A242391
Number of compositions of n in which each part has odd multiplicity.
Original entry on oeis.org
1, 1, 1, 4, 3, 10, 16, 28, 49, 91, 186, 266, 670, 884, 2350, 3028, 8259, 10536, 30241, 37382, 108628, 135550, 391202, 503750, 1429838, 1884659, 5222976, 7107138, 19119324, 27088726, 70366026, 103884570, 259884905, 399686188, 962312254, 1543116240, 3576132805
Offset: 0
a(0) = 1: the empty composition.
a(1) = 1: [1].
a(2) = 1: [2].
a(3) = 4: [3], [2,1], [1,2], [1,1,1].
a(4) = 3: [4], [3,1], [1,3].
a(5) = 10: [5], [4,1], [1,4], [3,2], [2,3], [2,1,1,1], [1,2,1,1], [1,1,2,1], [1,1,1,2], [1,1,1,1,1].
Cf.
A130495 (for even multiplicity).
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b:= proc(n, i, p) option remember; `if`(n=0, p!,
`if`(i<1, 0, add(`if`(j=0 or irem(j, 2)=1,
b(n-i*j, i-1, p+j)/j!, 0), j=0..n/i)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..45);
-
b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, Sum[If[j==0 || Mod[j, 2]==1, b[n-i*j, i-1, p+j]/j!, 0], {j, 0, n/i}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
A322132
Number of compositions of n that can be rearranged into a palindrome.
Original entry on oeis.org
1, 1, 2, 2, 6, 8, 18, 26, 56, 90, 202, 320, 730, 1154, 2592, 4130, 9522, 15208, 35330, 56746, 131352, 212074, 492570, 795920, 1855706, 3006482, 7016464, 11406930, 26635154, 43409752, 101387602, 165798282, 386965208, 635250986, 1480773866, 2439516656, 5678477866
Offset: 0
Case n=4: The 6 compositions are: 4, 211, 121, 112, 22, 1111.
Case n=5: The 8 compositions are: 5, 311, 131, 113, 122, 212, 221, 11111.
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b:= proc(n, i, p, t) option remember; `if`(n=0 or i=1,
`if`(t and n::odd, 0, (n+p)!/n!), add(`if`(t and j::odd,
0, b(n-i*j, i-1, p+j, t or j::odd))/j!, j=0..n/i))
end:
a:= n-> b(n$2, 0, false):
seq(a(n), n=0..40); # Alois P. Heinz, Dec 02 2018
-
b[n_, i_, p_, t_] := b[n, i, p, t] = If[n == 0 || i == 1, If[t && OddQ[n], 0, (n+p)!/n!], Sum[If[t && OddQ[j], 0, b[n-i*j, i-1, p+j, t || OddQ[j]]]/ j!, {j, 0, n/i}]];
a[n_] := b[n, n, 0, False];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 13 2019, after Alois P. Heinz *)
-
a(n)={sum(k=0, n\2, my(p=prod(i=1, k, 1 + sum(j=1, k\i, x^(i*j)*y^(2*j)/(2*j + (i==n-2*k))!) + O(x*x^k))); subst(serlaplace(polcoef(p,k)*y^(2*k
A322326
Number of compositions of n in which at most one part has even multiplicity.
Original entry on oeis.org
1, 1, 2, 4, 8, 16, 26, 64, 107, 226, 382, 859, 1488, 3124, 5628, 11326, 21629, 42274, 81420, 158002, 309129, 592798, 1181109, 2234319, 4501108, 8461706, 17211219, 32187953, 66018320, 122792362, 253549269, 469715744, 975300728, 1802165555, 3758679309, 6931995005
Offset: 0
a(6) = 26: 111111, 11112, 11121, 11211, 12111, 21111, 222, 1113, 1131, 1311, 3111, 123, 132, 213, 231, 312, 321, 33, 114, 141, 411, 24, 42, 15, 51, 6.
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b:= proc(n, i, p, t) option remember; `if`(n=0 or i=1, `if`(n>0 and
t and n::even, 0, (n+p)!/n!), b(n, i-1, p, t)+add(`if`(t and
j::even, 0, b(n-i*j, i-1, p+j, t or j::even))/j!, j=1..n/i))
end:
a:= n-> b(n$2, 0, false):
seq(a(n), n=0..40);
-
b[n_, i_, p_, t_] := b[n, i, p, t] = If[n == 0 || i == 1, If[n > 0 &&
t && EvenQ[n], 0, (n + p)!/n!], b[n, i - 1, p, t] + Sum[If[t &&
EvenQ[j], 0, b[n - i*j, i-1, p+j, t || EvenQ[j]]]/j!, {j, 1, n/i}]];
a[n_] := b[n, n, 0, False];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 05 2022, after Alois P. Heinz *)
Original entry on oeis.org
1, 4, 11, 37, 163, 907, 6067, 47107, 415027, 4084147, 44363827, 526994227, 6793931827, 94451224627, 1408352613427, 22418320792627, 379413423256627, 6802709918872627, 128803497755800627, 2568107879638168627, 53780695151756440627, 1180214324937540760627
Offset: 1
a(5) = 163 sum of row 5 terms of triangle A130478: (120 + 30 + 8 + 3 + 2); where (30, 8, 3, 2) = the first 4 reversed terms of A001048.
a(5) = 163 = 5! + A130495(4) = 120 + 43.
a(5) = 163 = 5! + (4! + 3!) + (3! + 2!) + (2! + 1!) + (1! + 1).
Showing 1-4 of 4 results.
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