cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A180356 a(n) = n^6 + 6n.

Original entry on oeis.org

0, 7, 76, 747, 4120, 15655, 46692, 117691, 262192, 531495, 1000060, 1771627, 2986056, 4826887, 7529620, 11390715, 16777312, 24137671, 34012332, 47045995, 64000120, 85766247, 113380036, 148036027, 191103120, 244140775, 308915932
Offset: 0

Views

Author

Odimar Fabeny, Aug 30 2010

Keywords

Links

Crossrefs

Cf. A079908, A132411, A180354, A180355.

Programs

  • Magma
    [n^6+6*n : n in [0..50]]; // Wesley Ivan Hurt, Jul 07 2025
  • Mathematica
    Table[n^6+6n,{n,0,30}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,7,76,747,4120,15655,46692},30] (* Harvey P. Dale, Feb 14 2023 *)

Formula

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). G.f.: x*(-7-27*x-362*x^2-242*x^3-87*x^4+5*x^5)/(x-1)^7. - R. J. Mathar, Sep 24 2010

Extensions

First term corrected by Odimar Fabeny, Sep 23 2010
a(0) corrected by R. J. Mathar, Sep 24 2010

A079910 Solution to the Dancing School Problem with 5 girls and n+5 boys: f(5,n).

Original entry on oeis.org

1, 6, 46, 212, 738, 2104, 5150, 11196, 22162, 40688, 70254, 115300, 181346, 275112, 404638, 579404, 810450, 1110496, 1494062, 1977588, 2579554, 3320600, 4223646, 5314012, 6619538, 8170704, 10000750, 12145796, 14644962, 17540488, 20877854
Offset: 0

Views

Author

Jaap Spies, Jan 28 2003

Keywords

Comments

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X (g+h) with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.
For fixed g, f(g,n) is polynomial in n for n >= g-2. See reference.

Links

Crossrefs

Cf. A079908-A079928.

Programs

  • Magma
    [1,6] cat [n^5-5*n^4+25*n^3-55*n^2+80*n-46: n in [2..30]]; // Vincenzo Librandi, Feb 17 2015
  • Mathematica
    CoefficientList[Series[(6 x^7 + 11 x^6 + 20 x^5 + 51 x^4 + 6 x^3 + 25 x^2 + 1) / (x - 1)^6, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 17 2015 *)
  • PARI
    Vec((6*x^7+11*x^6+20*x^5+51*x^4+6*x^3+25*x^2+1)/(x-1)^6 + O(x^100)) \\ Colin Barker, Jan 04 2015

Formula

a(0)=1, a(1)=6, a(2)=46, a(n) = n^5 - 5*n^4 + 25*n^3 - 55*n^2 + 80*n - 46.
G.f.: (6*x^7 + 11*x^6 + 20*x^5 + 51*x^4 + 6*x^3 + 25*x^2 + 1) / (x-1)^6. - Colin Barker, Jan 04 2015
E.g.f.: 47 + 6*x + exp(x)*(-46 + 46*x + 20*x^3 + 5*x^4 + x^5). - Stefano Spezia, Dec 18 2019

Extensions

More terms from Benoit Cloitre, Jan 29 2003

A079911 Solution to the Dancing School Problem with 6 girls and n+6 boys: f(6,n).

Original entry on oeis.org

1, 7, 79, 478, 2108, 7364, 21652, 55532, 127604, 268108, 523244, 960212, 1672972, 2788724, 4475108, 6948124, 10480772, 15412412, 22158844, 31223108, 43207004, 58823332, 78908852, 104437964, 136537108, 176499884, 225802892, 286122292
Offset: 0

Views

Author

Jaap Spies, Jan 28 2003

Keywords

Comments

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.
For fixed g, f(g,n) is polynomial in n for n >= g-2. See reference.

Links

Crossrefs

Cf. A079908-A079928.

