A241061 -id:A241061 - cp's OEIS frontend

A207642 Expansion of g.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + x^(n+k)).

1, 1, 2, 1, 2, 2, 2, 3, 3, 2, 4, 4, 4, 4, 5, 6, 6, 6, 6, 8, 9, 8, 10, 10, 10, 12, 14, 14, 14, 15, 16, 19, 20, 20, 22, 24, 24, 26, 28, 30, 34, 34, 35, 38, 40, 42, 46, 50, 50, 54, 58, 60, 63, 66, 70, 76, 80, 84, 88, 92, 96, 102, 108, 112, 120, 126, 131, 140, 146, 151

Offset: 0

Paul D. Hanna, Feb 19 2012

nonn

Conjecture: a(n) is the number of partitions p of n into distinct parts such that max(p) <= 1 + 2*min(p), for n >= 1 (as in the Mathematica program at A241061). - Clark Kimberling, Apr 16 2014

			G.f.: A(x) = 1 + x + 2*x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 3*x^8 + 2*x^9 + 4*x^10 + 4*x^11 + 4*x^12 + 4*x^13 + 5*x^14 + 6*x^15 + 6*x^16 + 6*x^17 + ...
such that, by definition,
A(x) = 1 + x*(1 + x) + x^2*(1 + x^2)*(1 + x^3) + x^3*(1 + x^3)*(1 + x^4)*(1 + x^5) + x^4*(1 + x^4)*(1 + x^5)*(1 + x^6)*(1 + x^7) + x^5*(1 + x^5)*(1 + x^6)*(1 + x^7)*(1 + x^8)*(1 + x^9) + ... + x^n*Product_{k=0..n-1} (1 + x^(n+k)) + ...
Also
A(x) = 1/(1 - x)  +  x^2/((1 - x^2)*(1 - x^3))  +  x^7/((1 - x^3)*(1 - x^4)*(1 - x^5))  +  x^15/((1 - x^4)*(1 - x^5)*(1 - x^6)*(1 - x^7))  +  x^26/((1 - x^5)*(1 - x^6)*(1 - x^7)*(1 - x^8)*(1 - x^9)) + ... + x^(n*(3*n+1)/2)/(Product_{k=0..n} 1 - x^(n+k+1)) + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Paul D. Hanna)

Cf. A053263, A087135, A241061.

m:=80; R:=PowerSeriesRing(Integers(), m); [1] cat Coefficients(R!( (&+[x^n*(&*[1+x^(n+j): j in [0..n-1]]) : n in [1..m]]) )); // G. C. Greubel, Jan 12 2019

With[{m = 80}, CoefficientList[Series[Sum[x^n*Product[1+x^(n+j), {j,0, n-1}], {n,0,m}], {x,0,m}], x]] (* G. C. Greubel, Jan 12 2019 *)
nmax = 100; pk = x + x^2; s = 1 + pk; Do[pk = Normal[Series[pk * x*(1 + x^(2*k - 2))*(1 + x^(2*k - 1))/(1 + x^(k - 1)), {x, 0, nmax}]]; s = s + pk, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Jun 18 2019 *)

{a(n)=polcoeff(sum(m=0,n,x^m*prod(k=0,m-1,1+x^(m+k) +x*O(x^n))),n)}
for(n=0,80,print1(a(n),", "))

R = PowerSeriesRing(ZZ, 'x')
m = 80
x = R.gen().O(m)
s = sum(x^n*prod(1+x^(n+j) for j in (0..n-1)) for n in (0..m))
s.coefficients() # G. C. Greubel, Jan 12 2019

G.f.: Sum_{n>=0} x^(n*(3*n+1)/2) / ( Product_{k=0..n} 1 - x^(n+k+1) ). - Paul D. Hanna, Oct 14 2020

a(n) ~ c * exp(r*sqrt(n)) / sqrt(n), where r = 0.926140105877... = 2*sqrt((3/2)*log(z)^2 - polylog(2, 1-z) + polylog(2, 1-z^2)), where z = (-1 + (44 - 3*sqrt(177))^(1/3) + (44 + 3*sqrt(177))^(1/3))/6 = 0.82948354095849703967... is the real root of the equation z^3*(1 - z)/(1 - z^2)^2 = 1 and c = 0.57862299312... - Vaclav Kotesovec, Jun 29 2019, updated Oct 09 2024

A241062 Number of partitions p of n into distinct parts such that max(p) = 1 + 2*min(p).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 2, 1, 2, 0, 1, 3, 2, 2, 2, 1, 2, 4, 4, 2, 3, 2, 3, 6, 4, 4, 6, 4, 4, 5, 6, 8, 8, 7, 6, 8, 7, 8, 12, 10, 10, 13, 12, 11, 12, 12, 14, 18, 18, 17, 18, 18, 18, 22, 20, 22, 26, 25, 28, 30, 29, 30, 32

Offset: 0

Clark Kimberling, Apr 16 2014

			a(10) counts these 2 partitions:  73, 532.

