A324969 -id:A324969 - cp's OEIS frontend

A348476 Number of compositions of n into exactly n nonnegative parts such that all positive parts are odd.

1, 1, 1, 4, 13, 36, 106, 323, 981, 2992, 9196, 28392, 87946, 273287, 851579, 2659764, 8324357, 26100560, 81969496, 257800532, 811862268, 2559731360, 8079294664, 25525787344, 80719066698, 255466082911, 809138591431, 2564605664428, 8134003910311, 25813957574292

Offset: 0

Alois P. Heinz, Oct 19 2021

nonn

			a(0) = 1: [].
a(1) = 1: [1].
a(2) = 1: [1,1].
a(3) = 4: [3,0,0], [0,3,0], [0,0,3], [1,1,1].
a(4) = 13: [3,1,0,0], [3,0,1,0], [3,0,0,1], [1,3,0,0], [0,3,1,0], [0,3,0,1],[1,0,3,0], [0,1,3,0], [0,0,3,1], [1,0,0,3], [0,1,0,3], [0,0,1,3], [1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..1968

Cf. A001700, A088218, A165817, A305161, A324969.

b:= proc(n, t) option remember; `if`(t=0, 1-signum(n),
      add(`if`(j=0 or j::odd, b(n-j, t-1), 0), j=0..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);

b[n_, t_] := b[n, t] = If[t == 0, 1 - Sign[n],
     Sum[If[j == 0 || OddQ[j], b[n - j, t - 1], 0], {j, 0, n}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)

a(n) ~ c * d^n / sqrt(Pi*n), where d = 3.22870495109450172934784925586... is largest positive root of the equation 4*d^4 - 12*d^3 + 4*d^2 - 24*d + 5 = 0 and c = 0.4302331663731241127284415754... is positive root of the equation 5824*c^8 - 32*c^4 - 4*c^2 - 5 = 0. - Vaclav Kotesovec, Nov 01 2021

A356605 Number of integer compositions of n into odd parts covering an interval of odd positive integers.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 10, 15, 26, 41, 65, 104, 164, 262, 424, 687, 1112, 1792, 2898, 4677, 7556, 12197, 19699, 31836, 51466, 83234, 134593, 217674, 352057, 569452, 921165, 1490173, 2410784, 3900288, 6310436, 10210358, 16521108, 26733020, 43258086, 69999295

Offset: 0

Gus Wiseman, Aug 31 2022

nonn

			The a(1) = 1 through a(8) = 15 compositions:
  (1)  (11)  (3)    (13)    (5)      (33)      (7)        (35)
             (111)  (31)    (113)    (1113)    (133)      (53)
                    (1111)  (131)    (1131)    (313)      (1133)
                            (311)    (1311)    (331)      (1313)
                            (11111)  (3111)    (11113)    (1331)
                                     (111111)  (11131)    (3113)
                                               (11311)    (3131)
                                               (13111)    (3311)
                                               (31111)    (111113)
                                               (1111111)  (111131)
                                                          (111311)
                                                          (113111)
                                                          (131111)
                                                          (311111)
                                                          (11111111)

These compositions are ranked by the intersection of A060142 and A356841.
Before restricting to odds we have A107428, initial A107429.
The not necessarily gapless version is A324969 (essentially A000045).
The strict case is A332032.
The initial case is A356604.
The case of partitions is A356737, initial A053251 (ranked by A356232).
A000041 counts partitions, compositions A011782.
A066208 lists numbers with all odd prime indices, counted by A000009.
A073491 lists numbers with gapless prime indices, initial A055932.
Cf. A001227, A066205, A137921, A333217, A356224, A356846.

nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@OddQ/@#&&nogapQ[(#+1)/2]&]],{n,0,15}]

More terms from Alois P. Heinz, Sep 01 2022

A348478 Number of compositions of n into exactly n nonnegative parts such that each positive i-th part has the same parity as i.