Programs

  • Maple
    seq(n^6-9*n^5+60*n^4-225*n^3+555*n^2-774*n+484,n=4..40);
  • Mathematica
    CoefficientList[Series[-(6 x^10 - 29 x^9 + 120 x^8 - 49 x^7 + 267 x^6 + 105 x^5 + 211 x^4 + 37 x^3 + 51 x^2 + 1)/(x - 1)^7, {x, 0, 28}], x] (* Michael De Vlieger, Dec 23 2019 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,7,79,478,2108,7364,21652,55532,127604,268108,523244},40] (* Harvey P. Dale, Jul 02 2022 *)
  • PARI
    Vec(-(6*x^10 -29*x^9 +120*x^8 -49*x^7 +267*x^6 +105*x^5 +211*x^4 +37*x^3 +51*x^2 +1) / (x -1)^7 + O(x^100)) \\ Colin Barker, Jan 04 2015

Formula

a(0)=1, a(2)=7, a(3)=79, a(n)=n^6-9*n^5+60*n^4-225*n^3+555*n^3-774*n+484.
G.f.: -(6*x^10 -29*x^9 +120*x^8 -49*x^7 +267*x^6 +105*x^5 +211*x^4 +37*x^3 +51*x^2 +1) / (x -1)^7. - Colin Barker, Jan 04 2015

A079912 Solution to the Dancing School Problem with 7 girls and n+7 boys: f(7,n).

Original entry on oeis.org

1, 8, 133, 1044, 5794, 24720, 86608, 260720, 693552, 1666000, 3675680, 7549488, 14591440, 26770832, 46955760, 79197040, 129067568, 204062160, 314062912, 471875120, 693838800, 1000520848, 1417492880, 1976199792, 2714924080
Offset: 0

Views

Author

Jaap Spies, Jan 28 2003

Keywords

Comments

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.
For fixed g, f(g,n) is polynomial in n for n >= g-2. See reference.

Links

Crossrefs

Cf. A079908-A079928.

Programs

  • Maple
    seq(n^7-14*n^6+126*n^5-700*n^4+2625*n^3-6342*n^2+9072*n-5840,n=5..20);
  • Mathematica
    Join[{1,8,133,1044,5794},Table[n^7-14n^6+126n^5-700n^4+2625n^3- 6342n^2 +9072n-5840,{n,5,30}]] (* Harvey P. Dale, May 03 2011 *)
  • PARI
    Vec(-(46*x^12 -340*x^11 +931*x^10 -1808*x^9 +727*x^8 -1400*x^7 -1506*x^6 -656*x^5 -788*x^4 -148*x^3 -97*x^2 -1) / (x -1)^8 + O(x^100)) \\ Colin Barker, Jan 04 2015

Formula

a(0) = 1, a(1) = 8, a(2) = 133, a(3) = 1044, a(4) = 5794; for n>4, a(n) = n^7-14*n^6+126*n^5-700*n^4+2625*n^3-6342*n^2+9072*n-5840.
G.f.: -(46*x^12 -340*x^11 +931*x^10 -1808*x^9 +727*x^8 -1400*x^7 -1506*x^6 -656*x^5 -788*x^4 -148*x^3 -97*x^2 -1) / (x -1)^8. - Colin Barker, Jan 04 2015

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003

A079926 Solution to the Dancing School Problem with n girls and n+7 boys: f(n,7).

Original entry on oeis.org

8, 57, 364, 2106, 11196, 55532, 260720, 1173240, 5112544, 21670160, 89700624, 363862092, 1450606028, 5697401802, 22088730348, 84669409935, 321307769052, 1208513572803, 4509661963752, 16709568237540
Offset: 1

Views

Author

Jaap Spies, Jan 28 2003

Keywords

Comments

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.

Links

Crossrefs

Cf. A079908-A079928.

Extensions

Corrected by Jaap Spies, Feb 01 2004
More terms Dec 14 2006

A180357 a(n) = n^7 + 7*n.

