A182736
Sum of parts in all partitions of 2n that do not contain 1 as a part.
Original entry on oeis.org
0, 2, 8, 24, 56, 120, 252, 476, 880, 1584, 2740, 4620, 7680, 12428, 19824, 31170, 48224, 73678, 111384, 166364, 246120, 360822, 524216, 755504, 1080912, 1535050, 2165592, 3036096, 4230632, 5861828, 8078820, 11076362, 15112384, 20523492, 27747128
Offset: 0
-
b:= proc(n,i) option remember; local p,q;
if n<0 then [0,0]
elif n=0 then [1,0]
elif i<2 then [0,0]
else p, q:= b(n,i-1), b(n-i,i);
[p[1]+q[1], p[2]+q[2]+q[1]*i]
fi
end:
a:= n-> b(2*n,2*n)[2]:
seq(a(n), n=0..34); # Alois P. Heinz, Dec 03 2010
-
b[n_] := (PartitionsP[n] - PartitionsP[n-1])*n; a[n_] := b[2n]; Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Oct 07 2015 *)
A182992
Number of parts of the n-th subshell of the head of the last section of the set of partitions of any even integer >= 2n.
Original entry on oeis.org
1, 2, 5, 9, 17, 34, 55, 96, 163, 260, 413, 658, 1000, 1528, 2304, 3405, 5003, 7302, 10497, 15022, 21330, 30010, 41980, 58395, 80607, 110740, 151334, 205623, 278119, 374513, 501842, 669828, 890461, 1178960, 1555314
Offset: 1
a(5)=17 because the 5th subshell of the head of the last section of any even integer >= 10 looks like this:
(10 . . . . . . . . . )
( 5 . . . . 5 . . . . )
( 6 . . . . . 4 . . . )
( 7 . . . . . . 3 . . )
( 4 . . . 3 . . 3 . . )
. (2 . )
. (2 . )
. (2 . )
. (2 . )
. (2 . )
. (2 . )
. (2 . )
The subshell has 17 parts, so a(5)=17.
A182735
Number of parts in all partitions of 2n+1 that do not contain 1 as a part.
Original entry on oeis.org
0, 1, 3, 8, 20, 41, 80, 153, 271, 469, 795, 1305, 2102, 3336, 5190, 7968, 12090, 18104, 26821, 39371, 57220, 82472, 117958, 167405, 235945, 330425, 459803, 636142, 875307, 1197983, 1631470, 2211377, 2983695, 4008386, 5362831, 7146335, 9486834, 12548085, 16538651
Offset: 0
-
b:= proc(n,i) option remember; local p,q;
if n<0 then [0,0]
elif n=0 then [1,0]
elif i<2 then [0,0]
else p, q:= b(n,i-1), b(n-i,i);
[p[1]+q[1], p[2]+q[2]+q[1]]
fi
end:
a:= n-> b(2*n+1, 2*n+1)[2]:
seq(a(n), n=0..35); # Alois P. Heinz, Dec 03 2010
-
b[n_, i_] := b[n, i] = Module[{p, q}, Which[n<0, {0, 0}, n == 0, {1, 0}, i < 2, {0, 0}, True, {p, q} = {b[n, i-1], b[n-i, i]}; {p[[1]] + q[[1]], p[[2]] + q[[2]] + q[[1]]}]]; a[n_] := b[2*n+1, 2*n+1][[2]]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Oct 29 2015, after Alois P. Heinz *)
A182812
Triangle read by rows in which row n lists the parts of the largest subshell of all partitions of 2n that do not contain 1 as a part.
Original entry on oeis.org
2, 4, 2, 6, 3, 3, 2, 2, 8, 4, 4, 5, 3, 2, 2, 2, 2, 10, 5, 5, 6, 4, 7, 3, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2, 12, 6, 6, 7, 5, 8, 4, 4, 4, 4, 9, 3, 5, 4, 3, 6, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 14, 7, 7, 8, 6, 9, 5
Offset: 1
For n=1 the unique partition of 2n that does not contains 1 as part is 2, so row 1 has an element = 2.
For n=2 there are 2 partitions of 2n that do not contain 1 as part:
4 ............ or ....... 4 . . .
2 + 2 ........ or .......(2). 2 .
These partitions contain (2), the row n-1 of triangle, so
the parts of the largest subshell are 4, 2.
For n=3 there are 4 partitions of 2n that do not contain 1 as part:
6 ............ or ....... 6 . . . . .
3 + 3 ........ or ....... 3 . . 3 . .
4 + 2 ........ or .......(4). . . 2 .
2 + 2 + 2 .... or .......(2).(2). 2 .
These partitions contain (4) and (2),(2), the parts of rows < n of triangle, so the parts of the largest subshell are 6, 3, 3, 2, 2.
And so on.
Triangle begins:
2,
4, 2,
6, 3, 3, 2, 2,
8, 4, 4, 5, 3, 2, 2, 2, 2,
10, 5, 5, 6, 4, 7, 3, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2,
Showing 1-4 of 4 results.
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