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.

Showing 1-4 of 4 results.

A256226 Number of partitions of 6n into 6 parts.

Original entry on oeis.org

0, 1, 11, 58, 199, 532, 1206, 2432, 4494, 7760, 12692, 19858, 29941, 43752, 62239, 86499, 117788, 157532, 207338, 269005, 344534, 436140, 546261, 677571, 832989, 1015691, 1229120, 1476997, 1763332, 2092435, 2468926, 2897747, 3384171, 3933815, 4552649, 5247008
Offset: 0

Views

Author

Colin Barker, Mar 19 2015

Keywords

Examples

			For n=2, the 11 partitions of 12 are Xs = [7,1,1,1,1,1], [6,2,1,1,1,1], [5,3,1,1,1,1], [4,4,1,1,1,1], [5,2,2,1,1,1], [4,3,2,1,1,1], [3,3,3,1,1,1], [4,2,2,2,1,1], [3,3,2,2,1,1], [3,2,2,2,2,1] and [2,2,2,2,2,2].

Links

Crossrefs

Cf. A001402, A077043, A238340, A256225.

Programs

  • Mathematica
    CoefficientList[Series[x (3 x^7 + 14 x^6 + 21 x^5 + 21 x^4 + 22 x^3 + 19 x^2 + 7 x + 1) / ((x - 1)^6 (x + 1) (x^4 + x^3 + x^2 + x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)
  • PARI
    concat(0, Vec(x*(3*x^7+14*x^6+21*x^5+21*x^4+22*x^3+19*x^2+7*x+1)/((x-1)^6*(x+1)*(x^4+x^3+x^2+x+1)) + O(x^100)))
  • PARI
    concat(0, vector(35, n, k=0; forpart(p=6*n, k++, , [6,6]); k)) \\ Colin Barker, Mar 21 2015

Formula

G.f.: x*(3*x^7+14*x^6+21*x^5+21*x^4+22*x^3+19*x^2+7*x+1) / ((x-1)^6*(x+1)*(x^4+x^3+x^2+x+1)).

A256235 Sum of all the parts in the partitions of 5n into 5 parts.

Original entry on oeis.org

0, 5, 70, 450, 1680, 4800, 11310, 23590, 44600, 78615, 130550, 207075, 315600, 465790, 667940, 935250, 1281520, 1723970, 2280330, 2972455, 3822500, 4857510, 6104560, 7596325, 9365400, 11450750, 13890760, 16731225, 20017060, 23801315, 28135800, 33081495
Offset: 0

Views

Author

Colin Barker, Mar 20 2015

Keywords

Examples

			For n=2 there are 7 partitions of 5*2 = 10, so a(2) = 7*10 = 70.

Links

Crossrefs

Cf. A235988, A238328, A256225, A256239.

Programs

  • Mathematica
    Plus @@ Total /@ IntegerPartitions[5 #, {5}] & /@ Range[0, 31] (* Michael De Vlieger, Mar 20 2015 *)
CoefficientList[Series[5 x (2 x^14 + 19 x^13 + 97 x^12 + 277 x^11 + 591 x^10 + 955 x^9 + 1267 x^8 + 1355 x^7 + 1217 x^6 + 880 x^5 + 520 x^4 + 231 x^3 + 75 x^2 + 13 x + 1) / ((x - 1)^6 (x + 1)^3 (x^2 + 1)^2 (x^2 + x + 1)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)
LinearRecurrence[{1,1,1,0,-4,-1,-1,4,4,-1,-1,-4,0,1,1,1,-1},{0,5,70,450,1680,4800,11310,23590,44600,78615,130550,207075,315600,465790,667940,935250,1281520},40] (* Harvey P. Dale, Jun 14 2016 *)
  • PARI
    concat(0, Vec(5*x*(2*x^14 +19*x^13 +97*x^12 +277*x^11 +591*x^10 +955*x^9 +1267*x^8 +1355*x^7 +1217*x^6 +880*x^5 +520*x^4 +231*x^3 +75*x^2 +13*x +1) / ((x -1)^6*(x +1)^3*(x^2 +1)^2*(x^2 +x +1)^2) + O(x^100)))

Formula

a(n) = 5*n*A256225(n).
G.f.: 5*x*(2*x^14 +19*x^13 +97*x^12 +277*x^11 +591*x^10 +955*x^9 +1267*x^8 +1355*x^7 +1217*x^6 +880*x^5 +520*x^4 +231*x^3 +75*x^2 +13*x +1) / ((x -1)^6*(x +1)^3*(x^2 +1)^2*(x^2 +x +1)^2).

A256239 Sum of all the parts in the partitions of 6n into 6 parts.

Original entry on oeis.org

0, 6, 132, 1044, 4776, 15960, 43416, 102144, 215712, 419040, 761520, 1310628, 2155752, 3412656, 5228076, 7784910, 11307648, 16068264, 22392504, 30666570, 41344080, 54953640, 72106452, 93504798, 119950416, 152353650, 191742720, 239273514, 296239776, 364083690
Offset: 0

Views

Author

Colin Barker, Mar 20 2015

Keywords

Examples

			For n=2 there are 11 partitions of 6*2 = 12, so a(2) = 11*12 = 132.

Links

Crossrefs

Cf. A235988, A238328, A256225, A256235.

Programs

  • Mathematica
    Plus @@ Total /@ IntegerPartitions[6 #, {6}] & /@ Range[0, 29] (* Michael De Vlieger, Mar 20 2015 *)
CoefficientList[Series[- 6 x (9 x^13 + 77 x^12 + 247 x^11 + 485 x^10 + 744 x^9 + 990 x^8 + 1109 x^7 + 1029 x^6 + 809 x^5 + 551 x^4 + 301 x^3 + 109 x^2 + 19 x + 1) / ((x - 1)^7 (x + 1)^2 (x^4 + x^3 + x^2 + x + 1)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)
LinearRecurrence[{3,-1,-5,5,3,-9,3,10,-10,-3,9,-3,-5,5,1,-3,1},{0,6,132,1044,4776,15960,43416,102144,215712,419040,761520,1310628,2155752,3412656,5228076,7784910,11307648},30] (* Harvey P. Dale, Mar 07 2025 *)
  • PARI
    concat(0, Vec(-6*x*(9*x^13 +77*x^12 +247*x^11 +485*x^10 +744*x^9 +990*x^8 +1109*x^7 +1029*x^6 +809*x^5 +551*x^4 +301*x^3 +109*x^2 +19*x +1) / ((x -1)^7*(x +1)^2*(x^4 +x^3 +x^2 +x +1)^2) + O(x^100)))

Formula

a(n) = 6*n*A256226(n).
G.f.: -6*x*(9*x^13 +77*x^12 +247*x^11 +485*x^10 +744*x^9 +990*x^8 +1109*x^7 +1029*x^6 +809*x^5 +551*x^4 +301*x^3 +109*x^2 +19*x +1) / ((x -1)^7*(x +1)^2*(x^4 +x^3 +x^2 +x +1)^2).

A256287 Number of partitions of 7n into 7 parts.

Original entry on oeis.org

0, 1, 15, 105, 436, 1367, 3539, 8033, 16475, 31275, 55748, 94425, 153192, 239691, 363446, 536375, 772909, 1090592, 1510201, 2056462, 2758123, 3648814, 4767088, 6157387, 7870067, 9962502, 12499033, 15552247, 19202869, 23541165, 28666799, 34690401, 41733315
Offset: 0

Views

Author

Colin Barker, Mar 21 2015

Keywords

Examples

			For n=2, the 15 partitions of 14 are [1,1,1,1,1,1,8], [1,1,1,1,1,2,7], ..., [1,2,2,2,2,2,3], [2,2,2,2,2,2,2].

Links

Crossrefs

Cf. A235988, A238340, A256225, A256226, A256288.

Programs

  • Mathematica
    Length /@ (Total /@ IntegerPartitions[7 #, {7}] & /@ Range[0, 24]) (* Michael De Vlieger, Mar 21 2015 *)
  • PARI
    concat(0, vector(35, n, k=0; forpart(p=7*n, k++, , [7,7]); k))
Showing 1-4 of 4 results.

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