Original entry on oeis.org

0, 8, 142, 2208, 16412, 78160, 279978, 823592, 2097208, 4783032, 10000070, 19487248, 35831892, 62748608, 105413602, 170859480, 268435568, 410338792, 612220158, 893871872, 1280000140, 1801088688, 2494358042
Offset: 0

Views

Author

Odimar Fabeny, Aug 30 2010

Keywords

Links

Crossrefs

Cf. A132411, A079908, A180354, A180355, A180356.

Programs

  • Mathematica
    Table[n^7+7n,{n,0,30}] (* Harvey P. Dale, Sep 10 2010 *)

Formula

From Chai Wah Wu, Oct 15 2016: (Start)
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) for n > 7.
G.f.: 2*x*(4*x^6 + 39*x^5 + 648*x^4 + 1138*x^3 + 648*x^2 + 39*x + 4)/(x - 1)^8. (End)

Extensions

First term corrected and additional terms from Harvey P. Dale, Sep 10 2010

A079913 Solution to the Dancing School Problem with 8 girls and n+8 boys: f(8,n).

Original entry on oeis.org

1, 9, 221, 2227, 15458, 80196, 334072, 1173240, 3598120, 9856552, 24553080, 56423032, 121013800, 244555560, 469343992, 860997880, 1517994792, 2583928360, 4262971000, 6839066232, 10699415080, 16362861352, 24513820920, 36042440440, 52091711272, 74112304680
Offset: 0

Views

Author

Jaap Spies, Jan 28 2003

Keywords

Comments

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.
For fixed g, f(g,n) is polynomial in n for n >= g-2. See reference.

Links

Crossrefs

Cf. A079908-A079928.

Programs

  • Maple
    A079913 := n->n^8 -20*n^7 +238*n^6 -1820*n^5 +9625*n^4 -35000*n^3 +84448*n^2 -122240*n +80680: (1,9,221,2227,15458,80196, seq(A079913(n), n=6..30)); # edited by Wesley Ivan Hurt, Sep 17 2015
  • Mathematica
    CoefficientList[Series[-(484*x^14 - 3902*x^13 + 13791*x^12 - 25930*x^11 + 32928*x^10 - 15756*x^9 + 14443*x^8 + 8652*x^7 + 8524*x^6 + 3690*x^5 + 2741*x^4 + 478*x^3 + 176*x^2 + 1)/(x - 1)^9, {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 17 2015 *)
  • PARI
    Vec(-(484*x^14 -3902*x^13 +13791*x^12 -25930*x^11 +32928*x^10 -15756*x^9 +14443*x^8 +8652*x^7 +8524*x^6 +3690*x^5 +2741*x^4 +478*x^3 +176*x^2 +1)/(x -1)^9 + O(x^100)) \\ Colin Barker, Jan 05 2015

Formula

a(0)=1, a(1)=9, a(2)=221, a(3)=2227, a(4)=15459, a(5)=80196, for n >= 6, a(n)= n^8 -20*n^7 +238*n^6 -1820*n^5 +9625*n^4 -35000*n^3 +84448*n^2 -122240*n +80680.
G.f.: -(484*x^14 -3902*x^13 +13791*x^12 -25930*x^11 +32928*x^10 -15756*x^9 +14443*x^8 +8652*x^7 +8524*x^6 +3690*x^5 +2741*x^4 +478*x^3 +176*x^2 +1) / (x -1)^9. - Colin Barker, Jan 05 2015
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), for n>8. - Wesley Ivan Hurt, Sep 17 2015

A079914 Solution to the Dancing School Problem with 9 girls and n+9 boys: f(9,n).

Original entry on oeis.org

1, 10, 364, 4664, 40296, 253072, 1249768, 5112544, 17990600, 56010096, 157175032, 403579328, 959942664, 2136701200, 4488418616, 8961185952, 17105944648, 31378295984, 55549351800, 95256535936, 158727963272, 257719103568
Offset: 0

Views

Author

Jaap Spies, Jan 28 2003

Keywords

Comments

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.
For fixed g, f(g,n) is polynomial in n for n >= g-2. See reference.

Links

Crossrefs

Cf. A079908-A079928.

Programs

  • Maple
    f := n->n^9-27*n^8+414*n^7-4158*n^6+29421*n^5-148743*n^4+530796*n^3-1276992*n^2+1866384*n-1255608; seq(f(i),i=7..21);
  • Mathematica
    CoefficientList[Series[-(5840 x^16 - 52960 x^15 + 210480 x^14 - 481464 x^13 + 671100 x^12 - 619882 x^11 + 258311 x^10 - 123144 x^9 - 98197 x^8 - 57276 x^7 - 46818 x^6 - 18160 x^5 - 9046 x^4 - 1354 x^3 - 309 x^2 - 1)/(x - 1)^10, {x, 0, 21}], x] (* Michael De Vlieger, Dec 23 2019 *)
  • PARI
    Vec(-(5840*x^16 -52960*x^15 +210480*x^14 -481464*x^13 +671100*x^12 -619882*x^11 +258311*x^10 -123144*x^9 -98197*x^8 -57276*x^7 -46818*x^6 -18160*x^5 -9046*x^4 -1354*x^3 -309*x^2 -1) / (x -1)^10 + O(x^100)) \\ Colin Barker, Jan 05 2015

Formula

a(0)=1, a(1)=10, a(2)=364, a(3)=4664, a(4)=40296, a(5)=253072, a(6)=1249768, for n >= 7: a(n)=n^9-27n^8+414n^7-4158n^6+29421n^5-148743n^4+530796n^3-1276992n^2+1866384n-1255608.
G.f.: -(5840*x^16 -52960*x^15 +210480*x^14 -481464*x^13 +671100*x^12 -619882*x^11 +258311*x^10 -123144*x^9 -98197*x^8 -57276*x^7 -46818*x^6 -18160*x^5 -9046*x^4 -1354*x^3 -309*x^2 -1) / (x -1)^10.- Colin Barker, Jan 05 2015

A079915 Solution to the Dancing School Problem with 10 girls and n+10 boys: f(10,n).

Original entry on oeis.org

1, 11, 596, 9627, 103129, 780902, 4557284, 21670160, 87396728, 308055528, 971055240, 2780440664, 7324967640, 17945144328, 41249101928, 89635336440, 185317652664, 366517590440, 696695849928
Offset: 0

Views

Author

Jaap Spies, Jan 28 2003

Keywords

Comments

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.
For fixed g, f(g,n) is polynomial in n for n >= g-2. See reference.

References

  • Jaap Spies, Dancing School Problems, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.

Links

Crossrefs

Cf. A079908-A079928.

Programs

  • Maple
    f:= n-> n^10 -35*n^9 +675*n^8 -8610*n^7 +78435*n^6 -523467*n^5 +2562525*n^4 -9008160*n^3 +21623220*n^2 -31840760*n +21750840: seq(f(i), i=8..21);

Formula

for n>=8: a(n) = n^10 -35*n^9 +675*n^8 -8610*n^7 +78435*n^6 -523467*n^5 +2562525*n^4 -9008160*n^3 +21623220*n^2 -31840760*n +21750840.

Extensions

Corrected by Jaap Spies, Feb 01 2004

A079918 Solution to the Dancing School Problem with 13 girls and n+13 boys: f(13,n).

Original entry on oeis.org

1, 14, 2567, 79544, 1596800, 20789082, 196933710, 1450606028, 8719846960, 44321202192, 195717772000, 767025716736, 2713659864832, 8787898861568
Offset: 0

Views

Author

Jaap Spies, Jan 28 2003

Keywords

Comments

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h. See A079908 for more information.
For fixed g, f(g,n) is polynomial in n for n >= g-2. See reference.

References

  • Jaap Spies, Dancing School Problems, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.

Links

Crossrefs

Cf. A079908-A079928.

Extensions

Corrected by Jaap Spies, Feb 01 2004
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