A238342 -id:A238342 - cp's OEIS frontend

A241864 Number of compositions of n such that the smallest part has multiplicity four.

1, 0, 5, 5, 21, 35, 85, 175, 366, 730, 1481, 2925, 5726, 11110, 21375, 40766, 77266, 145495, 272290, 506836, 938783, 1730725, 3176920, 5808020, 10578162, 19197898, 34725765, 62616485, 112574807, 201827366, 360885835, 643679795, 1145341756, 2033369086

Offset: 4

Joerg Arndt and Alois P. Heinz, Apr 30 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 4..1000

Column k=4 of A238342.

b:= proc(n, s) option remember; `if`(n=0, 1,
      `if`(nadd(coeff(p, x, i)*binomial(i+k, k),
       i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)
    end:
seq(a(n), n=4..40);

b[n_, s_] := b[n, s] = If[n == 0, 1, If[nJean-François Alcover, Feb 09 2015, after Maple *)

a(n) ~ (1/2-1/sqrt(5))^3 / 15 * n^4 * ((1+sqrt(5))/2)^n. - Vaclav Kotesovec, May 01 2014

A241865 Number of compositions of n such that the smallest part has multiplicity five.

Original entry on oeis.org

1, 0, 6, 6, 27, 49, 125, 258, 579, 1202, 2512, 5157, 10463, 20949, 41627, 81912, 159834, 309641, 595836, 1139211, 2165502, 4094219, 7701857, 14420351, 26880988, 49902183, 92279657, 170020844, 312173822, 571307477, 1042310911, 1896039086, 3439404321, 6222483152

Offset: 5

Joerg Arndt and Alois P. Heinz, Apr 30 2014

nonn

Joerg Arndt, Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 5..1500 (first 1000 terms from Joerg Arndt and Alois P. Heinz)

Column k=5 of A238342.

b:= proc(n, s) option remember; `if`(n=0, 1,
      `if`(nadd(coeff(p, x, i)*binomial(i+k, k),
       i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)
    end:
seq(a(n), n=5..40);

b[n_, s_] := b[n, s] = If[n == 0, 1, If[n < s, 0, Expand[Sum[b[n - j, s]*x, {j, s, n}]]]]; a[n_] := With[{k = 5}, Sum[Function[{p}, Sum[Coefficient[p, x, i] * Binomial[i+k, k], {i, 0, Exponent[p, x]}]][b[n-j*k, j+1]], {j, 1, n/k}]]; Table[ a[n], {n, 5, 40}] (* Jean-François Alcover, Feb 09 2015, after Maple *)

a(n) ~ n^5 * ((1+sqrt(5))/2)^(n-11) / (5^3 * 5!). - Vaclav Kotesovec, May 02 2014

A241866 Number of compositions of n such that the smallest part has multiplicity six.

Original entry on oeis.org

1, 0, 7, 7, 35, 63, 176, 371, 861, 1862, 4032, 8512, 17851, 36848, 75286, 152334, 305466, 607313, 1198443, 2348388, 4571728, 8846314, 17021480, 32579029, 62048589, 117627699, 222018034, 417326148, 781395064, 1457684326, 2709797693, 5020734691, 9273107977

Offset: 6

Joerg Arndt and Alois P. Heinz, Apr 30 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=6 of A238342.

b:= proc(n, s) option remember; `if`(n=0, 1,
      `if`(nadd(coeff(p, x, i)*binomial(i+k, k),
       i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)
    end:
seq(a(n), n=6..40);

b[n_, s_] := b[n, s] = If[n == 0, 1, If[n < s, 0, Expand[Sum[b[n - j, s]*x, {j, s, n}]]]]; a[n_] := With[{k = 6}, Sum[Function[{p}, Sum[Coefficient[p, x, i]*Binomial[i + k, k], {i, 0, Exponent[p, x]}]][b[n - j*k, j + 1]], {j, 1, n/k}]]; Table[a[n], {n, 6, 40}] (* Jean-François Alcover, Feb 09 2015, after Maple *)

a(n) ~ n^6 * ((1+sqrt(5))/2)^(n-13) / (5^(7/2) * 6!). - Vaclav Kotesovec, May 02 2014

A241867 Number of compositions of n such that the smallest part has multiplicity seven.

