A256225
Number of partitions of 5n into 5 parts.
Original entry on oeis.org
0, 1, 7, 30, 84, 192, 377, 674, 1115, 1747, 2611, 3765, 5260, 7166, 9542, 12470, 16019, 20282, 25337, 31289, 38225, 46262, 55496, 66055, 78045, 91606, 106852, 123935, 142979, 164147, 187572, 213429, 241860, 273052, 307156, 344370, 384855, 428821, 476437, 527925
Offset: 0
For n=2, the 7 partitions of 10 are [6,1,1,1,1], [5,2,1,1,1], [4,3,1,1,1], [4,2,2,1,1], [3,3,2,1,1], [3,2,2,2,1] and [2,2,2,2,2].
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-2,2,0,1,0,-2,1).
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Length /@ (Length /@ IntegerPartitions[5 #, {5}] & /@ Range@ 39) (* Michael De Vlieger, Mar 20 2015 *)
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concat(0, Vec(-x* (x^8+5*x^7+16*x^6+25*x^5+31*x^4+25*x^3+16*x^2+5*x+1) / ((x-1)^5*(x+1)^2*(x^2+1)*(x^2+x+1)) + O(x^100)))
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concat(0, vector(40, n, k=0; forpart(p=5*n, k++, , [5,5]); k)) \\ Colin Barker, Mar 21 2015
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
For n=2 there are 11 partitions of 6*2 = 12, so a(2) = 11*12 = 132.
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,3,-9,3,10,-10,-3,9,-3,-5,5,1,-3,1).
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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 *)
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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)))
A256288
Sum of all the parts in the partitions of 7n into 7 parts.
Original entry on oeis.org
0, 7, 210, 2205, 12208, 47845, 148638, 393617, 922600, 1970325, 3902360, 7270725, 12868128, 21811881, 35617708, 56319375, 86565808, 129780448, 190285326, 273509446, 386137220, 536375658, 734131552, 991339307, 1322171256, 1743437850, 2274824006, 2939374683
Offset: 0
For n=2 there are 15 partitions of 7*2 = 14, so a(2) = 15*14 = 210.
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Plus @@ Total /@ IntegerPartitions[7 #, {7}] & /@ Range[0, 24] (* Michael De Vlieger, Mar 21 2015 *)
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concat(0, vector(35, n, k=0; forpart(p=7*n, k++, , [7,7]); 7*n*k))
Showing 1-3 of 3 results.