A077229 -id:A077229 - cp's OEIS frontend

A098124 Number of compositions of n in which the largest part is equal to the number of parts.

1, 0, 2, 1, 3, 6, 10, 15, 30, 54, 92, 160, 282, 492, 859, 1490, 2570, 4428, 7627, 13098, 22421, 38290, 65265, 111018, 188475, 319380, 540266, 912397, 1538371, 2589858, 4353820, 7309362, 12255474, 20523307, 34328731, 57357184, 95733131, 159626049

Offset: 1

Vladeta Jovovic, Sep 25 2004

			a(7)=10 because we have 223, 232, 322, 133, 313, 331, 1114, 1141, 1411 and 4111.

Cf. A077229, A047993.

G:=sum(((x^(k+1)-x)^k-(x^k-x)^k)/(x-1)^k,k=1..25):Gser:=series(G,x=0,45):seq(coeff(Gser,x^n),n=1..42); # Emeric Deutsch, Apr 16 2005

G.f.: Sum_{k>=1} ((x^(k+1)-x)^k - (x^k-x)^k)/(x-1)^k.

More terms from Emeric Deutsch, Apr 16 2005

A077227 Triangle of compositions of n into exactly k parts each no more than k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 6, 4, 1, 0, 0, 7, 10, 5, 1, 0, 0, 6, 20, 15, 6, 1, 0, 0, 3, 31, 35, 21, 7, 1, 0, 0, 1, 40, 70, 56, 28, 8, 1, 0, 0, 0, 44, 121, 126, 84, 36, 9, 1, 0, 0, 0, 40, 185, 252, 210, 120, 45, 10, 1, 0, 0, 0, 31, 255, 456, 462, 330, 165, 55, 11, 1, 0, 0, 0, 20

Offset: 1

Henry Bottomley, Oct 29 2002

			T(6,3)=7 since 6 can be written as 1+2+3, 1+3+2, 2+1+3, 2+2+2, 2+3+1, 3+1+2, or 3+2+1.
Triangle begins:
1;
0, 1;
0, 2, 1;
0, 1, 3, 1;
0, 0, 6, 4, 1;
0, 0, 7, 10, 5, 1;
0, 0, 6, 20, 15, 6, 1;
0, 0, 3, 31, 35, 21, 7, 1;
0, 0, 1, 40, 70, 56, 28, 8, 1;
0, 0, 0, 44, 121, 126, 84, 36, 9, 1;
0, 0, 0, 40, 185, 252, 210, 120, 45, 10, 1; ...
where column sums are k^k (A000312).

Index entries for sequences related to compositions

Column sums are A000312. Row sums are A077229. Central diagonal is A000984 offset. Right hand side is right hand side of A007318. Cf. A077228.

T(n,k)=polcoeff(((1-x^k)/(1-x +x*O(x^n)))^k,n-k)
for(n=1,12,for(k=1,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Jan 25 2013

T(n, k) = A077228(n, k) - A077228(n-1, k).
If n>=k^2, T(n, k) = 0. If k<=n<2k, T(n, k) = C(n-1, k-1).
G.f. of column k is: x^k*(1-x^k)^k/(1-x)^k for k>=1. - Paul D. Hanna, Jan 25 2013

A098131 Number of compositions of n where the smallest part is greater than or equal to the number of parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 33, 41, 51, 64, 81, 103, 131, 166, 209, 261, 323, 397, 486, 594, 726, 888, 1087, 1331, 1629, 1991, 2428, 2952, 3577, 4320, 5202, 6249, 7493, 8973, 10736, 12838, 15345, 18334, 21894, 26127, 31149, 37092, 44107, 52368

Offset: 0

Vladeta Jovovic, Sep 27 2004

			a(7)=5 because we have 7, 4+3, 3+4, 5+2 and 2+5.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003114, A077229.

G:=sum(x^(k^2)/(1-x)^k, k=0..20): Gser:=series(G,x=0,56): seq( coeff( Gser,x^n), n=0..54);  # Emeric Deutsch

nmax = 60; Flatten[{1, Rest[CoefficientList[Series[Sum[x^(k^2)/(1-x)^k, {k, 1, Sqrt[nmax]}], {x, 0, nmax}], x]]}] (* Vaclav Kotesovec, Nov 11 2018 *)

my(N=66,x='x+O('x^N)); Vec(sum(n=1,N,x^(n*n)*(1)/(1-x)^n)) \\ Joerg Arndt, Jan 23 2024

G.f.: Sum_{k>=0} x^(k^2)/(1-x)^k.

