A238341 -id:A238341 - cp's OEIS frontend

A243740 Number of compositions of n with exactly five occurrences of the largest part.

1, 0, 0, 0, 0, 1, 6, 21, 56, 126, 253, 468, 819, 1385, 2317, 3928, 6908, 12803, 25079, 51415, 108598, 232784, 500780, 1073700, 2285703, 4822956, 10082161, 20884730, 42892750, 87405633, 176867184, 355685658, 711460052, 1416584340, 2809770487, 5555833511

Offset: 5

Joerg Arndt and Alois P. Heinz, Jun 09 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 5..650

Column k=5 of A238341.

b:= proc(n, p, i) option remember; `if`(n=0, p!,
      `if`(i<1, 0, add(b(n-i*j, p+j, i-1)/j!, j=0..n/i)))
    end:
a:= proc(n) local k; k:=5;
      add(b(n-i*k, k, i-1)/k!, i=1..n/k)
    end:
seq(a(n), n=5..50);

b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!, If[i<1, 0, Sum[b[n-i*j, p+j, i-1]/j!, {j, 0, n/i}]]]; a[n_] := (k=5; Sum[b[n-i*k, k, i-1]/k!, {i, 1, n/k}]); Table[a[n], {n, 5, 40}] (* Jean-François Alcover, Feb 10 2015, after Maple *)

A243741 Number of compositions of n with exactly six occurrences of the largest part.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 7, 28, 84, 210, 462, 925, 1723, 3038, 5145, 8498, 13930, 23143, 39859, 72500, 140266, 286678, 609217, 1323197, 2897511, 6339636, 13787488, 29726648, 63472185, 134190162, 280988786, 583076160, 1199816068, 2449963412, 4967798570, 10009806563

Offset: 6

Joerg Arndt and Alois P. Heinz, Jun 09 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 6..650

Column k=6 of A238341.

b:= proc(n, p, i) option remember; `if`(n=0, p!,
      `if`(i<1, 0, add(b(n-i*j, p+j, i-1)/j!, j=0..n/i)))
    end:
a:= proc(n) local k; k:=6;
      add(b(n-i*k, k, i-1)/k!, i=1..n/k)
    end:
seq(a(n), n=6..50);

b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!, If[i<1, 0, Sum[b[n-i*j, p+j, i-1]/j!, {j, 0, n/i}]]]; a[n_] := (k=6; Sum[b[n-i*k, k, i-1]/k!, {i, 1, n/k}]); Table[a[n], {n, 6, 40}] (* Jean-François Alcover, Feb 10 2015, after Maple *)

A243742 Number of compositions of n with exactly seven occurrences of the largest part.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 8, 36, 120, 330, 792, 1716, 3433, 6443, 11484, 19640, 32550, 52860, 85296, 139249, 235001, 418473, 795544, 1610418, 3421514, 7489962, 16625389, 37003313, 82024320, 180421399, 393126594, 848051064, 1811227670, 3831269241, 8030748161

Offset: 7

Joerg Arndt and Alois P. Heinz, Jun 09 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 7..650

Column k=7 of A238341.

b:= proc(n, p, i) option remember; `if`(n=0, p!,
      `if`(i<1, 0, add(b(n-i*j, p+j, i-1)/j!, j=0..n/i)))
    end:
a:= proc(n) local k; k:=7;
      add(b(n-i*k, k, i-1)/k!, i=1..n/k)
    end:
seq(a(n), n=7..50);

b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!, If[i<1, 0, Sum[b[n-i*j, p+j, i-1]/j!, {j, 0, n/i}]]]; a[n_] := (k=7; Sum[b[n-i*k, k, i-1]/k!, {i, 1, n/k}]); Table[a[n], {n, 7, 40}] (* Jean-François Alcover, Feb 10 2015, after Maple *)

A243743 Number of compositions of n with exactly eight occurrences of the largest part.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 9, 45, 165, 495, 1287, 3003, 6435, 12871, 24319, 43812, 75837, 127005, 207252, 332343, 529617, 851797, 1410484, 2456794, 4572624, 9116790, 19248417, 42237738, 94608183, 213426424, 480788350, 1076330078, 2388681769, 5249788389

Offset: 8

Joerg Arndt and Alois P. Heinz, Jun 09 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 8..650

Column k=8 of A238341.

