A232665 -id:A232665 - cp's OEIS frontend

A238342 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with exactly k occurrences of the smallest part, n>=0, 0<=k<=n.

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 3, 4, 0, 1, 0, 8, 3, 4, 0, 1, 0, 11, 10, 5, 5, 0, 1, 0, 20, 18, 14, 5, 6, 0, 1, 0, 34, 35, 24, 21, 6, 7, 0, 1, 0, 59, 60, 59, 35, 27, 7, 8, 0, 1, 0, 96, 121, 108, 85, 49, 35, 8, 9, 0, 1, 0, 167, 217, 213, 175, 125, 63, 44, 9, 10, 0, 1, 0, 282, 391, 419, 366, 258, 176, 80, 54, 10, 11, 0, 1

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Conjecture: Generally, for k > 0 is a(n) ~ n^k * ((1+sqrt(5))/2)^(n-2*k-1) / (5^((k+1)/2) * k!). Holds for all k<=10. - Vaclav Kotesovec, May 02 2014
G.f.: 1 + Sum_{i>0} (-y*(x^i)*(x - 1)^2)/( (x^(i+1) + x - 1)*((x^i)*(x*(y - 1) - y) - x + 1) ). - John Tyler Rascoe, Oct 15 2024
Sum_{k=0..n} k * T(n,k) = A097941(n). - Alois P. Heinz, Oct 15 2024

			Triangle starts:
00:  1;
01:  0,    1;
02:  0,    1,    1;
03:  0,    3,    0,    1;
04:  0,    3,    4,    0,    1;
05:  0,    8,    3,    4,    0,    1;
06:  0,   11,   10,    5,    5,    0,    1;
07:  0,   20,   18,   14,    5,    6,    0,    1;
08:  0,   34,   35,   24,   21,    6,    7,    0,   1;
09:  0,   59,   60,   59,   35,   27,    7,    8,   0,   1;
10:  0,   96,  121,  108,   85,   49,   35,    8,   9,   0,   1;
11:  0,  167,  217,  213,  175,  125,   63,   44,   9,  10,   0,  1;
12:  0,  282,  391,  419,  366,  258,  176,   80,  54,  10,  11,  0,  1;
13:  0,  475,  709,  808,  730,  579,  371,  236,  99,  65,  11, 12,  0,  1;
14:  0,  800, 1281, 1522, 1481, 1202,  861,  513, 309, 120,  77, 12, 13,  0, 1;
15:  0, 1352, 2283, 2872, 2925, 2512, 1862, 1238, 684, 395, 143, 90, 13, 14, 0, 1;
...

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Cf. A238341 (the same for largest part).
Columns k=0-10 give: A000007, A105039, A241862, A241863, A241864, A241865, A241866, A241867, A241868, A241869, A241870.
Row sums are A011782.
T(2*n,n) gives A232665(n).
Cf. A097941.

b:= proc(n, s) option remember;`if`(n=0, 1,
      `if`(n`if`(k=0, `if`(n=0, 1, 0), add((p->add(coeff(p, x, i)*
     binomial(i+k, k), i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)):
seq(seq(T(n, k), k=0..n), n=0..15);

b[n_, s_] := b[n, s] = If[n == 0, 1, If[nJean-François Alcover, Nov 07 2014, translated from Maple *)

T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N), h=1+sum(i=1,N,(-y*(x^i)*(x-1)^2)/((x^(i+1)+x-1)*((x^i)*(x*(y-1)-y)-x+1)))); for(i=0,N-1, print(Vecrev(polcoef(h,i))))}
T_xy(15) \\ John Tyler Rascoe, Oct 15 2024

A242447 Number T(n,k) of compositions of n in which the maximal multiplicity of parts equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 3, 4, 0, 1, 0, 5, 6, 4, 0, 1, 0, 11, 10, 5, 5, 0, 1, 0, 13, 21, 18, 5, 6, 0, 1, 0, 19, 40, 34, 21, 6, 7, 0, 1, 0, 27, 87, 59, 40, 27, 7, 8, 0, 1, 0, 57, 121, 132, 100, 49, 35, 8, 9, 0, 1, 0, 65, 219, 272, 210, 131, 63, 44, 9, 10, 0, 1

Offset: 0

Alois P. Heinz, May 15 2014

T(0,0) = 1 by convention. T(n,k) counts the compositions of n in which at least one part has multiplicity k and no part has a multiplicity larger than k.

