A304797 -id:A304797 - cp's OEIS frontend

A339351 Irregular triangle read by rows in which row n lists the compositions (ordered partitions) of n into distinct parts in lexicographic order.

1, 2, 1, 2, 2, 1, 3, 1, 3, 3, 1, 4, 1, 4, 2, 3, 3, 2, 4, 1, 5, 1, 2, 3, 1, 3, 2, 1, 5, 2, 1, 3, 2, 3, 1, 2, 4, 3, 1, 2, 3, 2, 1, 4, 2, 5, 1, 6, 1, 2, 4, 1, 4, 2, 1, 6, 2, 1, 4, 2, 4, 1, 2, 5, 3, 4, 4, 1, 2, 4, 2, 1, 4, 3, 5, 2, 6, 1, 7, 1, 2, 5, 1, 3, 4, 1, 4, 3, 1, 5, 2, 1, 7, 2, 1, 5, 2, 5, 1, 2, 6, 3, 1, 4, 3, 4, 1, 3, 5, 4, 1, 3, 4, 3, 1, 5, 1, 2, 5, 2, 1, 5, 3, 6, 2, 7, 1, 8

Offset: 1

Ilya Gutkovskiy, Dec 01 2020

			Triangle begins:
[1],
[2],
[1, 2], [2, 1], [3],
[1, 3], [3, 1], [4],
[1, 4], [2, 3], [3, 2], [4, 1], [5],
...

Index entries for sequences related to compositions

Cf. A026793, A066099, A097910 (row lengths), A118457, A228369, A246688, A304797 (row sums), A339178.

Table[Sort[Join @@ Permutations /@ Select[IntegerPartitions[n], UnsameQ @@ # &], OrderedQ[PadRight[{#1, #2}]] &], {n, 8}] // Flatten

A304908 Expansion of x * (d/dx) 1/(1 - Sum_{k>=0} x^(2^k)).

Original entry on oeis.org

0, 1, 4, 9, 24, 50, 108, 217, 448, 882, 1740, 3366, 6504, 12428, 23660, 44745, 84352, 158270, 296064, 551950, 1026360, 1903524, 3522596, 6504998, 11990160, 22061700, 40528748, 74343096, 136183488, 249145148, 455265420, 830985473, 1515201792, 2760087990, 5023154832, 9133857670

Offset: 0

Ilya Gutkovskiy, May 20 2018

nonn

Sum of all parts of all compositions (ordered partitions) of n into powers of 2.

Index entries for sequences related to compositions

Cf. A000079, A001787, A023359, A036987, A281811, A304797, A303666, A304909.

nmax = 35; CoefficientList[Series[x D[1/(1 - Sum[x^2^k, {k, 0, Floor[Log[nmax]/Log[2]] + 1}]), x], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Boole[k == 2^IntegerExponent[k, 2]] a[n - k], {k, 1, n}]; Table[n a[n], {n, 0, 35}]

a(n) = n*A023359(n).

A339178 Irregular triangle read by rows in which row n lists the compositions (ordered partitions) of n into distinct parts in reverse lexicographic order.

Original entry on oeis.org

1, 2, 3, 2, 1, 1, 2, 4, 3, 1, 1, 3, 5, 4, 1, 3, 2, 2, 3, 1, 4, 6, 5, 1, 4, 2, 3, 2, 1, 3, 1, 2, 2, 4, 2, 3, 1, 2, 1, 3, 1, 5, 1, 3, 2, 1, 2, 3, 7, 6, 1, 5, 2, 4, 3, 4, 2, 1, 4, 1, 2, 3, 4, 2, 5, 2, 4, 1, 2, 1, 4, 1, 6, 1, 4, 2, 1, 2, 4, 8, 7, 1, 6, 2, 5, 3, 5, 2, 1, 5, 1, 2, 4, 3, 1, 4, 1, 3, 3, 5, 3, 4, 1, 3, 1, 4, 2, 6, 2, 5, 1, 2, 1, 5, 1, 7, 1, 5, 2, 1, 4, 3, 1, 3, 4, 1, 2, 5

Offset: 1

Ilya Gutkovskiy, Nov 26 2020

			Triangle begins:
[1],
[2],
[3], [2, 1], [1, 2],
[4], [3, 1], [1, 3],
[5], [4, 1], [3, 2], [2, 3], [1, 4],
...

Index entries for sequences related to compositions

Cf. A026793, A066099, A097910 (row lengths), A118457, A228369, A246688, A304797 (row sums).

Table[Sort[Join @@ Permutations /@ Select[IntegerPartitions[n], UnsameQ @@ # &], OrderedQ[PadRight[{#2, #1}]] &], {n, 8}] // Flatten

cp's OEIS Frontend

A339351 Irregular triangle read by rows in which row n lists the compositions (ordered partitions) of n into distinct parts in lexicographic order.

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A304908 Expansion of x * (d/dx) 1/(1 - Sum_{k>=0} x^(2^k)).

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Keywords

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Programs

Mathematica

Formula

A339178 Irregular triangle read by rows in which row n lists the compositions (ordered partitions) of n into distinct parts in reverse lexicographic order.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica