A238344 -id:A238344 - cp's OEIS frontend

A241632 Number of compositions of n with exactly seven descents.

21, 259, 1932, 10898, 51083, 208936, 768594, 2594957, 8158005, 24140097, 67794864, 181889780, 468678716, 1164905882, 2803070038, 6549961919, 14901997455, 33085162997, 71822232573, 152710556875, 318509850193, 652536555773, 1314728710475, 2607856668543

Offset: 21

Joerg Arndt and Alois P. Heinz, Apr 26 2014

nonn

			a(21) = 21: [3,2,1,2,1,3,2,1,3,2,1], [3,2,1,3,2,1,2,1,3,2,1], [3,2,1,3,2,1,3,2,1,2,1], [2,1,3,2,1,3,2,1,3,2,1], [2,1,2,1,2,1,3,2,1,3,2,1], [3,2,1,3,2,1,2,1,2,1,2,1], [2,1,2,1,3,2,1,2,1,3,2,1], [2,1,3,2,1,2,1,3,2,1,2,1], [3,2,1,2,1,3,2,1,2,1,2,1], [3,2,1,2,1,2,1,2,1,3,2,1], [2,1,3,2,1,2,1,2,1,3,2,1], [3,2,1,2,1,2,1,3,2,1,2,1], [2,1,2,1,3,2,1,3,2,1,2,1], [2,1,3,2,1,3,2,1,2,1,2,1], [3,2,1,2,1,2,1,2,1,2,1,2,1], [2,1,2,1,2,1,2,1,2,1,3,2,1], [2,1,3,2,1,2,1,2,1,2,1,2,1], [2,1,2,1,2,1,2,1,3,2,1,2,1], [2,1,2,1,2,1,3,2,1,2,1,2,1], [2,1,2,1,3,2,1,2,1,2,1,2,1], [2,1,2,1,2,1,2,1,2,1,2,1,2,1].

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 21..1000

Column k=7 of A238343 and of A238344.

b:= proc(n, i) option remember;
      `if`(n=0, 1, convert(series(add(b(n-j, j)*
      `if`(j coeff(b(n, 0), x, 7):
seq(a(n), n=21..50);

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, j]*
   If[j < i, x, 1], {j, 1, n}] // Expand];
a[n_] := Coefficient[b[n, 0], x, 7];
Table[a[n], {n, 21, 50}] (* Jean-François Alcover, Nov 28 2023, from Maple code *)

A241633 Number of compositions of n with exactly eight descents.

Original entry on oeis.org

34, 474, 3914, 24222, 123713, 548540, 2178219, 7909965, 26662725, 84357961, 252677505, 721404913, 1974020281, 5200561960, 13240895310, 32684520969, 78435463950, 183422986603, 418849888662, 935640152746, 2047838246581, 4397763452093, 9278220501698

Offset: 24

Joerg Arndt and Alois P. Heinz, Apr 26 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 24..1000

Column k=8 of A238343 and of A238344.

b:= proc(n, i) option remember;
      `if`(n=0, 1, convert(series(add(b(n-j, j)*
      `if`(j coeff(b(n, 0), x, 8):
seq(a(n), n=24..50);

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, j]*
   If[j < i, x, 1], {j, 1, n}] // Expand];
a[n_] := Coefficient[b[n, 0], x, 8];
Table[a[n], {n, 24, 50}] (* Jean-François Alcover, Nov 28 2023, from Maple code *)

A241634 Number of compositions of n with exactly nine descents.

Original entry on oeis.org

55, 856, 7752, 52203, 288439, 1377331, 5868260, 22792116, 81944050, 275852636, 877202199, 2653544633, 7679263550, 21360356057, 57331068005, 148970797526, 375812977280, 922708833921, 2209559805490, 5170203452802, 11840994145688, 26581854857244, 58569145064714

Offset: 27

Joerg Arndt and Alois P. Heinz, Apr 26 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 27..1000

Column k=9 of A238343 and of A238344.

b:= proc(n, i) option remember;
      `if`(n=0, 1, convert(series(add(b(n-j, j)*
      `if`(j coeff(b(n, 0), x, 9):
seq(a(n), n=27..50);

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, j]*
   If[j < i, x, 1], {j, 1, n}] // Expand];
a[n_] := Coefficient[b[n, 0], x, 9];
Table[a[n], {n, 27, 50}] (* Jean-François Alcover, Nov 28 2023, from Maple code *)

A241635 Number of compositions of n with exactly ten descents.

Original entry on oeis.org

89, 1529, 15072, 109700, 651755, 3333143, 15159290, 62673818, 239267852, 853417204, 2869778022, 9163554647, 27947044811, 81799126374, 230699706092, 629085587197, 1663426352840, 4275866388599, 10708451601656, 26178837135330, 62580214195713, 146503017803381

Offset: 30

Joerg Arndt and Alois P. Heinz, Apr 26 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 30..1000

Column k=10 of A238343 and of A238344.

b:= proc(n, i) option remember;
      `if`(n=0, 1, convert(series(add(b(n-j, j)*
      `if`(j coeff(b(n, 0), x, 10):
seq(a(n), n=30..55);

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, j]*
   If[j < i, x, 1], {j, 1, n}] // Expand];
a[n_] := Coefficient[b[n, 0], x, 10];
Table[a[n], {n, 30, 55}] (* Jean-François Alcover, Nov 28 2023, from Maple code *)

cp's OEIS Frontend

A241632 Number of compositions of n with exactly seven descents.

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A241633 Number of compositions of n with exactly eight descents.

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A241634 Number of compositions of n with exactly nine descents.

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Maple

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A241635 Number of compositions of n with exactly ten descents.

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Maple

Mathematica