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Bisect A053445 then calculate the first differences of the resulting sequence.
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0, 0, 0, 1, 1, 1, 4, 4, 6, 11, 15, 20, 33, 43, 60, 88, 119, 160, 226, 300, 404, 549, 727, 961, 1283, 1680, 2201, 2887, 3750, 4857, 6301, 8105, 10410, 13357, 17050, 21714, 27625, 34992, 44240, 55840, 70261, 88220, 110600, 138274, 172558, 214984, 267234
COMMENTS
a(n) counts the following subset of the partitions (cf. A000041): the number being partitioned is odd, the minimum part is two and the three largest parts match.
EXAMPLE
A161921 begins: 0, 0, 1, 2, 3, 7, 11, 17, 28, 43, 63, 96, 139, 199, 287, 406, 566, ...
Therefore a(n) begins 0, 0, 0, 1, 1, 1, 4, 4, 6, ..., counting 333; 3332; 33322; 555, 4443, 333222, 33333; etc.
MATHEMATICA
Join[{0}, Differences[Take[Differences[Table[PartitionsP[n], {n, 0, 100}], 2], {2, -1, 2}]]] (* Harvey P. Dale, Sep 02 2013 *)
0, 1, 1, 1, 3, 3, 4, 8, 10, 13, 22, 28, 39, 58, 77, 104, 148, 197, 265, 363, 481, 638, 858, 1126, 1480, 1953, 2544, 3309, 4312, 5566, 7175, 9246, 11843, 15136, 19328, 24564, 31158, 39466, 49811, 62737, 78900, 98931, 123817, 154707, 192830, 239911, 298013
COMMENTS
Second differences count a subset of unrestricted partitions; cf. A160648.
EXAMPLE
A160644 begins 1, 1, 2, 3, 4, 7, 10, 14, 22, 32, 45, 67, 95, 134, 192, ... so a(n) begins 0, 1, 1, 1, 3, 3, 4, 8, 10, 13, 22, 28, 39, 58, ...
Second differences of sequence A160644.
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0, 1, 0, 0, 2, 0, 1, 4, 2, 3, 9, 6, 11, 19, 19, 27, 44, 49, 68, 98, 118, 157, 220, 268, 354, 473, 591, 765, 1003, 1254, 1609, 2071, 2597, 3293, 4192, 5236, 6594, 8308, 10345, 12926, 16163, 20031, 24886, 30890, 38123, 47081, 58102, 71381, 87704, 107643
COMMENTS
A160644 bisects sequence A053445 which counts unrestricted partitions such that the two largest values match and that no part is less than two.
Conjecture: a(n) counts unrestricted partitions of even numbers such that
the three largest values match and that, after "222", no part is less than three.
EXAMPLE
a(n) begins 0 1 0 0 2 0 1 4 2 3 9 ... and counts 222; 444,3333;
666,5553,444433,333333; 5555,44444; 6664,55543,444433;
888,6666,7773,55554,66633,444444,555333,4443333,33333333; ...
0, 0, 0, 1, 2, 3, 7, 11, 17, 28, 43, 63, 96, 139, 199, 287, 406, 566, 792, 1092, 1496, 2045, 2772, 3733, 5016, 6696, 8897, 11784, 15534, 20391, 26692, 34797, 45207, 58564, 75614, 97328, 124953, 159945, 204185, 260025, 330286, 418506, 529106, 667380, 839938
MATHEMATICA
Take[Differences[Table[PartitionsP[n], {n, 0, 100}], 2], {2, -1, 2}] (* Harvey P. Dale, Sep 02 2013 *)
CROSSREFS
Cf. A160644 (the other bisection), A160643 (first differences of a(n)).
Table with the mapped A125106(p) in row n where p runs through the partitions counted by A160644(n).
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2, 6, 12, 14, 24, 26, 30, 48, 50, 54, 62, 56, 60, 96, 98, 102, 110, 126, 104, 108, 114, 122, 192, 194, 198, 206, 222, 254, 120, 200, 204, 210, 218, 230, 246, 384, 386, 390, 398, 414, 446, 510, 216, 224, 228, 236, 242, 252, 392, 396, 402, 410, 422, 438, 462, 494, 768, 770
COMMENTS
A160644(n) with n > 0 counts the partitions of 2n such that all parts are > 1 and the largest part occurs more than once. If n=7, these are 10 partitions of 14: 2^7 = (2^4;3^2) = (2^1;3^4) = (2^3;4^2) = (3^2;4^2) = (2^1;4^3) = (2^2;5^2) = (4^1;5^2) = (2^1;6^2) = 7^2, for example.
For each of these admitted partitions p of 2n, p is mapped to a binary and the decimal rep. of this binary is added to row n of this table here, sorting the row according to the natural order of integers (not according to any property of partitions).
EXAMPLE
The partition 4+4+4+4 = 16 and maps to 120 = 64 + 32 + 16 + 8 as described in A125106, so 120 is in the 8th row. The table has A160644(n) integers in row n and starts 2,
6,.......[2,2]->6
12,14,..........[3,3]->12, [2,2,2]->14
24,26,30,...........[4,4]->24, [2,3,3]->26, [2,2,2,2] ->30
48,50,54,62, ....... [5,5]->48, [2,4,4]->50, [2,2,3,3]->54, [2,2,2,2,2]->62
56,60,96,98,102,110,126,.....[4,4,4]->56, [3,3,3,3]->60, [6,6]->96, [2,5,5]->98, [2,2,4,4]->102, [2,2,2,3,3]->110
104,108,114,122,192,194,198,206,222,254,...[4,5,5]->104, [3,3,4,4]->108, [2,4,4,4]->114, [2,3,3,3,3]->122
MAPLE
A125106m := proc(par) local c, dgs, p ; c := 1 ; dgs := [] ; for p in par do if p = c then dgs := [op(dgs), 1] ; else dgs := [op(dgs), seq(0, j=1..p-c), 1] ; fi; c := p ; od: add(op(i, dgs) *2^(i-1), i=1..nops(dgs)) ; end:
A161922 := proc(n) r := {} ; prts := combinat[partition](2*n) ; for p in prts do convert(p, set) intersect {1}; if % = {} then if nops(p) < 2 then ; elif op(-1, p) = op(-2, p) then r := r union {A125106m(p)} ; fi; fi; od: sort(r) ; end:
EXTENSIONS
Detailed description and examples and rows n >= 8 completed by R. J. Mathar, Sep 11 2009
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