Original entry on oeis.org

1, 0, 8, 8, 44, 80, 236, 513, 1238, 2744, 6160, 13384, 28846, 61228, 128513, 266668, 548185, 1116580, 2255452, 4521198, 8998844, 17792361, 34962224, 68305274, 132724871, 256587512, 493665604, 945497642, 1803122075, 3424720416, 6479635254, 12214748337

Offset: 7

Joerg Arndt and Alois P. Heinz, Apr 30 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 7..1000

Column k=7 of A238342.

b:= proc(n, s) option remember; `if`(n=0, 1,
      `if`(nadd(coeff(p, x, i)*binomial(i+k, k),
       i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)
    end:
seq(a(n), n=7..40);

b[n_, s_] := b[n, s] = If[n == 0, 1, If[n < s, 0, Expand[Sum[b[n - j, s]*x, {j, s, n}]]]]; a[n_] := With[{k = 7}, Sum[Function[{p}, Sum[Coefficient[p, x, i]*Binomial[i + k, k], {i, 0, Exponent[p, x]}]][b[n - j*k, j + 1]], {j, 1, n/k}]]; Table[a[n], {n, 7, 40}] (* Jean-François Alcover, Feb 09 2015, after Maple *)

a(n) ~ n^7 * ((1+sqrt(5))/2)^(n-15) / (5^4 * 7!). - Vaclav Kotesovec, May 02 2014

A241868 Number of compositions of n such that the smallest part has multiplicity eight.

Original entry on oeis.org

1, 0, 9, 9, 54, 99, 309, 684, 1720, 3909, 9036, 20178, 44676, 97191, 209151, 444498, 935002, 1947729, 4021429, 8234244, 16732173, 33758283, 67656843, 134751630, 266817214, 525411981, 1029271671, 2006453683, 3893241810, 7521104292, 14468931402, 27724579185

Offset: 8

Joerg Arndt and Alois P. Heinz, Apr 30 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 8..1000

Column k=8 of A238342.

b:= proc(n, s) option remember; `if`(n=0, 1,
      `if`(nadd(coeff(p, x, i)*binomial(i+k, k),
       i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)
    end:
seq(a(n), n=8..40);

b[n_, s_] := b[n, s] = If[n == 0, 1, If[n < s, 0, Expand[Sum[b[n - j, s]*x, {j, s, n}]]]]; a[n_] := With[{k = 8}, Sum[Function[{p}, Sum[Coefficient[p, x, i]*Binomial[i + k, k], {i, 0, Exponent[p, x]}]][b[n - j*k, j + 1]], {j, 1, n/k}]]; Table[a[n], {n, 8, 40}] (* Jean-François Alcover, Feb 09 2015, after Maple *)

a(n) ~ n^8 * ((1+sqrt(5))/2)^(n-17) / (5^(9/2) * 8!). - Vaclav Kotesovec, May 02 2014

A241869 Number of compositions of n such that the smallest part has multiplicity nine.

Original entry on oeis.org

1, 0, 10, 10, 65, 120, 395, 890, 2320, 5401, 12847, 29380, 66735, 148630, 327270, 711247, 1529020, 3252775, 6855276, 14320645, 29672905, 61018010, 124587120, 252694835, 509337682, 1020610708, 2033777830, 4031514561, 7951981550, 15611183177, 30510678865

Offset: 9

Joerg Arndt and Alois P. Heinz, Apr 30 2014

nonn

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 9..1000

Column k=9 of A238342.

b:= proc(n, s) option remember; `if`(n=0, 1,
      `if`(nadd(coeff(p, x, i)*binomial(i+k, k),
       i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)
    end:
seq(a(n), n=9..40);

b[n_, s_] := b[n, s] = If[n == 0, 1, If[nJean-François Alcover, Feb 09 2015, after Maple *)

a(n) ~ n^9 * ((1+sqrt(5))/2)^(n-19) / (5^5 * 9!). - Vaclav Kotesovec, May 02 2014

A241870 Number of compositions of n such that the smallest part has multiplicity ten.

Original entry on oeis.org

1, 0, 11, 11, 77, 143, 495, 1133, 3058, 7271, 17777, 41580, 96701, 220187, 495528, 1099626, 2412927, 5236308, 11251449, 23952841, 50556265, 105852923, 219975999, 453933348, 930544912, 1895736986, 3839424644, 7732852963, 15492659226, 30884561378, 61276442019

Offset: 10

Joerg Arndt and Alois P. Heinz, Apr 30 2014

nonn

Conjecture: Generally, for k > 0 is column k of A238342 asymptotic to n^k * ((1+sqrt(5))/2)^(n-2*k-1) / (5^((k+1)/2) * k!). - Vaclav Kotesovec, May 02 2014

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 10..1000

Column k=10 of A238342.

b:= proc(n, s) option remember; `if`(n=0, 1,
      `if`(nadd(coeff(p, x, i)*binomial(i+k, k),
       i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)
    end:
seq(a(n), n=10..40);

b[n_, s_] := b[n, s] = If[n == 0, 1, If[n < s, 0, Expand[Sum[b[n - j, s]*x, {j, s, n}]]]]; a[n_] := With[{k = 10}, Sum[Function[{p}, Sum[Coefficient[p, x, i] * Binomial[i + k, k], {i, 0, Exponent[p, x]}]][b[n - j*k, j + 1]], {j, 1, n/k}]]; Table[ a[n], {n, 10, 40}] (* Jean-François Alcover, Feb 09 2015, after Maple *)

a(n) ~ n^10 * ((1+sqrt(5))/2)^(n-21) / (5^(11/2) * 10!). - Vaclav Kotesovec, May 02 2014

A383870 Number of compositions of n such that none of the smallest parts are adjacent.

Original entry on oeis.org

1, 1, 1, 3, 4, 9, 15, 29, 53, 98, 180, 336, 618, 1142, 2110, 3899, 7197, 13283, 24509, 45218, 83396, 153769, 283463, 522449, 962732, 1773742, 3267417, 6018030, 11082693, 20407174, 37572633, 69169726, 127326924, 234362474, 431343281, 793831500, 1460854117

Offset: 0

John Tyler Rascoe, May 13 2025

			a(5) = 9 counts: (1,2,2), (1,3,1), (1,4), (2,1,2), (2,2,1), (2,3), (3,2), (4,1), (5).

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A003242, A007318, A011782, A074909, A105039, A238342.

b:= proc(n, i, p) option remember; `if`(i<1, 0,
     `if`(irem(n, i, 'r')=0, p!*binomial(p+1, r), 0)+
      add(b(n-i*j, min(n-i*j, i-1), p+j)/j!, j=0..n/i))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..36);  # Alois P. Heinz, May 13 2025

A_x(N) ={Vec(1+sum(j=0,N, sum(i=j+1,N-j, (binomial(i,i-j-1) * x^(j+1) * (x^2/(1-x))^(i-1) )/(1-x^(i+j))))+O('x^N))}
A_x(50)

G.f.: 1 + Sum_{j>=0} Sum_{i>j} (binomial(i,i-j-1) * x^(j+1) * (x^2/(1 - x))^(i-1))/(1 - x^(i+j)).

cp's OEIS Frontend

A241864 Number of compositions of n such that the smallest part has multiplicity four.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A241865 Number of compositions of n such that the smallest part has multiplicity five.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A241866 Number of compositions of n such that the smallest part has multiplicity six.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A241867 Number of compositions of n such that the smallest part has multiplicity seven.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A241868 Number of compositions of n such that the smallest part has multiplicity eight.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A241869 Number of compositions of n such that the smallest part has multiplicity nine.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A241870 Number of compositions of n such that the smallest part has multiplicity ten.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A383870 Number of compositions of n such that none of the smallest parts are adjacent.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs