansion of 1/(1-x-x^2-x^3+x^5+x^7).

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 22, 38, 67, 118, 207, 363, 638, 1119, 1964, 3447, 6049, 10615, 18629, 32691, 57369, 100676, 176674, 310041, 544085, 954802, 1675561, 2940405, 5160051, 9055258, 15890871, 27886534, 48937456, 85879249, 150707576, 264473359, 464118392, 814471000, 1429296968

Offset: 0

David Neil McGrath, May 13 2015

This sequence counts partially ordered partitions of (n) into parts (1,2,3,4) where the position (order) of 3's is unimportant.

			a(6)=22; these are (42),(24),(411),(141),(114),(33),(321=231=213),(312=132=123),(3111=1311=1131=1113),(222),(2211),(1122),(1221),(2112),(2121),(1212),(21111),(12111),(11211),(11121),(11112),(111111).

Robert Israel, Table of n, a(n) for n = 0..4090
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,-1,0,-1).

f:= gfun:-rectoproc({a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-5) - a(n-7), seq(a(i)=[1, 1, 2, 4, 7, 12,22][i+1],i=0..6)},a(n),remember):
map(f, [$0..50]); # Robert Israel, Apr 26 2017

LinearRecurrence[{1, 1, 1, 0, -1, 0, -1}, {1, 1, 2, 4, 7, 12, 22}, 39] (* Robert P. P. McKone, Feb 08 2021 *)

Vec(1/((x-1)*(x+1)*(x^2+x+1)*(x^3-x^2+2*x-1)) + O(x^100)) \\ Colin Barker, May 17 2015

a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-5) - a(n-7).

G.f.: 1 / ((x-1)*(x+1)*(x^2+x+1)*(x^3-x^2+2*x-1)). - Colin Barker, May 17 2015

A257934 Expansion of 1/(1-x-x^2-x^3-x^4+x^5+x^6+x^7).

Original entry on oeis.org

1, 1, 2, 4, 8, 14, 26, 48, 89, 163, 300, 552, 1016, 1868, 3436, 6320, 11625, 21381, 39326, 72332, 133040, 244698, 450070, 827808, 1522577, 2800455, 5150840, 9473872, 17425168, 32049880, 58948920, 108423968, 199422769, 366795657, 674642394, 1240860820, 2282298872, 4197802086, 7720961778

Offset: 0

David Neil McGrath, May 13 2015

This sequence counts partially ordered partitions of (n) into parts (1,2,3,4) in which the position (order) of the 4's are unimportant. For example the permutations of (43421) are counted as permutations of (321)=6.

			a(6)=26; these are (42=24),(411=141=114),(33),(321=six),(3111=four),(222),(2211=six),(21111=five),(111111).

Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,-1,-1,-1).

Cf. A258000, A257863.

Vec(1 / ((x-1)*(x+1)*(x^2+1)*(x^3+x^2+x-1)) + O(x^100)) \\ Colin Barker, May 17 2015

a(n)= a(n-1) + a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7).

G.f.: 1 / ((x-1)*(x+1)*(x^2+1)*(x^3+x^2+x-1)). - Colin Barker, May 17 2015

Missing term (6320) inserted by Colin Barker, May 17 2015

A257792 Expansion of 1/(1-x-x^2-x^3-x^5+x^8-x^9).

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 49, 92, 174, 328, 618, 1166, 2197, 4143, 7811, 14726, 27764, 52344, 98687, 186058, 350784, 661347, 1246865, 2350768, 4432000, 8355837, 15753609, 29700940, 55996428, 105572414, 199040101, 375258649, 707490872, 1333862213, 2514786376

Offset: 0

David Neil McGrath, May 08 2015

This sequence counts partially ordered partitions of (n) in two distinct ways. It partitions (n) into parts containing (1,2,3,5,9) where the adjacent order of 3's and 5's are unimportant, example (1), and it partitions (n) into parts containing (1,2,3,4,5,6) where the adjacent order of the odd numbers is unimportant, example (2). The sign "=" is used within a bracket to indicate that the arrangements are counted as one.