Cf. A207642, A241061, A241037, A241064, A000009.

z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
  Table[Count[f[n], p_ /; Max[p] < 1 + 2*Min[p]], {n, 0, z}] (* A241061 *)
  Table[Count[f[n], p_ /; Max[p] <= 1 + 2*Min[p]], {n, 0, z}](* A207642 *)
  Table[Count[f[n], p_ /; Max[p] == 1 + 2*Min[p]], {n, 0, z}](* A241062 *)
  Table[Count[f[n], p_ /; Max[p] >= 1 + 2*Min[p]], {n, 0, z}](* A241037 *)
  Table[Count[f[n], p_ /; Max[p] > 1 + 2*Min[p]], {n, 0, z}] (* A241064 *)

a(n) + A241061(n) + A241064(n) = A000009(n) for n >= 1.

a(n) = A241037(n) - A241064(n) = A207642(n) - A241061(n) for n >= 0.

A241064 Number of partitions p of n into distinct parts such that max(p) > 1 + 2*min(p).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 4, 4, 6, 8, 11, 13, 16, 21, 26, 32, 38, 45, 56, 66, 79, 94, 110, 128, 151, 178, 207, 240, 277, 320, 370, 426, 488, 561, 642, 732, 834, 948, 1079, 1225, 1388, 1570, 1774, 2002, 2254, 2540, 2856, 3206, 3598, 4034, 4516, 5050, 5642, 6298

Offset: 0

Clark Kimberling, Apr 16 2014

			a(10) counts these 6 partitions:  91, 82, 721, 631, 541, 4321

Cf. A207642, A241061, A241062, A241064, A000009.

z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 1 + 2*Min[p]], {n, 0, z}]  (* A241061 *)
Table[Count[f[n], p_ /; Max[p] <= 1 + 2*Min[p]], {n, 0, z}] (* A207642 *)
Table[Count[f[n], p_ /; Max[p] == 1 + 2*Min[p]], {n, 0, z}] (* A241062 *)
Table[Count[f[n], p_ /; Max[p] >= 1 + 2*Min[p]], {n, 0, z}] (* A241037 *)
Table[Count[f[n], p_ /; Max[p] > 1 + 2*Min[p]], {n, 0, z}]  (* A241064 *)
Table[Count[IntegerPartitions[n],?(Length[#]==Length[Union[#]]&&#[[1]]>2#[[-1]]+1&)],{n,0,60}]//Quiet (* _Harvey P. Dale, Sep 25 2024 *)

a(n) + A241061(n) + A241062(n) = A000009(n) for n >= 1.

a(n) = A241037(n) - A241062(n) for n>= 0.

A363221 Number of strict integer partitions of n such that (length) * (maximum) <= 2n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 19, 23, 26, 29, 37, 39, 49, 55, 62, 71, 84, 93, 108, 118, 141, 149, 188, 193, 217, 257, 279, 318, 369, 376, 441, 495, 572, 587, 692, 760, 811, 960, 1046, 1065, 1307, 1387, 1550, 1703, 1796, 2041, 2295, 2456, 2753, 3014

Offset: 1

Gus Wiseman, May 23 2023

nonn

Also strict partitions such that (maximum) <= 2*(mean).

These are strict partitions whose complement (see A361851) has size <= n.

			The partition y = (4,3,1) has length 3 and maximum 4, and 3*4 <= 2*8, so y is counted under a(8). The complement of y has size 4, which is less than or equal to n = 8.

The equal case for median is A361850, non-strict A361849 (ranks A361856).
The non-strict version is A361851, A361848 for median.
The equal case is A361854, non-strict A361853 (ranks A361855).
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean.
Cf. A111907, A237984, A240219, A241061, A241086, A324521, A324562, A349156, A360068, A360241, A361394, A361852, A361906.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Max@@#<=2*Mean[#]&]],{n,30}]

cp's OEIS Frontend

A207642 Expansion of g.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + x^(n+k)).

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A241062 Number of partitions p of n into distinct parts such that max(p) = 1 + 2*min(p).

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A241064 Number of partitions p of n into distinct parts such that max(p) > 1 + 2*min(p).

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Mathematica

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A363221 Number of strict integer partitions of n such that (length) * (maximum) <= 2n.

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