Original entry on oeis.org

1, 1, 1, 4, 7, 23, 55, 164, 407, 1235, 3051, 9432, 23431, 72989, 182624, 571384, 1436855, 4511979, 11387467, 35866100, 90782837, 286622226, 727226578, 2300578392, 5848776767, 18533394763, 47197285045, 149769168304, 381956145802, 1213526310665, 3098742448230

Offset: 0

Alois P. Heinz, Oct 20 2021

nonn

			a(0) = 1: [].
a(1) = 1: [1].
a(2) = 1: [0,2].
a(3) = 4: [1,2,0], [0,2,1], [3,0,0], [0,0,3].
a(4) = 7: [1,2,1,0], [1,0,1,2], [3,0,1,0], [1,0,3,0], [0,2,0,2], [0,4,0,0], [0,0,0,4].

Alois P. Heinz, Table of n, a(n) for n = 0..2169

Cf. A001700, A062200, A088218, A122514, A165817, A305161, A324969, A348476.

b:= proc(n, t) option remember; `if`(t=0, 1-signum(n),
      add(`if`(j=0 or (t-j)::even, b(n-j, t-1), 0), j=0..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..33);

b[n_, t_] := b[n, t] = If[t == 0, 1 - Sign[n],
     Sum[If[j == 0 || EvenQ[t - j], b[n - j, t - 1], 0], {j, 0, n}]];
a[n_] :=  b[n, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz *)

A330795 Evaluation of the polynomials given by the Riordan square of the Fibonacci sequence with a(0) = 1 (A193737) at 1/2 and normalized with 2^n.

Original entry on oeis.org

1, 3, 9, 39, 153, 615, 2457, 9831, 39321, 157287, 629145, 2516583, 10066329, 40265319, 161061273, 644245095, 2576980377, 10307921511, 41231686041, 164926744167, 659706976665, 2638827906663, 10555311626649, 42221246506599, 168884986026393, 675539944105575

Offset: 0

Peter Luschny, Jan 10 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,4).

Cf. A006131, A015521, A193737, A321620, A324969 (Fibonacci with a(0)=1).

[1] cat [3*(4^n -(-1)^n)/5: n in [1..30]]; // G. C. Greubel, Sep 14 2023

gf := (4*x^2 - 1)/(x*(4*x + 3) - 1): ser := series(gf, x, 32):
seq(coeff(ser, x, n), n=0.. 25);
# Alternative:
gf:= (3/5)*exp(-x)*(exp(5*x) - 1) + 1: ser := series(gf, x, 32):
seq(n!*coeff(ser, x, n), n=0.. 25);
# Or:
a := proc(n) option remember; if n < 3 then return [1, 3, 9][n + 1] fi;
4*a(n-2) + 3*a(n-1) end: seq(a(n), n=0..25);

LinearRecurrence[{3,4}, {1,3,9}, 31] (* G. C. Greubel, Sep 14 2023 *)

[3*(4^n -(-1)^n)//5 + int(n==0) for n in range(31)] # G. C. Greubel, Sep 14 2023

a(n) = 2^n*Sum_{k=0..n} A193737(n,k)/2^k.
a(n) = [x^n] (1 - 4*x^2)/(1 - x*(3 + 4*x)).
a(n) = n! [x^n] (3/5)*exp(-x)*(exp(5*x) - 1) + 1.
a(n) = 4*a(n-2) + 3*a(n-1).
a(n) = 3*A015521(n), n>0. - R. J. Mathar, Aug 19 2022

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A348476 Number of compositions of n into exactly n nonnegative parts such that all positive parts are odd.

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A356605 Number of integer compositions of n into odd parts covering an interval of odd positive integers.

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A348478 Number of compositions of n into exactly n nonnegative parts such that each positive i-th part has the same parity as i.

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A330795 Evaluation of the polynomials given by the Riordan square of the Fibonacci sequence with a(0) = 1 (A193737) at 1/2 and normalized with 2^n.

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