A026807 -id:A026807 - cp's OEIS frontend

A185329 Number of partitions of n with parts >= 9.

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 24, 26, 30, 34, 39, 43, 50, 55, 63, 71, 80, 89, 102, 113, 128, 143, 161, 179, 203, 225, 253, 282, 316, 351, 395, 437, 489, 544, 607, 673, 752, 832, 927, 1028, 1143

Offset: 0

Jason Kimberley, Feb 01 2012

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 9 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.
By removing a single part of size 9, an A026802 partition of n becomes an A185329 partition of n - 9. Hence this sequence is essentially the same as A026802.
In general, if g>=1 and g.f. = Product_{m>=g} 1/(1-x^m), then a(n,g) ~ Pi^(g-1) * (g-1)! * exp(Pi*sqrt(2*n/3)) / (2^((g+3)/2) * 3^(g/2) * n^((g+1)/2)) ~ p(n) * Pi^(g-1) * (g-1)! / (6*n)^((g-1)/2), where p(n) is the partition function A000041(n). - Vaclav Kotesovec, Jun 02 2018

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), this sequence (g=9).

Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+9): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(1/mul(1-x^(m+9), m = 0..80), x, n+1), x, n), n = 0..70); # G. C. Greubel, Nov 03 2019

CoefficientList[Series[x^9/QPochhammer[x^9, x], {x,0,75}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^70)); Vec(1/prod(m=0,80, 1-x^(m+9))) \\ G. C. Greubel, Nov 03 2019

def A185329_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+9)) for m in (0..80)) ).list()
A185329_list(70) # G. C. Greubel, Nov 03 2019

G.f.: Product_{m>=9} 1/(1-x^m).
a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + p(n-7) + p(n-9) - p(n-11) - 2*p(n-12) - p(n-13) - p(n-15) + p(n-16) + p(n-17) + 2*p(n-18) + p(n-19) + p(n-20) - p(n-21) - p(n-23) - 2*p(n-24) - p(n-25) + p(n-27) + p(n-29) + p(n-31) - p(n-34) - p(n-35) + p(n-36) where p(n)=A000041(n). - Shanzhen Gao
This sequence is the Euler transformation of A185119.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 70*Pi^8 / (9*sqrt(3)*n^5). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=0} x^(9*k) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 28 2020
G.f.: 1 + Sum_{n >= 1} x^(n+8)/Product_{k = 0..n-1} (1 - x^(k+9)). - Peter Bala, Dec 01 2024

A026799 Number of partitions of n in which the least part is 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 16, 17, 21, 24, 29, 32, 40, 44, 53, 60, 71, 80, 96, 107, 126, 143, 167, 188, 221, 248, 288, 326, 376, 424, 491, 552, 634, 716, 819, 922, 1056, 1187, 1353, 1523, 1730, 1944, 2209, 2478, 2806, 3151

Offset: 0

Clark Kimberling

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth exactly 6 (all such graphs are simple). Each integer part i corresponds to an i-cycle; the addition of integers corresponds to the disconnected union of cycles.

			a(0)=0 because there does not exist a least part of the empty partition.
The  a(6)=1 partition is 6.
The a(12)=1 partition is 6+6.
The a(13)=1 partition is 6+7.
.............................
The a(17)=1 partition is 6+11.
The a(18)=2 partitions are 6+6+6 and 6+12.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Essentially the same as A185326.
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), this sequence (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 04 2011

p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A026799 := func< n | p(n-6)-p(n-7)-p(n-8)+p(n-11)+p(n-12)+p(n-13)- p(n-14)-p(n-15)-p(n-16)+p(n-19)+p(n-20)-p(n-21) >; // Jason Kimberley, Feb 04 2011

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0,0,0] cat Coefficients(R!( x^6/(&*[1-x^(m+6): m in [0..70]]) )); // G. C. Greubel, Nov 03 2019