b:= proc(n, p, i) option remember; `if`(n=0, p!,
      `if`(i<1, 0, add(b(n-i*j, p+j, i-1)/j!, j=0..n/i)))
    end:
a:= proc(n) local k; k:=8;
      add(b(n-i*k, k, i-1)/k!, i=1..n/k)
    end:
seq(a(n), n=8..50);

b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!, If[i<1, 0, Sum[b[n-i*j, p+j, i-1]/j!, {j, 0, n/i}]]]; a[n_] := (k=8; Sum[b[n-i*k, k, i-1]/k!, {i, 1, n/k}]); Table[a[n], {n, 8, 50}] (* Jean-François Alcover, Feb 10 2015, after Maple *)

A243744 Number of compositions of n with exactly nine occurrences of the largest part.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48621, 92388, 168025, 294260, 498850, 822712, 1327029, 2107325, 3324750, 5280276, 8595025, 14648920, 26637015, 52076915, 108669142, 237787000, 535084341, 1219016810, 2782344676

Offset: 9

Joerg Arndt and Alois P. Heinz, Jun 09 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 9..650

Column k=9 of A238341.

b:= proc(n, p, i) option remember; `if`(n=0, p!,
      `if`(i<1, 0, add(b(n-i*j, p+j, i-1)/j!, j=0..n/i)))
    end:
a:= proc(n) local k; k:=9;
      add(b(n-i*k, k, i-1)/k!, i=1..n/k)
    end:
seq(a(n), n=9..50);

b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!, If[i<1, 0, Sum[b[n-i*j, p+j, i-1]/j!, {j, 0, n/i}]]]; a[n_] := (k=9; Sum[b[n-i*k, k, i-1]/k!, {i, 1, n/k}]); Table[a[n], {n, 9, 50}] (* Jean-François Alcover, Feb 10 2015, after Maple *)

A243745 Number of compositions of n with exactly ten occurrences of the largest part.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184757, 352727, 646723, 1144484, 1963181, 3276625, 5341050, 8537815, 13454155, 21056035, 33090201, 53057236, 88592087, 157261599, 300524147, 616878471, 1339454952, 3015109174

Offset: 10

Joerg Arndt and Alois P. Heinz, Jun 09 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 10..650

Column k=10 of A238341.

b:= proc(n, p, i) option remember; `if`(n=0, p!,
      `if`(i<1, 0, add(b(n-i*j, p+j, i-1)/j!, j=0..n/i)))
    end:
a:= proc(n) local k; k:=10;
      add(b(n-i*k, k, i-1)/k!, i=1..n/k)
    end:
seq(a(n), n=10..50);

b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!, If[i<1, 0, Sum[b[n-i*j, p+j, i-1]/j!, {j, 0, n/i}]]]; a[n_] := (k=10; Sum[b[n-i*k, k, i-1]/k!, {i, 1, n/k}]); Table[ a[n], {n, 10, 50}] (* Jean-François Alcover, Feb 10 2015, after Maple *)

A243746 Number of compositions of n^2 with exactly n occurrences of the largest part.

Original entry on oeis.org

1, 1, 1, 21, 686, 108598, 134190162, 581266801787, 7792898359869376, 343349252968004533986, 60917528224825622999788393, 57691110936849283646013592507915, 280564704602761525363382338982479319450, 5619591974217690324311922622790661532819536973

Offset: 0

Alois P. Heinz, Jun 09 2014

nonn

			a(3) = 21: 333, 111222, 112122, 112212, 112221, 121122, 121212, 121221, 122112, 122121, 122211, 211122, 211212, 211221, 212112, 212121, 212211, 221112, 221121, 221211, 222111.

Alois P. Heinz, Table of n, a(n) for n = 0..33

Cf. A238341.

b:= proc(n, p, i) option remember; `if`(n=0, p!,
      `if`(i<1, 0, add(b(n-i*j, p+j, i-1)/j!, j=0..n/i)))
    end:
a:= n-> add(b(n^2-i*n, n, i-1)/n!, i=0..n):
seq(a(n), n=0..15);

b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!,
     If[i < 1, 0, Sum[b[n - i*j, p + j, i - 1]/j!, {j, 0, n/i}]]];
a[n_] := Sum[b[n^2 - i*n, n, i - 1]/n!, {i, 0, n}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Mar 04 2022, after Alois P. Heinz *)

a(n) = A238341(n^2,n).

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A243740 Number of compositions of n with exactly five occurrences of the largest part.

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A243741 Number of compositions of n with exactly six occurrences of the largest part.

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A243742 Number of compositions of n with exactly seven occurrences of the largest part.

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A243743 Number of compositions of n with exactly eight occurrences of the largest part.

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A243744 Number of compositions of n with exactly nine occurrences of the largest part.

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A243745 Number of compositions of n with exactly ten occurrences of the largest part.

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Maple

Mathematica

A243746 Number of compositions of n^2 with exactly n occurrences of the largest part.

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