			Example (1):Partial order of (n) into parts (1,2,3,5,9) where the adjacent order of 3's and 5's is unimportant. a(8)=92 These are (53=35)=1,(521)=6,(5111)=4,(332)=3,(3311)=6,(3221)=12,(32111)=20,(311111)=6,(2222)=1,(22211)=10,(221111)=15,(2111111)=7,(11111111)=1.
Example (2):Partial order of (n) into parts (1,2,3,4,5,6) where the adjacent order of all odd numbers (i.e. 1,3,5) is unimportant. a(6)=26 These are (6),(51=15),(42),(24),(411),(141),(114),(33),(321),(123),(231=213),(312=132),(3111=1311=1131=1113),(222),(2211),(1122),(1221),(2112),(2121),(1212),(21111),(12111),(11211),(11121),(11112),(111111).

Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,1,0,0,-1,1).

I:=[1,1,2,4,7,14,26,49,92]; [n le 9 select I[n] else Self(n-1)+Self(n-2)+Self(n-3)+Self(n-5)-Self(n-8)+Self(n-9): n in [1..40]]; // Vincenzo Librandi, May 09 2015

CoefficientList[Series[1/(1 - x - x^2 - x^3 - x^5 + x^8 - x^9), {x, 0, 80}], x] (* Vincenzo Librandi, May 09 2015 *)
LinearRecurrence[{1,1,1,0,1,0,0,-1,1},{1,1,2,4,7,14,26,49,92},36] (* Ray Chandler, Jul 14 2015 *)

m = 40; L. = PowerSeriesRing(ZZ, m); f = 1/(1-x-x^2-x^3-x^5+x^8-x^9); print(f.coefficients()) # Bruno Berselli, May 12 2015

G.f.: 1/(1-x-x^2-x^3-x^5+x^8-x^9).

a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-5) - a(n-8) + a(n-9).

More terms from Vincenzo Librandi, May 09 2015

A257863 Expansion of 1/(1 - x - x^2 + x^5 - x^6).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 18, 29, 45, 72, 112, 178, 279, 441, 693, 1094, 1721, 2714, 4273, 6735, 10607, 16715, 26329, 41485, 65352, 102965, 162209, 255560, 402613, 634306, 999306, 1574368, 2480323, 3907638, 6156268, 9698906, 15280112, 24073063, 37925860, 59750293

Offset: 0

David Neil McGrath, May 11 2015

This sequence counts partially ordered partitions of (n) into parts (1,2,3,4) where only the position (order) of the 4's are important. The 4's behave like placeholders for the unordered 1's, 2's and 3's. (See example.)

			a(8)=29 These are (44),(341),(143),(431=413),(314=134),(422),(242),(224),(4211=4121=4112),(2114=1214=1124),(1421=1412),(2141=1241),(2411),(1142),(41111),(14111),(11411),(11141),(11114),(332=323=233),(3311=1133=1331=3113=1313=3131),(3221=twelve),(32111=twenty),(311111=six),(2222),(22211=ten),(221111=fifteen),(2111111=seven),(11111111)

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

[n le 6 select NumberOfPartitions(n-1) else Self(n-1)+Self(n-2)-Self(n-5)+Self(n-6): n in [1..50]]; // Vincenzo Librandi, May 12 2015

RecurrenceTable[{a[n] == a[n - 1] + a[n - 2] - a[n - 5] + a[n - 6], a[1] == 1, a[2] == 1, a[3] == 2, a[4] == 3, a[5] == 5, a[6] == 7}, a, {n, 43}] (* Michael De Vlieger, May 11 2015 *)
CoefficientList[Series[1/(1 - x - x^2 + x^5 - x^6), {x, 0, 80}], x] (* or *) LinearRecurrence[{1, 1, 0, 0, -1, 1}, {1, 1, 2, 3, 5, 7}, 50] (* Vincenzo Librandi, May 12 2015 *)

m = 50; L. = PowerSeriesRing(ZZ, m); f = 1/(1-x-x^2+x^5-x^6); print(f.coefficients()) # Bruno Berselli, May 12 2015

G.f.: 1/(1-x-x^2+x^5-x^6).

a(n) = a(n-1) + a(n-2) - a(n-5) + a(n-6).

