A318156
Expansion of (1/(1 - x)) * Sum_{k>=1} x^(k*(2*k-1)) / Product_{j=1..2*k-1} (1 - x^j).
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 27, 35, 44, 55, 69, 85, 104, 127, 154, 186, 224, 268, 320, 381, 452, 534, 630, 741, 869, 1017, 1187, 1382, 1606, 1862, 2155, 2489, 2869, 3301, 3792, 4349, 4979, 5692, 6497, 7405, 8429, 9581, 10876, 12331, 13963, 15792, 17840, 20131, 22691
Offset: 0
From _Gus Wiseman_, Jul 18 2021: (Start)
Also the number of strict integer partitions of 2n+1 of even length with exactly one odd part. For example, the a(1) = 1 through a(8) = 12 partitions are:
(2,1) (3,2) (4,3) (5,4) (6,5) (7,6) (8,7) (9,8)
(4,1) (5,2) (6,3) (7,4) (8,5) (9,6) (10,7)
(6,1) (7,2) (8,3) (9,4) (10,5) (11,6)
(8,1) (9,2) (10,3) (11,4) (12,5)
(10,1) (11,2) (12,3) (13,4)
(12,1) (13,2) (14,3)
(6,4,2,1) (14,1) (15,2)
(6,4,3,2) (16,1)
(8,4,2,1) (6,5,4,2)
(8,4,3,2)
(8,6,2,1)
(10,4,2,1)
Also the number of integer partitions of 2n+1 covering an initial interval and having even maximum and alternating sum 1.
(End)
The following relate to strict integer partitions of 2n+1 of even length with exactly one odd part.
- Allowing any length gives
A036469.
- The non-strict version is
A306145.
- Allowing any number of odd parts gives
A343942 (odd bisection of
A067661).
A103919 counts partitions by sum and alternating sum (reverse:
A344612).
Cf.
A000070,
A030229,
A035294,
A058696,
A078616,
A087447,
A152146,
A236559,
A343941,
A344611,
A344739.
-
b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
end:
a:= proc(n) option remember; b(n$2, 0)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..60);
-
nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k (2 k - 1))/Product[(1 - x^j), {j, 1, 2 k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 53; CoefficientList[Series[(QPochhammer[-x, x] - QPochhammer[x])/(2 (1 - x)), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&EvenQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,0,15}] (* _Gus Wiseman, Jul 18 2021 *)
A078407
Number of ways to partition 4*n+2 into distinct positive integers.
Original entry on oeis.org
1, 4, 10, 22, 46, 89, 165, 296, 512, 864, 1426, 2304, 3658, 5718, 8808, 13394, 20132, 29927, 44046, 64234, 92864, 133184, 189586, 267968, 376256, 525016, 728260, 1004544, 1378304, 1881578, 2556284, 3457027, 4654670, 6240974, 8334326
Offset: 0
a(n) = t(4*n+2, 0), t as defined in
A079211.
A078409
Number of ways to partition 4*n+1 into distinct positive integers.
Original entry on oeis.org
1, 3, 8, 18, 38, 76, 142, 256, 448, 760, 1260, 2048, 3264, 5120, 7917, 12076, 18200, 27130, 40026, 58499, 84756, 121792, 173682, 245920, 345856, 483330, 671418, 927406, 1274118, 1741521, 2368800, 3207086, 4322816, 5802008, 7755776
Offset: 0
A318155
Expansion of (1/(1 - x)) * Sum_{k>=0} x^(k*(2*k+1)) / Product_{j=1..2*k} (1 - x^j).
Original entry on oeis.org
1, 1, 1, 2, 3, 5, 7, 10, 13, 17, 22, 28, 35, 44, 55, 68, 84, 103, 126, 153, 185, 223, 268, 320, 381, 452, 535, 631, 742, 870, 1018, 1188, 1383, 1607, 1863, 2155, 2489, 2869, 3301, 3792, 4348, 4978, 5691, 6496, 7404, 8428, 9580, 10875, 12330, 13962, 15791, 17840, 20131, 22691
Offset: 0
A000070 counts partitions with alternating sum 1.
A103919 counts partitions by sum and alternating sum (reverse:
A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf.
A027193,
A035294,
A067659,
A087447,
A236559,
A236914,
A239829,
A306145,
A344611,
A344739,
A346634.
-
nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k (2 k + 1))/Product[(1 - x^j), {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 53; CoefficientList[Series[(QPochhammer[-x, x] + QPochhammer[x])/(2 (1 - x)), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&OddQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,0,15}] (* _Gus Wiseman, Jul 29 2021 *)
A323891
a(n) is the number of partitions of 72*n + 42 into 10 odd squares.
Original entry on oeis.org
2, 9, 22, 41, 68, 106, 154, 212, 285, 368, 477, 598, 741, 898, 1076, 1286, 1524, 1785, 2068, 2379, 2741, 3131, 3554, 4002, 4497, 5044, 5644, 6274, 6939, 7653, 8445, 9295, 10186, 11117, 12113, 13192, 14355, 15556, 16807, 18147, 19570, 21089, 22673, 24300, 26029, 27865, 29821, 31822, 33894, 36088
Offset: 0
For n=0, 72*0+42 = 42 = 25+9+1+1+1+1+1+1+1+1 = 9+9+9+9+1+1+1+1+1+1, so a(0)=2.
For n=1, 72*1+42 = 114 = 81+25+1+1+1+1+1+1+1+1 = 81+9+9+9+1+1+1+1+1+1 = 49+49+9+1+1+1+1+1+1+1 = 49+25+25+9+1+1+1+1+1+1 = 49+25+9+9+9+9+1+1+1+1 = 49+9+9+9+9+9+9+9+1+1 = 25+25+25+25+9+1+1+1+1+1 = 25+25+25+9+9+9+9+1+1+1 = 25+25+9+9+9+9+9+9+9+1, so a(1)=9.
- Laurențiu Panaitopol, Alexandru Gica, Arithmetic problems and number theory, Ed. Gil, Zalău, (2006), ch. 14, p. 85, pr. 32. (in Romanian).
Cf.
A000041,
A001156,
A016754,
A025425,
A025434,
A033461,
A035294,
A078406,
A090677,
A167661,
A167700.
-
[#RestrictedPartitions(72*n+42, 10, {(2*d+1)^2:d in [0..100]}): n in [0..100]];
-
S:= proc(n, k, m)
option remember;
local p,j;
if k = 0 then if n = 0 then return 1 else return 0 fi
elif m < 1 then return 0
elif n < k then return 0
elif n > k*m^2 then return 0
fi;
if m^2 > n then
p:= floor(sqrt(n));
if p::even then p:= p-1 fi;
return procname(n, k, p)
fi;
add(procname(n-j*m^2,k-j,m-2), j=0..n/m^2)
end proc:
seq(S(72*n+42, 10, 72*n+42), n=0..100); # Robert Israel, Feb 24 2019
-
a[n_] := IntegerPartitions[72n+42, {10}, Select[ Range[1, 72n+42, 2], IntegerQ@Sqrt@#&]] // Length;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 19 2022 *)
A168431
General number of transpositions of n objects with repetitions.
Original entry on oeis.org
1, 2, 9, 46, 245, 1601
Offset: 1
a(1)=1!/1!=1 because A035294(1)=1; a(2)=2!/(1!*1!)=2 because A035294(2)=1; a(3)=3!/(1!*1!*1!)+3!/(2!*1!)=9 because A035294(3)=2; a(4)=4!/(1!*1!*1!*1!)+4!/(2!*1!*1!)+4!/(2!*2!)+4!/(3!*1!)=46 because A035294(4)=4; a(5)=5!/(1!*1!*1!*1!*1!)+5!/(2!*1!*1!*1!)+5!/(2!*2!*1!)+5!/(3!*1!*1!)+5!/(3!*2!)+5!/(4!*1!)=245 because A035294(5)=6; a(6)=6!/(1!*1!*1!*1!*1!*1!)+6!/(2!*1!*1!*1!*1!)+6!/(2!*2!*1!*1!)+6!/(2!*2!*2!)+6!/(3!*1!*1!*1!)+6!/(3!*2!*1!)+6!/(3!*3!)+6!/(4!*1!*1!)+6!/(4!*2!)+6!/(5!*1!)=1601 because A035294(6)=10.
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