			T(6,1) = 11: [1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2], [3,2,1], [2,4], [4,2], [1,5], [5,1], [6].
T(6,2) = 10: [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1], [3,3], [1,1,4], [1,4,1], [4,1,1].
T(6,3) = 5: [2,2,2], [1,1,1,3], [1,1,3,1], [1,3,1,1], [3,1,1,1].
T(6,4) = 5: [1,1,1,1,2], [1,1,1,2,1], [1,1,2,1,1], [1,2,1,1,1], [2,1,1,1,1].
T(6,6) = 1: [1,1,1,1,1,1].
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,   1;
  0,  3,   0,   1;
  0,  3,   4,   0,   1;
  0,  5,   6,   4,   0,  1;
  0, 11,  10,   5,   5,  0,  1;
  0, 13,  21,  18,   5,  6,  0, 1;
  0, 19,  40,  34,  21,  6,  7, 0, 1;
  0, 27,  87,  59,  40, 27,  7, 8, 0, 1;
  0, 57, 121, 132, 100, 49, 35, 8, 9, 0, 1;

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A032020 (for n>0), A243119, A243120, A243121, A243122, A243123, A243124, A243125, A243126, A243127.
T(2n,n) = A232665(n).
Row sums give A011782.
Cf. A242451 (the same for minimal multiplicity).

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, k)/j!, j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(n$2, 0, k) -`if`(k=0, 0, b(n$2, 0, k-1)):
seq(seq(T(n, k), k=0..n), n=0..14);

b[n_, i_, p_, k_] := b[n, i, p, k] = If[n == 0, p!, If[i<1, 0, Sum[b[n - i*j, i-1, p + j, k]/j!, {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[n, n, 0, k] - If[k == 0, 0, b[n, n, 0, k-1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 22 2015, after Alois P. Heinz *)

A232605 Number of compositions of 2n into parts with multiplicity <= n.

Original entry on oeis.org

1, 1, 7, 26, 114, 459, 1892, 7660, 31081, 125464, 506025, 2036706, 8189555, 32894825, 132033140, 529614616, 2123365038, 8509634259, 34092146068, 136546197412, 546774790297, 2189060331762, 8762770476060, 35072837719356, 140363923730474, 561697985182654

Offset: 0

Alois P. Heinz, Nov 26 2013

nonn

a(n) = A243081(2n,n) = Sum_{i=0..n} A242447(2n,i).

			a(1) = 1: [2].
a(2) = 7: [4], [3,1], [2,2], [1,3], [2,1,1], [1,2,1], [1,1,2].
a(3) = 26: [6], [5,1], [4,2], [3,3], [2,4], [1,5], [3,2,1], [2,3,1], [1,4,1], [3,1,2], [2,2,2], [1,3,2], [2,1,3], [1,2,3], [1,1,4], [4,1,1], [2,1,2,1], [1,2,2,1], [1,1,3,1], [3,1,1,1], [2,2,1,1], [1,1,2,2], [1,1,1,3], [1,3,1,1], [2,1,1,2], [1,2,1,2].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A232623, A232665.

a:= proc(n) option remember;
     `if`(n<5, [1, 1, 7, 26, 114][n+1],
      (2*(n-1)*(11092322562903*n^3 -66692687083623*n^2
       +117736395568913*n -51473509383358) *a(n-1)
      -(17386283060104*n^4 -178154697569624*n^3 +652039987731328*n^2
       -984836231488344*n +485931992440304) *a(n-2)
      -(89948343833304*n^4 -664733317200192*n^3 +1662507315916082*n^2
       -1594206267597886*n +485625773146800) *a(n-3)
      +(92866735410328*n^4 -1047423564207444*n^3 +4160804083968884*n^2
       -6634447008138888*n +3217864137236880) *a(n-4)
      -16*(n-5)*(2*n-9)*(310469340359*n^2 -847919784312*n
       +494768703748) *a(n-5)) / (5*n*(n-1)*
      (681426847222*n^2 -3587414825361*n +4663189129034)))
    end:
seq(a(n), n=0..40);

b[n_, i_, p_, k_] := b[n, i, p, k] = If[n == 0, p!, If[i < 1, 0, Sum[b[n - i*j, i - 1, p + j, k]/j!, {j, 0, Min[n/i, k]}]]];
A[n_, k_] := If[k >= n, If[n == 0, 1, 2^(n - 1)], b[n, n, 0, k]];
a[n_] := A[2 n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 31 2017, after Alois P. Heinz *)

Recurrence: 5*(n-2)*(n-1)*n*(1258*n^4 - 11230*n^3 + 37013*n^2 - 53645*n + 28764)*a(n) = 2*(n-2)*(n-1)*(17612*n^5 - 159736*n^4 + 538872*n^3 - 824111*n^2 + 541051*n - 107568)*a(n-1) - 4*(n-2)*(5032*n^5 - 44925*n^4 + 134332*n^3 - 137541*n^2 - 6614*n + 52596)*a(n-2) - 2*(83028*n^7 - 1074550*n^6 + 5758938*n^5 - 16516699*n^4 + 27297714*n^3 - 25934731*n^2 + 13070460*n - 2661120)*a(n-3) + 8*(n-4)*(n-1)*(2*n-7)*(1258*n^4 - 6198*n^3 + 10871*n^2 - 8277*n + 2160)*a(n-4). - Vaclav Kotesovec, Nov 27 2013

a(n) ~ 2^(2*n-1). - Vaclav Kotesovec, Nov 27 2013

A332051 Number of compositions of 2n where the multiplicity of the first part equals n.

Original entry on oeis.org

1, 1, 3, 4, 15, 36, 126, 372, 1239, 3910, 12848, 41581, 136578, 447188, 1473342, 4855704, 16053831, 53138244, 176233968, 585202262, 1945964080, 6478043121, 21588979877, 72016891509, 240452892570, 803489258286, 2686964354376, 8991840800137, 30110638705890

Offset: 0

Alois P. Heinz, Feb 06 2020

nonn

			a(0) = 1: the empty composition.
a(1) = 1: 2.
a(2) = 3: 22, 112, 121.
a(3) = 4: 222, 1113, 1131, 1311.
a(4) = 15: 2222, 11114, 11141, 11411, 14111, 111122, 111212, 111221, 112112, 112121, 112211, 121112, 121121, 121211, 122111.

Alois P. Heinz, Table of n, a(n) for n = 0..1882

Cf. A232665, A232697, A255197, A331332.

b:= proc(n, i) option remember; `if`(n=0, x, add(expand(
     `if`(i=j, x, 1)*b(n-j, `if`(n `if`(n=0, 1, coeff(add(b(2*n-j, j), j=1..2*n), x, n)):
seq(a(n), n=0..35);

b[n_, i_] := b[n, i] = If[n == 0, x, Sum[Expand[If[i == j, x, 1] b[n - j, If[n < i + j, 0, i]]], {j, 1, n}]];
a[n_] := If[n == 0, 1, Coefficient[Sum[b[2 n - j, j], {j, 1, 2 n}], x, n]];
a /@ Range[0, 35] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

a(n) = A331332(2n,n).
a(n) ~ c * d^n / sqrt(Pi*n), where d = 3.40869819984215108586487649733361214893... is the root of the equation 4 - 32*d - 8*d^2 + 5*d^3 = 0, and c = 0.34930509632919368540449993196290415079... is the root of the equation 5 - 4*c^2 - 592*c^4 + 2368*c^6 = 0. - Vaclav Kotesovec, Feb 08 2020
Recurrence: 5*(n-1)*n*(2294*n^5 - 31267*n^4 + 168064*n^3 - 445121*n^2 + 580494*n - 297864)*a(n) = (n-1)*(29822*n^6 - 415647*n^5 + 2327634*n^4 - 6668807*n^3 + 10238782*n^2 - 7910608*n + 2368800)*a(n-1) + 2*(27528*n^7 - 434848*n^6 + 2851985*n^5 - 10024036*n^4 + 20278349*n^3 - 23438626*n^2 + 14189888*n - 3420000)*a(n-2) - 2*(41292*n^7 - 647684*n^6 + 4218357*n^5 - 14743832*n^4 + 29759871*n^3 - 34533464*n^2 + 21199620*n - 5259600)*a(n-3) + 2*(n-4)*(2*n - 7)*(2294*n^5 - 19797*n^4 + 65936*n^3 - 105591*n^2 + 80846*n - 23400)*a(n-4). - Vaclav Kotesovec, Feb 08 2020

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A238342 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with exactly k occurrences of the smallest part, n>=0, 0<=k<=n.

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A242447 Number T(n,k) of compositions of n in which the maximal multiplicity of parts equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A232605 Number of compositions of 2n into parts with multiplicity <= n.

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A332051 Number of compositions of 2n where the multiplicity of the first part equals n.

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