A254685 Number of partially ordered partitions of n into parts less than or equal to 3, in which the order of adjacent 2's and 3's is unimportant.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 22, 39, 69, 123, 219, 389, 692, 1231, 2189, 3893, 6924, 12314, 21900, 38949, 69270, 123195, 219100, 389665, 693011, 1232506, 2191987, 3898404, 6933232, 12330612, 21929742, 39001599, 69363549, 123361658, 219396194, 390191659, 693947912

Offset: 0

David Neil McGrath, May 04 2015

Also number of compositions of n into parts 1, 2, 3, and 5.

			a(7)=39. These are (331),(313),(133),(322=232=223),(3211=2311),(1123=1132),(1231=1321),(3112),(2113),(1312),(1213),(3121),(2131),(31111),(13111),(11311),(11131),(11113),(2221),(2212),(2122),(1222),(22111),(21211),(12211),(12121),(11221),(11212),(11122),(12112),(21112),(21121),(211111),(121111),(112111),(111211),(111121),(111112),(1111111).

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

Cf. A001399.

I:=[1,2,4,7,12]; [n le 5 select I[n] else Self(n-1)+Self(n-2)+Self(n-3)-Self(n-5): n in [1..40]]; // Vincenzo Librandi, May 06 2015

CoefficientList[Series[1/(x^5 - x^3 - x^2 - x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, May 06 2015 *)

G.f.: 1/(x^5 - x^3 - x^2 - x + 1).

a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-5).

Corrected g.f. and more terms from Vincenzo Librandi, May 06 2015

a(0) added and g.f. adapted from Alois P. Heinz, May 08 2015

A245369 Number of compositions of n into parts 3, 5 and 8.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 3, 1, 1, 5, 1, 5, 7, 2, 13, 9, 8, 25, 12, 26, 41, 22, 64, 62, 56, 130, 96, 146, 233, 174, 340, 391, 376, 703, 661, 862, 1327, 1211, 1905, 2379, 2449, 3935, 4251, 5216, 7641, 7911, 11056, 14271, 15576, 22632, 26433, 31848, 44544, 49920, 65536, 85248, 97344, 132712, 161601, 194728, 262504, 308865

Offset: 0

David Neil McGrath, Aug 23 2014

			a(19)=25. The compositions of 19 into parts 3, 5, and 8 are the permutations of (883) (these are 3!/2!=3), (8533) (these are 4!/2!=12), and (55333) (these are 5!/3!2!=10).

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

Cf. A079957, A245367.

LinearRecurrence[{0,0,1,0,1,0,0,1},{1,0,0,1,0,1,1,0},70] (* Harvey P. Dale, Sep 05 2022 *)

Vec( 1/(1-x^3-x^5-x^8) +O(x^66) ) \\ Joerg Arndt, Aug 25 2014

G.f.: 1/(1-x^3-x^5-x^8).

a(n) = a(n-3) + a(n-5) + a(n-8).

A245370 Number of compositions of n into parts 3, 5 and 9.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 2, 2, 1, 3, 3, 3, 6, 5, 6, 11, 10, 13, 19, 19, 27, 35, 37, 52, 65, 74, 100, 121, 145, 192, 230, 282, 365, 440, 548, 695, 843, 1058, 1327, 1621, 2035, 2535, 3119, 3910, 4851, 5997, 7503, 9297, 11528, 14389, 17829, 22150, 27596, 34208, 42536, 52928, 65655, 81660, 101525, 126020, 156738, 194776, 241888

Offset: 0

David Neil McGrath, Aug 24 2014

			a(28)=100 The compositions of n into parts 3,5 and 9 are the permutations of (9955)(these are 4!/2!2!=6), (555553) (these are 6!/5!=6), (955333) (these are 6!/3!2!=60), (55333333) (these are 8!/6!2!=28).

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

Cf. A079957, A245367, A245369.

Vec( 1/(1-x^3-x^5-x^9) +O(x^66) ) \\ Joerg Arndt, Aug 24 2014

G.f.: 1/(1-x^3-x^5-x^9).

a(n) = a(n-3) + a(n-5) + a(n-9).

cp's OEIS Frontend

User: David Neil McGrath

A260710 Expansion of 1/(1 - x - x^2 - x^4 + x^5 + x^7).

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A260917 Expansion of 1/(1 - x - x^2 - x^3 + x^6 + x^7).

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A258000 Expansion of 1/(1-x-x^2-x^3-x^4+x^5+x^6+x^7-x^9).

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A257932 Expansion of 1/(1-x-x^2-x^3+x^5+x^7).

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A257934 Expansion of 1/(1-x-x^2-x^3-x^4+x^5+x^6+x^7).

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A257792 Expansion of 1/(1-x-x^2-x^3-x^5+x^8-x^9).

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A257863 Expansion of 1/(1 - x - x^2 + x^5 - x^6).

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