ZL := [ B,{B=Set(Set(Z, card>=6))}, unlabeled ]: 0,0,0,0,0,0, seq(combstruct[count](ZL, size=n), n=0..63); # Zerinvary Lajos, Mar 13 2007
seq(coeff(series(x^6/mul(1-x^(m+6), m=0..70), x, n+1), x, n), n = 0..65); # G. C. Greubel, Nov 03 2019

f[1, 1]=f[0, k_]=1; f[n_, k_]:= f[n, k] = If[n<0, 0, If[k>n, 0, If[k==n, 1, f[n, k+1] +f[n-k, k]]]]; Join[{0,0,0,0,0,0}, Table[f[n, 6], {n, 0, 65}]] (* Robert G. Wilson v, Jan 31 2011 *)
CoefficientList[Series[x^6/QPochhammer[x^6, x], {x,0,70}], x] (* G. C. Greubel, Nov 03 2019 *)
Join[{0},Table[Count[IntegerPartitions[n][[;;,-1]],6],{n,70}]] (* Harvey P. Dale, Dec 27 2023 *)

my(x='x+O('x^60)); concat([0,0,0,0,0,0], Vec(x^6/prod(m=0,70, 1-x^(m+6)))) \\ G. C. Greubel, Nov 03 2019

def A026799_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^6/product((1-x^(m+6)) for m in (0..70)) ).list()
A026799_list(65) # G. C. Greubel, Nov 03 2019

G.f.: x^6 * Product_{m>=6} 1/(1-x^m).
a(n) = p(n-6) -p(n-7) -p(n-8) +p(n-11) +p(n-12) +p(n-13) -p(n-14) -p(n-15) -p(n-16) +p(n-19) +p(n-20) -p(n-21) for n>0 where p(n) = A000041(n). - Shanzhen Gao, Oct 28 2010
a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^5 / (18*sqrt(2)*n^(7/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(6*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

A026805 Number of partitions of n in which the least part is even.

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 2, 6, 5, 9, 9, 16, 17, 26, 28, 42, 48, 66, 77, 105, 122, 160, 189, 245, 290, 368, 436, 547, 650, 804, 954, 1174, 1390, 1693, 2004, 2425, 2865, 3445, 4060, 4858, 5716, 6802, 7986, 9468, 11087, 13088, 15298, 17995, 20987, 24604, 28631, 33464

Offset: 1

Clark Kimberling

nonn

Also number of partitions of n in which the largest part occurs an even number of times. Example: a(6)=3 because we have [3,3],[2,2,1,1] and [1,1,1,1,1,1]. - Emeric Deutsch, Apr 04 2006

			a(6)=3 because we have [6],[4,2] and [2,2,2].

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

g:=sum(x^(2*k)/(1-x^(2*k))/product(1-x^j,j=1..k-1),k=1..40): gser:=series(g,x=0,52): seq(coeff(gser,x,n),n=1..49); # Emeric Deutsch, Apr 04 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n<1 or i<1, 0, b(n, i-1)+
      `if`(n=i, 1-irem(n, 2), 0)+`if`(i>n, 0, b(n-i, i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=1..60);  # Alois P. Heinz, Jul 26 2015

b[n_, i_] := b[n, i] = If[n<1 || i<1, 0, b[n, i-1] + If[n==i, 1-Mod[n, 2], 0] + If[i>n, 0, b[n-i, i]]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *)

From Vladeta Jovovic, Aug 26 2003: (Start)
G.f.: Sum_{k>=2} ((-1)^k*(-1+1/Product_{i>=k} (1-x^i))).
a(n) = Sum_{k=2..n} (-1)^k*A026807(n, k) = A000041(n)-A026804(n). (End)
From Emeric Deutsch, Apr 04 2006: (Start)
G.f.: Sum_{k>=1}(x^(2k)/Product_{j>=2k}(1-x^j)).
G.f.: Sum_{k>=1}(x^(2k)/((1-x^(2k))*Product_{j=1..k-1}(1-x^j))). (End)
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + 61*Pi/(24*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Jul 06 2019, extended Nov 02 2019

A185327 Number of partitions of n into parts >= 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 18, 20, 24, 27, 32, 36, 42, 48, 56, 63, 73, 83, 96, 108, 125, 141, 162, 183, 209, 236, 270, 304, 346, 390, 443, 498, 565, 635, 719, 807, 911, 1022, 1153, 1291, 1453, 1628, 1829, 2045

Offset: 0

Jason Kimberley, Feb 03 2011

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 7 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.

By removing a single part of size 7, an A026800 partition of n becomes an A185327 partition of n - 7. Hence this sequence is essentially the same as A026800.

			The  a(0)=1 empty partition vacuously has each part >= 7.
The  a(7)=1 partition is 7.
The  a(8)=1 partition is 8.
............................
The a(13)=1 partition is 13.
The a(14)=2 partitions are 7+7 and 14.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

2-regular simple graphs with girth at least 7: A185117 (connected), A185227 (disconnected), this sequence (not necessarily connected).
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), this sequence (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).

p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A185327 := func< n | p(n)-p(n-1)-p(n-2)+p(n-5)+2*p(n-7)-p(n-9)-p(n-10)- p(n-11)-p(n-12)+2*p(n-14)+p(n-16)-p(n-19)-p(n-20)+p(n-21) >;

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+7): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(1/mul(1-x^(m+7), m = 0..80), x, n+1), x, n), n = 0..70); # G. C. Greubel, Nov 03 2019

f[1, 1] = f[0, k_] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 7], {n, 0, 65}] (* Robert G. Wilson v, Jan 31 2011 *) (* moved from A026800 by Jason Kimberley, Feb 03 2011 *)
Join[{1},Table[Count[IntegerPartitions[n],?(Min[#]>=7&)],{n,0,70}]] (* _Harvey P. Dale, Oct 16 2011 *)
CoefficientList[Series[1/QPochhammer[x^7, x], {x, 0, 75}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^70)); Vec(1/prod(m=0,80, 1-x^(m+7))) \\ G. C. Greubel, Nov 03 2019

def A185327_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+7)) for m in (0..80)) ).list()
A185327_list(70) # G. C. Greubel, Nov 03 2019

G.f.: Product_{m>=7} 1/(1-x^m).
a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + 2*p(n-7) - p(n-9) - p(n-10) - p(n-11) - p(n-12) + 2*p(n-14) + p(n-16) - p(n-19) - p(n-20) + p(n-21) where p(n)=A000041(n). - Shanzhen Gao, Oct 28 2010 [moved/copied from A026800 by Jason Kimberley, Feb 03 2011]
This sequence is the Euler transformation of A185117.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^6 / (6*sqrt(3)*n^4). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=0} x^(7*k) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 28 2020
G.f.: 1 + Sum_{n >= 1} x^(n+6)/Product_{k = 0..n-1} (1 - x^(k+7)). - Peter Bala, Dec 01 2024

A185328 Number of partitions of n with parts >= 8.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 21, 23, 27, 30, 36, 39, 46, 51, 60, 66, 77, 85, 99, 110, 126, 140, 162, 179, 205, 228, 260, 289, 329, 365, 415, 461, 521, 579, 655, 726, 818, 909, 1022, 1134, 1273, 1411

Offset: 0

Jason Kimberley, Jan 31 2012

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth at least 8 (all such graphs are simple). The integer i corresponds to the i-cycle; addition of integers corresponds to disconnected union of cycles.

By removing a single part of size 8, an A026801 partition of n becomes an A185328 partition of n - 8. Hence this sequence is essentially the same as A026801.

Robert Israel, Table of n, a(n) for n = 0..2000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), this sequence (g=8), A185329 (g=9).

Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800(g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/(&*[1-x^(m+8): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

N:= 100: # for a(0)..a(N)
g:= mul(1/(1-x^m),m=8..N):
S:= series(g,x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Dec 19 2017

CoefficientList[Series[1/QPochhammer[x^8, x], {x,0,75}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^70)); Vec(1/prod(m=0,80, 1-x^(m+8))) \\ G. C. Greubel, Nov 03 2019

def A185328_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+8)) for m in (0..80)) ).list()
A185328_list(70) # G. C. Greubel, Nov 03 2019

G.f.: Product_{m>=8} 1/(1-x^m).
a(n) = p(n) - p(n-1) - p(n-2) + p(n-5) + p(n-7) + p(n-8) - p(n-10) - p(n-11) - 2*p(n-12) + 2*p(n-16) + p(n-17) + p(n-18) - p(n-20) - p(n-21) - p(n-23) + p(n-26) + p(n-27) - p(n-28) where p(n)=A000041(n). - Shanzhen Gao
This sequence is the Euler transformation of A185118.
a(n) ~ exp(Pi*sqrt(2*n/3)) * 35*Pi^7 / (18*sqrt(2)*n^(9/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=0} x^(8*k) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Nov 28 2020
G.f.: 1 + Sum_{n >= 1} x^(n+7)/Product_{k = 0..n-1} (1 - x^(k+8)). - Peter Bala, Dec 01 2024

A026800 Number of partitions of n in which the least part is 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 18, 20, 24, 27, 32, 36, 42, 48, 56, 63, 73, 83, 96, 108, 125, 141, 162, 183, 209, 236, 270, 304, 346, 390, 443, 498, 565, 635, 719, 807, 911, 1022, 1153, 1291, 1453, 1628, 1829, 2045

Offset: 0

Clark Kimberling

From Jason Kimberley, Feb 03 2011: (Start)
a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth exactly 7 (all such graphs are simple). The integer i corresponds to the i-cycle; the addition of integers corresponds to the disconnected union of cycles.
By removing a single part of size 7, an A026800 partition of n becomes an A185327 partition of n - 7. (End)

			a(0)=0 because there does not exist a least part of the empty partition.
The  a(7)=1 partition is 7.
The a(14)=1 partition is 7+7.
The a(15)=1 partition is 7+8.
.............................
The a(20)=1 partition is 7+13.
The a(21)=2 partitions are 7+7+7 and 7+14.

Robert Israel, Table of n, a(n) for n = 0..10000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Cf. A185327 (Mathematica code)
Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).
Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), this sequence (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 03 2011

p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A026800 := func< n | p(n-7)-p(n-8)-p(n-9)+p(n-12)+2*p(n-14)-p(n-16)- p(n-17)-p(n-18)-p(n-19)+2*p(n-21)+p(n-23)-p(n-26)-p(n-27)+p(n-28) >; // Jason Kimberley, Feb 03 2011

R:=PowerSeriesRing(Integers(), 75); [0,0,0,0,0,0,0] cat Coefficients(R!( x^7/(&*[1-x^(m+7): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

N:= 100: # for a(0)..a(N)
S:= series(x^7/mul(1-x^i,i=7..N-7),x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Jul 04 2019

CoefficientList[Series[x^7/QPochhammer[x^7, x], {x, 0, 75}], x] (* G. C. Greubel, Nov 03 2019 *)
Join[{0},Table[Count[IntegerPartitions[n],?(#[[-1]]==7&)],{n,80}]] (* _Harvey P. Dale, Apr 05 2025 *)

my(x='x+O('x^75)); concat([0,0,0,0,0,0,0], Vec(x^7/prod(m=0,80, 1-x^(m+7)))) \\ G. C. Greubel, Nov 03 2019

def A026800_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^7/product((1-x^(m+7)) for m in (0..80)) ).list()
A026800_list(75) # G. C. Greubel, Nov 03 2019

G.f.: x^7 * Product_{m>=7} 1/(1-x^m).
a(n) = p(n-7) -p(n-8) -p(n-9) +p(n-12) +2*p(n-14) -p(n-16) -p(n-17) -p(n-18) -p(n-19) +2*p(n-21) +p(n-23) -p(n-26) -p(n-27) +p(n-28) where p(n)=A000041(n) including the implicit p(n)=0 for negative n. - Shanzhen Gao, Oct 28 2010; offset corrected / made explicit by Jason Kimberley, Feb 03 2011
a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^6 / (6*sqrt(3)*n^4). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(7*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001

A026801 Number of partitions of n in which the least part is 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 9, 10, 12, 13, 16, 17, 21, 23, 27, 30, 36, 39, 46, 51, 60, 66, 77, 85, 99, 110, 126, 140, 162, 179, 205, 228, 260, 289, 329, 365, 415, 461, 521, 579, 655, 726, 818, 909, 1022, 1134, 1273, 1411

Offset: 1

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).

Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800 (g=7), this sequence (g=8), A026802 (g=9), A026803 (g=10).

R:=PowerSeriesRing(Integers(), 70); [0,0,0,0,0,0,0] cat Coefficients(R!( x^8/(&*[1-x^(m+8): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(x^8/mul(1-x^(m+8), m = 0..80), x, n+1), x, n), n = 1..70); # G. C. Greubel, Nov 03 2019

Rest@CoefficientList[Series[x^8/QPochhammer[x^8, x], {x, 0, 75}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^70)); concat(vector(7), Vec(x^8/prod(m=0,80, 1-x^(m+8)))) \\ G. C. Greubel, Nov 03 2019

def A026801_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^8/product((1-x^(m+8)) for m in (0..80)) ).list()
a=A026801_list(71); a[1:] # G. C. Greubel, Nov 03 2019

G.f.: x^8 * Product_{m>=8} 1/(1-x^m).
a(n+8) = p(n) -p(n-1) -p(n-2) +p(n-5) +p(n-7) +p(n-8) -p(n-10) -p(n-11) -2*p(n-12) +2*p(n-16) +p(n-17) +p(n-18) -p(n-20) -p(n-21) -p(n-23) +p(n-26) +p(n-27) -p(n-28) where p(n)=A000041(n). - Shanzhen Gao, Oct 28 2010
a(n) ~ exp(Pi*sqrt(2*n/3)) * 35*Pi^7 / (18*sqrt(2)*n^(9/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(8*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001

A026802 Number of partitions of n in which the least part is 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 24, 26, 30, 34, 39, 43, 50, 55, 63, 71, 80, 89, 102, 113, 128, 143, 161, 179, 203, 225, 253, 282, 316, 351, 395, 437, 489, 544, 607, 673, 752, 832, 927, 1028, 1143

Offset: 1

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).

Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800 (g=7), A026801 (g=8), this sequence (g=9), A026803 (g=10).

R:=PowerSeriesRing(Integers(), 80); [0,0,0,0,0,0,0,0] cat Coefficients(R!( x^9/(&*[1-x^(m+9): m in [0..85]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(x^9/mul(1-x^(m+9), m = 0..85), x, n+1), x, n), n = 1..80); # G. C. Greubel, Nov 03 2019

Table[Count[IntegerPartitions[n],?(Min[#]==9&)],{n,80}] (* _Harvey P. Dale, May 09 2013 *)
Rest@CoefficientList[Series[x^9/QPochhammer[x^9, x], {x,0,80}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^70)); concat(vector(8), Vec(x^9/prod(m=0,85, 1-x^(m+9)))) \\ G. C. Greubel, Nov 03 2019

def A026802_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^9/product((1-x^(m+9)) for m in (0..85)) ).list()
a=A026802_list(81); a[1:] # G. C. Greubel, Nov 03 2019

G.f.: x^9 * Product_{m>=9} 1/(1-x^m).
a(n+9) = p(n) -p(n-1) -p(n-2) +p(n-5) +p(n-7) +p(n-9) -p(n-11) -2*p(n-12) -p(n-13) -p(n-15) +p(n-16) +p(n-17) +2*p(n-18) +p(n-19) +p(n-20) -p(n-21) -p(n-23) -2*p(n-24) -p(n-25) +p(n-27) +p(n-29) +p(n-31) -p(n-34) -p(n-35) +p(n-36) where p(n)=A000041(n). - Shanzhen Gao, Oct 28 2010
a(n) ~ exp(Pi*sqrt(2*n/3)) * 70*Pi^8 / (9*sqrt(3)*n^5). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(9*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001

A026803 Number of partitions of n in which the least part is 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 10, 11, 13, 14, 17, 18, 21, 23, 27, 29, 34, 37, 43, 47, 54, 59, 68, 74, 85, 93, 106, 116, 132, 145, 164, 180, 203, 223, 252, 276, 310, 341, 382, 420, 470, 516, 576, 633, 706, 775, 863

Offset: 1

Clark Kimberling

In general, if g>=1 and g.f. = x^g * Product_{m>=g} 1/(1-x^m), then a(n,g) ~ Pi^(g-1) * (g-1)! * exp(Pi*sqrt(2*n/3)) / (2^((g+3)/2) * 3^(g/2) * n^((g+1)/2)) ~ p(n) * Pi^(g-1) * (g-1)! / (6*n)^((g-1)/2), where p(n) is the partition function A000041(n). - Vaclav Kotesovec, Jun 02 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).

Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), A026800 (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10).

R:=PowerSeriesRing(Integers(), 80); [0,0,0,0,0,0,0,0,0] cat Coefficients(R!( x^10/(&*[1-x^(m+10): m in [0..85]]) )); // G. C. Greubel, Nov 03 2019

seq(coeff(series(x^10/mul(1-x^(m+10), m = 0..85), x, n+1), x, n), n = 1..80); # G. C. Greubel, Nov 03 2019

Rest@CoefficientList[Series[x^10/QPochhammer[x^10, x], {x,0,80}], x] (* G. C. Greubel, Nov 03 2019 *)

my(x='x+O('x^80)); concat(vector(9), Vec(x^10/prod(m=0,85, 1-x^(m+10)))) \\ G. C. Greubel, Nov 03 2019

def A026803_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^10/product((1-x^(m+10)) for m in (0..85)) ).list()
a=A026803_list(71); a[1:] # G. C. Greubel, Nov 03 2019

G.f.: x^10 * Product_{m>=10} 1/(1-x^m).
a(n) ~ exp(Pi*sqrt(2*n/3)) * 35*sqrt(2)*Pi^9 / (3*n^(11/2)). - Vaclav Kotesovec, Jun 02 2018
G.f.: Sum_{k>=1} x^(10*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001

A231599 T(n,k) is the coefficient of x^k in Product_{i=1..n} (1-x^i); triangle T(n,k), n >= 0, 0 <= k <= A000217(n), read by rows.

Original entry on oeis.org

1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 0, 2, 0, 0, -1, -1, 1, 1, -1, -1, 0, 0, 1, 1, 1, -1, -1, -1, 0, 0, 1, 1, -1, 1, -1, -1, 0, 0, 1, 0, 2, 0, -1, -1, -1, -1, 0, 2, 0, 1, 0, 0, -1, -1, 1, 1, -1, -1, 0, 0, 1, 0, 1, 1, 0, -1, -1, -2, 0

Offset: 0

Marc Bogaerts, Nov 11 2013

From Tilman Piesk, Feb 21 2016: (Start)
The sum of each row is 0. The even rows are symmetric; in the odd rows numbers with the same absolute value and opposed signum are symmetric to each other.
The odd rows where n mod 4 = 3 have the central value 0.
The even rows where n mod 4 = 0 have positive central values. They form the sequence A269298 and are also the rows maximal values.
A086376 contains the maximal values of each row, A160089 the maximal absolute values, and A086394 the absolute parts of the minimal values.
Rows of this triangle can be used to efficiently calculate values of A026807.
(End)

			For n=2 the corresponding polynomial is (1-x)*(1-x^2) = 1 -x - x^2 + x^3.
Irregular triangle starts:
  k    0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
n
0      1
1      1  -1
2      1  -1  -1   1
3      1  -1  -1   0   1   1  -1
4      1  -1  -1   0   0   2   0   0  -1  -1   1
5      1  -1  -1   0   0   1   1   1  -1  -1  -1   0   0   1   1  -1

Alois P. Heinz, Rows n = 0..40, flattened
Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11.
Tilman Piesk, Rows n = 0..40 as left-aligned and centered table

Cf. A000217 (triangular numbers).
Cf. A086376, A160089, A086394 (maxima, etc.).
Cf. A269298 (central nonzero values).

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))
        (expand(mul(1-x^i, i=1..n))):
seq(T(n), n=0..10);  # Alois P. Heinz, Dec 22 2013

Table[If[k == 0, 1, Coefficient[Product[(1 - x^i), {i, n}], x^k]], {n, 0, 6}, {k, 0, (n^2 + n)/2}] // Flatten (* Michael De Vlieger, Mar 04 2018 *)

row(n) = pol = prod(i=1, n, 1 - x^i); for (i=0, poldegree(pol), print1(polcoeff(pol, i), ", ")); \\ Michel Marcus, Dec 21 2013

from sympy import poly, symbols
def a231599_row(n):
    if n == 0:
        return [1]
    x = symbols('x')
    p = 1
    for i in range(1, n+1):
        p *= poly(1-x**i)
    p = p.all_coeffs()
    return p[::-1]
# Tilman Piesk, Feb 21 2016

T(n,k) = [x^k] Product_{i=1..n} (1-x^i).

T(n,k) = T(n-1, k) + (-1)^n*T(n-1, n*(n+1)/2-k), n > 1. - Gevorg Hmayakyan, Feb 09 2017 [corrected by Giuliano Cabrele, Mar 02 2018]

cp's OEIS Frontend

A185329 Number of partitions of n with parts >= 9.

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A026799 Number of partitions of n in which the least part is 6.

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A026805 Number of partitions of n in which the least part is even.

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A185327 Number of partitions of n into parts >= 7.

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A185328 Number of partitions of n with parts >= 8.

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A026800 Number of partitions of n in which the least part is 7.

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