More terms from Emeric Deutsch, Mar 29 2005

Prepended a(0)=1 to match g.f., Joerg Arndt, Apr 22 2014

A098132 Number of compositions of n where the smallest part is greater than the number of parts.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 34, 42, 51, 62, 76, 94, 117, 146, 182, 226, 279, 342, 416, 503, 606, 729, 877, 1056, 1273, 1536, 1854, 2237, 2696, 3243, 3891, 4655, 5553, 6607, 7844, 9297, 11006, 13019, 15393, 18195, 21503, 25407, 30010

Offset: 1

Vladeta Jovovic, Sep 27 2004

			a(11)=7 because we have: 11, 8+3, 3+8, 7+4, 4+7, 6+5 and 5+6.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Hùng Việt Chu, Nurettin Irmak, Steven J. Miller, László Szalay, and Sindy Xin Zhang, Schreier Multisets and the s-step Fibonacci Sequences, arXiv:2304.05409 [math.CO], 2023. See also Integers (2024) Vol. 24A, Art. No. A7, p. 4.

Cf. A003106, A003114, A077229.

G:=sum(x^(k^2+k)/(1-x)^k,k=0..20): Gser:=series(G,x=0,67): seq(coeff(Gser,x^n),n=1..65); # Emeric Deutsch, Mar 29 2005

nmax = 60; Rest[CoefficientList[Series[Sum[x^(k*(k+1))/(1-x)^k, {k, 1, Sqrt[nmax] + 1}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 11 2018 *)

G.f.: Sum_{n>=0} x^(n*(n+1)) / (1-x)^n.

More terms from Emeric Deutsch, Mar 29 2005

A098125 Number of compositions of n where the largest part is less than the number of parts.

Original entry on oeis.org

0, 1, 1, 4, 8, 17, 38, 83, 174, 367, 771, 1606, 3324, 6849, 14054, 28743, 58605, 119161, 241717, 489345, 988945, 1995604, 4021710, 8095815, 16281400, 32716231, 65694106, 131833462, 264423116, 530128036, 1062424867, 2128513095

Offset: 1

Vladeta Jovovic, Sep 25 2004

			a(5)=8 because we have 1112, 1121, 1211, 2111, 122, 212, 221 and 11111.

Cf. A077229, A047993, A064173, A064174.

G:=sum(((x^k-x)/(x-1))^k,k=0..45): Gser:=series(G,x=0,40): seq(coeff(Gser,x^n),n=1..36); # Emeric Deutsch, Apr 16 2005

G.f.: Sum_{k>=0} ((x^k-x)/(x-1))^k.

More terms from Emeric Deutsch, Apr 16 2005

A262007 G.f.: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n / (1 - x)^n.

Original entry on oeis.org

1, 2, 1, 8, 7, 27, 45, 102, 194, 439, 844, 1775, 3608, 7342, 14891, 30283, 61113, 123625, 249355, 502430, 1011305, 2034028, 4086860, 8206874, 16469851, 33035697, 66234208, 132746099, 265961186, 532718115, 1066778721, 2135822309, 4275459594, 8557335615, 17125445126, 34268966022, 68568212859, 137187104632

Offset: 1

Paul D. Hanna, Sep 21 2015

nonn

Compare to the curious identity: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n = 0.

Compare also to the g.f. of A077229, where A077229(n) equals the number of compositions of n where the largest part is <= the number of parts.

			G.f.: A(x) = x + 2*x^2 + x^3 + 8*x^4 + 7*x^5 + 27*x^6 + 45*x^7 + 102*x^8 + 194*x^9 + 439*x^10 + 844*x^11 + 1775*x^12 +...
such that A(x) = N(x) + P(x) where
N(x) = Sum_{n>=1} (-1)^n * x^(n^2-n) * (1 - x)^n / (1 - x^n)^n
P(x) = Sum_{n>=0} x^n * (1 - x^n)^n / (1 - x)^n.
Explicitly,
N(x) = -1 + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 4*x^6 - 3*x^7 + 4*x^8 - 10*x^9 + 18*x^10 - 19*x^11 + 9*x^12 + 2*x^13 + x^14 - 22*x^15 + 50*x^16 +...
P(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 11*x^5 + 23*x^6 + 48*x^7 + 98*x^8 + 204*x^9 + 421*x^10 + 863*x^11 + 1766*x^12 + 3606*x^13 + 7341*x^14 + 14913*x^15 + 30233*x^16 +...+ A077229(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A077229, A260147.

{a(n) = local(A=1);
A = sum(k=-n-1, n+1, x^k * (1-x^k)^k / (1-x +x*O(x^n))^k); polcoeff(A, n)}
for(n=1, 31, print1(a(n), ", "))

{a(n) = local(A=1);
A = sum(k=-n-1, n+1, (-1)^k * x^(k^2-k) * (1 - x)^k / (1 - x^k +x*O(x^n))^k); polcoeff(A, n)}
for(n=1, 31, print1(a(n), ", "))

G.f.: Sum_{n=-oo..+oo} (-1)^n * x^(n^2-n) * (1 - x)^n / (1 - x^n)^n.
Limit a(n)^(1/n) = 2.
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 03 2017

A348125 Number of compositions of n where the largest part is larger than the number of parts.

Original entry on oeis.org

0, 1, 1, 3, 5, 9, 16, 30, 52, 91, 161, 282, 490, 851, 1471, 2535, 4361, 7483, 12800, 21845, 37210, 63258, 107329, 181775, 307341, 518821, 874492, 1471869, 2473969, 4153018, 6963137, 11661191, 19507566, 32599451, 54423609, 90772692, 151263317, 251849623

Offset: 1

R. J. Mathar, Oct 01 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A011782, A098124, A098125, A077229.

a(n) + A077229(n) = 2^(n-1).

a(22)-a(38) from Alois P. Heinz, Oct 01 2021

A386474 Number of sets of lists of [n] such that no list is longer than than the total number of lists.

Original entry on oeis.org

1, 1, 1, 7, 25, 141, 1171, 9913, 85233, 907273, 11010691, 143824341, 1988010553, 29605763773, 475664908083, 8284952367721, 153508912353121, 2997209814190353, 61485486404453443, 1326994255131585373, 30144049509450774441, 718905298680190094341, 17940822818538396541843

Offset: 0

John Tyler Rascoe, Jul 23 2025

Here sets of lists are set partitions of [n] such that the elements within each block are ordered but the blocks themselves are unordered.

			a(3) = 7 counts: {(1),(2),(3)}, {(1),(2,3)}, {(1),(3,2)}, {(1,2),(3)}, {(1,3),(2)}, {(2),(3,1)}, {(2,1),(3)}.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A000262, A077229, A088009, A088311, A088312, A088313, A113775, A351884.

b:= proc(n, m, l) option remember; `if`(m>n+l, 0, `if`(n=0, 1,
      add(b(n-j, max(m, j), l+1)*(n-1)!*j/(n-j)!, j=1..n)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..22);  # Alois P. Heinz, Jul 23 2025

With[{m = 22}, CoefficientList[1 + Series[Sum[((x - x^(i + 1))/(1 - x))^i/i!, {i, 1, m}], {x, 0, m}], x] * Range[0, m]!] (* Amiram Eldar, Jul 24 2025 *)

R_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(sum(i=0,N,((x-x^(i+1))/(1-x))^i/i!)))}

E.g.f.: Sum_{i>=0} ((x - x^(i+1))/(1 - x))^i / i!.

A078228 a(n) = A078227(n+1)/A078227(n).

Original entry on oeis.org

2, 2, 2, 2, 3, 7, 68, 61, 97, 1876, 53771, 5178128, 1533415117

Offset: 1

Amarnath Murthy, Nov 23 2002

Cf. A078221, A078222, A078223, A078224, A078225, A078226, A078227, A077229.

a(8)-a(13) from Donovan Johnson, Nov 11 2008

A221834 G.f.: Sum_{n>=1} x^n * (1-x^n)^(n-1) / (1-x)^(n-1).

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 27, 54, 111, 225, 456, 926, 1877, 3796, 7671, 15483, 31212, 62859, 126484, 254296, 510892, 1025765, 2058395, 4128578, 8277344, 16589180, 33237163, 66574351, 133318484, 266924608, 534335692, 1069492787, 2140370294, 4283071475, 8570061106

Offset: 1

Paul D. Hanna, Jan 26 2013

nonn

Conjecture: a(n) is the number of compositions of n if all single instances of the part 1 are frozen ([1]). Example: The compositions enumerated by a(5) = 13 are 5; 4,[1]; 3,2; 2,3; 3,1,1; 1,3,1; 1,1,3; 2,2,[1]; 2,1,1,1; 1,2,1,1; 1,1,2,1; 1,1,1,2; 1,1,1,1,1. - Gregory L. Simay, Oct 27 2022

			G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 7*x^5 + 13*x^6 + 27*x^7 + 54*x^8 + ...
where
A(x) = x + x^2*(1-x^2)/(1-x) + x^3*(1-x^3)^2/(1-x)^2 + x^4*(1-x^4)^3/(1-x)^3 + ...
or, equivalently,
A(x) = x + x^2*(1+x) + x^3*(1+x+x^2)^2 + x^4*(1+x+x^2+x^3)^3 + ...

Cf. A221833, A077229.

{a(n)=polcoeff(sum(k=1,n,x^k*((1-x^k)/(1-x) +x*O(x^n))^(k-1)),n)}
for(n=1,40,print1(a(n),", "))

Equals row sums of triangle A221833.

cp's OEIS Frontend

A098124 Number of compositions of n in which the largest part is equal to the number of parts.

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A077227 Triangle of compositions of n into exactly k parts each no more than k.

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A098131 Number of compositions of n where the smallest part is greater than or equal to the number of parts.

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A098132 Number of compositions of n where the smallest part is greater than the number of parts.

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A098125 Number of compositions of n where the largest part is less than the number of parts.

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A262007 G.f.: Sum_{n=-oo..+oo} x^n * (1 - x^n)^n / (1 - x)^n.

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A348125 Number of compositions of n where the largest part is larger than the number of parts.

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A386474 Number of sets of lists of [n] such that no list is longer than than the total number of lists.

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