A270867 -id:A270867 - cp's OEIS frontend

A270868 a(n) = n^4 + 3n^3 + 8n^2 + 9*n + 2.

2, 23, 92, 263, 614, 1247, 2288, 3887, 6218, 9479, 13892, 19703, 27182, 36623, 48344, 62687, 80018, 100727, 125228, 153959, 187382, 225983, 270272, 320783, 378074, 442727, 515348, 596567, 687038, 787439, 898472, 1020863, 1155362, 1302743, 1463804, 1639367

Offset: 0

Vincenzo Librandi, Apr 01 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015 (page 19, 5th row; page 21, 4th row).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A033816, A270867.

Magma
```
[n^4+3*n^3+8*n^2+9*n+2: n in [0..40]];
```

A270868:=n->n^4 + 3*n^3 + 8*n^2 + 9*n + 2: seq(A270868(n), n=0..50); # Wesley Ivan Hurt, Apr 01 2016

Table[n^4 + 3 n^3 + 8 n^2 + 9 n + 2, {n, 0, 40}]

G.f.: (2+13*x-3*x^2+13*x^3-x^4)/(1-x)^5.
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5).
From G. C. Greubel, Apr 01 2016: (Start)
a(2*m) == 0 (mod 2).
a(4*m + 2) == 0 (mod 4).
E.g.f.: (2 +21*x +24*x^2 +9*x^3 +x^4)*exp(x). (End)
a(n)+a(n+2)-2*a(n+1) = 6*A033816(n+1). - Wesley Ivan Hurt, Apr 02 2016

A270869 a(n) = n^5 + 4n^4 + 13n^3 + 23n^2 + 25n + 3.

Original entry on oeis.org

3, 69, 345, 1203, 3351, 7953, 16749, 32175, 57483, 96861, 155553, 239979, 357855, 518313, 732021, 1011303, 1370259, 1824885, 2393193, 3095331, 3953703, 4993089, 6240765, 7726623, 9483291, 11546253, 13953969, 16747995, 19973103, 23677401, 27912453

Offset: 0

Vincenzo Librandi, Apr 03 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015. (page 19, 4th row; page 21, 5th row).
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A270867, A270868.

[n^5+4*n^4+13*n^3+23*n^2+25*n+3: n in [0..40]];

Table[n^5 + 4 n^4 + 13 n^3 + 23 n^2 + 25 n + 3, {n, 0, 40}]

x='x+O('x^99); Vec((3+51*x-24*x^2+108*x^3-27*x^4+9*x^5)/(1-x)^6) \\ Altug Alkan, Apr 03 2016

O.g.f.: (3 + 51*x - 24*x^2 + 108*x^3 - 27*x^4 + 9*x^5)/(1-x)^6.
E.g.f.: (3 + 66*x + 105*x^2 + 62*x^3 + 14*x^4 + x^5)*exp(x).
a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6).

A270870 a(n) = n^6 + 5n^5 + 19n^4 + 44n^3 + 72n^2 + 69*n + 5.

Original entry on oeis.org

5, 215, 1311, 5531, 18329, 50775, 122675, 266411, 531501, 989879, 1741895, 2923035, 4711361, 7335671, 11084379, 16315115, 23465045, 33061911, 45735791, 62231579, 83422185, 110322455, 144103811, 186109611, 237871229, 301124855, 377829015, 470182811

Offset: 0

Vincenzo Librandi, Apr 03 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015. (page 19, 4th row; page 21, 6th row).
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A270867, A270868, A270869.

[n^6+5*n^5+19*n^4+44*n^3+72*n^2+69*n+5: n in [0..40]];

Table[n^6 + 5 n^5 + 19 n^4 + 44 n^3 + 72 n^2 + 69 n + 5, {n, 0, 40}]

x='x+O('x^99); Vec((5+180*x-89*x^2+694*x^3-207*x^4+158*x^5-21*x^6)/(1-x)^7) \\ Altug Alkan, Apr 03 2016

O.g.f.: (5 +180*x -89*x^2 +694*x^3 -207*x^4 +158*x^5 -21*x^6)/(1-x)^7.
E.g.f.: (5 +210*x +443*x^2 +373*x^3 +134*x^4 +20*x^5 +x^6)*exp(x).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).

A270871 a(n) = n^7 + 6n^6 + 26n^5 + 73n^4 + 152n^3 + 222n^2 + 203n + 8.

Original entry on oeis.org

8, 691, 5030, 25511, 100372, 324323, 898706, 2206135, 4914656, 10116467, 19506238, 35604071, 62028140, 103822051, 167841962, 263208503, 401828536, 598991795, 874047446, 1251165607, 1760188868, 2437578851, 3327462850, 4482785591, 5966571152, 7853300083

Offset: 0

Vincenzo Librandi, Apr 03 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015. (page 19, 4th row; page 21, 7th row).
Index entries for linear recurrences with constant coefficients, signature (8,-28, 56,-70,56,-28,8,-1).

Cf. A270867, A270868. A270869, A270870.

[n^7+6*n^6+26*n^5+73*n^4+152*n^3+222*n^2+203*n+8: n in [0..40]];

Table[n^7 + 6 n^6 + 26 n^5 + 73 n^4 + 152 n^3 + 222 n^2 + 203 n + 8, {n, 0, 40}]
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{8,691,5030,25511,100372,324323,898706,2206135},30] (* Harvey P. Dale, Dec 17 2023 *)

x='x+O('x^99); Vec((8+627*x-274*x^2+4171*x^3-1012*x^4+1897*x^5-450*x^6+73*x^7) / (1-x)^8) \\ Altug Alkan, Apr 03 2016

G.f.: (8 + 627*x - 274*x^2 + 4171*x^3 - 1012*x^4 + 1897*x^5 - 450*x^6 + 73*x^7)/(1 - x)^8.

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).

A270872 a(n) = n^8 + 7n^7 + 34n^6 + 111n^5 + 275n^4 + 511n^3 + 703n^2 + 623*n + 13.

Original entry on oeis.org

13, 2278, 19439, 117910, 550009, 2072078, 6584443, 18269614, 45445445, 103390294, 218437543, 433677158, 816642289, 1469399230, 2541499379, 4246292158, 6881138173, 10852102214, 16703746015, 25154681014, 37139581673, 53858400238, 76833564139, 107975977550

Offset: 0

Vincenzo Librandi, Apr 04 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015. (page 19, 4th row; page 21, 8th row).
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A270867, A270868, A270869, A270870, A270871.

[n^8+7*n^7+34*n^6+111*n^5+275*n^4+511*n^3+703*n^2+623*n+13: n in [0..40]];

Table[n^8 + 7 n^7 + 34 n^6 + 111 n^5 + 275 n^4 + 511 n^3 + 703 n^2 + 623 n + 13, {n, 0, 40}]
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{13,2278,19439,117910,550009,2072078,6584443,18269614,45445445},30] (* Harvey P. Dale, Jan 14 2023 *)

x='x+O('x^99); Vec((13+2161*x-595*x^2+23875*x^3-1091*x^4+19271*x^5-4997*x^6+1909*x^7-226*x^8)/(1-x)^9) \\ Altug Alkan, Apr 04 2016

G.f.: (13+2161*x-595*x^2+23875*x^3-1091*x^4+19271*x^5-4997*x^6+1909*x^7-226*x^8)/(1-x)^9.

a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).

A270873 a(n) = n^9 + 8n^8 + 43n^7 + 159n^6 + 452n^5 + 997n^4 + 1725n^3 + 2272n^2 + 1990n + 21.

Original entry on oeis.org

21, 7668, 75545, 545730, 3015021, 13239896, 48243393, 151298070, 420233285, 1056651996, 2446142121, 5282430218, 10751650845, 20796493440, 38483939921, 68504620446, 117836491893, 196610583620, 319221957945, 505734798546, 783636668621, 1190003472168

Offset: 0

Vincenzo Librandi, Apr 04 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015. (page 19, 4th row; page 21, 9th row).
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A270867, A270868, A270869, A270870, A270871, A270872.

[n^9+8*n^8+43*n^7+159*n^6+452*n^5+997*n^4+1725*n^3+2272*n^2+1990*n+21: n in [0..40]];

Table[n^9 + 8 n^8 + 43 n^7 + 159 n^6 + 452 n^5 + 997 n^4 + 1725 n^3 + 2272 n^2 + 1990 n + 21, {n, 0, 40}]
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{21,7668,75545,545730,3015021,13239896,48243393,151298070,420233285,1056651996},30] (* Harvey P. Dale, Dec 02 2018 *)

my(x='x+O('x^99)); Vec((21+7458*x-190*x^2+132820*x^3+41496*x^4+187124*x^5-30698*x^6+30660*x^7-6565*x^8+754*x^9)/(1-x)^10) \\ Altug Alkan, Apr 04 2016

G.f.: (21+7458*x-190*x^2+132820*x^3+41496*x^4+187124*x^5-30698*x^6+30660*x^7-6565*x^8+754*x^9)/(1-x)^10.

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10).

A270874 a(n) = n^10 + 9n^9 + 53n^8 + 218n^7 + 695n^6 + 1754n^5 + 3572n^4 + 5854n^3 + 7510n^2 + 6559*n + 34.

Original entry on oeis.org

34, 26259, 294888, 2528263, 16531326, 84603579, 353479684, 1252968303, 3885899418, 10799026531, 27392790624, 64342966359, 141552806518, 294334006923, 582732259836, 1105171977919, 2017898582034, 3562049183283, 6100587181528, 10167796877991, 16534554287214

Offset: 0

Vincenzo Librandi, Apr 05 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015. (page 19, 4th row; page 21, 10th row).
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A270867, A270868, A270869, A270870, A270871, A270872, A270873.

[n^10 +9*n^9 +53*n^8 +218*n^7 +695*n^6 +1754*n^5 +3572*n^4 +5854*n^3 +7510*n^2 +6559*n +34: n in [0..30]];

Table[n^10 + 9 n^9 + 53 n^8 + 218 n^7 + 695 n^6 + 1754 n^5 + 3572 n^4 + 5854 n^3 + 7510 n^2 + 6559 n + 34, {n, 0, 30}]
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{34,26259,294888,2528263,16531326,84603579,353479684,1252968303,3885899418,10799026531,27392790624},30] (* Harvey P. Dale, Apr 10 2017 *)

x='x+O('x^99); Vec((34+25885*x+7909*x^2+723130*x^3+617758*x^4+1806700*x^5+ 96940*x^6+428806*x^7-101360*x^8+25527*x^9-2529*x^10)/(1-x)^11) \\ Altug Alkan, Apr 05 2016

G.f.: (34+25885*x+7909*x^2+723130*x^3+617758*x^4+1806700*x^5+ 96940*x^6+428806*x^7-101360*x^8+25527*x^9-2529*x^10)/(1-x)^11.

a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11).

cp's OEIS Frontend

A270868 a(n) = n^4 + 3*n^3 + 8*n^2 + 9*n + 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A270869 a(n) = n^5 + 4*n^4 + 13*n^3 + 23*n^2 + 25*n + 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A270870 a(n) = n^6 + 5*n^5 + 19*n^4 + 44*n^3 + 72*n^2 + 69*n + 5.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A270871 a(n) = n^7 + 6*n^6 + 26*n^5 + 73*n^4 + 152*n^3 + 222*n^2 + 203*n + 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A270872 a(n) = n^8 + 7*n^7 + 34*n^6 + 111*n^5 + 275*n^4 + 511*n^3 + 703*n^2 + 623*n + 13.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A270873 a(n) = n^9 + 8*n^8 + 43*n^7 + 159*n^6 + 452*n^5 + 997*n^4 + 1725*n^3 + 2272*n^2 + 1990*n + 21.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A270874 a(n) = n^10 + 9*n^9 + 53*n^8 + 218*n^7 + 695*n^6 + 1754*n^5 + 3572*n^4 + 5854*n^3 + 7510*n^2 + 6559*n + 34.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A270868 a(n) = n^4 + 3n^3 + 8n^2 + 9*n + 2.

A270869 a(n) = n^5 + 4n^4 + 13n^3 + 23n^2 + 25n + 3.

A270870 a(n) = n^6 + 5n^5 + 19n^4 + 44n^3 + 72n^2 + 69*n + 5.

A270871 a(n) = n^7 + 6n^6 + 26n^5 + 73n^4 + 152n^3 + 222n^2 + 203n + 8.

A270872 a(n) = n^8 + 7n^7 + 34n^6 + 111n^5 + 275n^4 + 511n^3 + 703n^2 + 623*n + 13.

A270873 a(n) = n^9 + 8n^8 + 43n^7 + 159n^6 + 452n^5 + 997n^4 + 1725n^3 + 2272n^2 + 1990n + 21.

A270874 a(n) = n^10 + 9n^9 + 53n^8 + 218n^7 + 695n^6 + 1754n^5 + 3572n^4 + 5854n^3 + 7510n^2 + 6559*n + 34.