A026800 -id:A026800 - cp's OEIS frontend

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

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

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

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

A339170 Number of compositions (ordered partitions) of n into distinct parts, the least being 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 8, 8, 14, 14, 20, 20, 26, 26, 32, 32, 62, 62, 92, 116, 146, 170, 224, 248, 302, 350, 404, 572, 650, 818, 1016, 1328, 1526, 1958, 2300, 2852, 3314, 4010, 4592, 6248, 6974, 8750, 10436, 13196, 15722, 19442, 22952

Offset: 0

Ilya Gutkovskiy, Nov 25 2020

nonn

			a(24) = 8 because we have [17, 7], [9, 8, 7], [9, 7, 8], [8, 9, 7], [8, 7, 9], [7, 17], [7, 9, 8] and [7, 8, 9].

Index entries for sequences related to compositions

Cf. A026800, A026828, A339108, A339162, A339163, A339164, A339165, A339166, A339169, A339171, A339172.

nmax = 64; CoefficientList[Series[Sum[k! x^(k (k + 13)/2)/Product[1 - x^j, {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=1} k! * x^(k*(k + 13)/2) / Product_{j=1..k-1} (1 - x^j).

A199086 T(n,k) = Number of partitions of n+2k-2 into parts >= k.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 2, 5, 1, 2, 2, 4, 7, 1, 2, 2, 3, 4, 11, 1, 2, 2, 3, 4, 7, 15, 1, 2, 2, 3, 3, 5, 8, 22, 1, 2, 2, 3, 3, 5, 6, 12, 30, 1, 2, 2, 3, 3, 4, 5, 9, 14, 42, 1, 2, 2, 3, 3, 4, 5, 7, 10, 21, 56, 1, 2, 2, 3, 3, 4, 4, 6, 8, 13, 24, 77, 1, 2, 2, 3, 3, 4, 4, 6, 7, 11, 17, 34, 101, 1, 2, 2, 3, 3, 4, 4, 5

Offset: 1

R. H. Hardin, Nov 06 2011

Row n goes to floor(n/2)+1
Table starts
...1...1...1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1
...2...2...2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2
...3...2...2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2..2
...5...4...3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3
...7...4...4..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3..3
..11...7...5..5..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4
..15...8...6..5..5..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4..4
..22..12...9..7..6..6..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5
..30..14..10..8..7..6..6..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5..5
..42..21..13.11..9..8..7..7..6..6..6..6..6..6..6..6..6..6..6..6..6..6..6..6..6
..56..24..17.12.10..9..8..7..7..6..6..6..6..6..6..6..6..6..6..6..6..6..6..6..6
..77..34..21.16.13.11.10..9..8..8..7..7..7..7..7..7..7..7..7..7..7..7..7..7..7
.101..41..25.18.15.12.11.10..9..8..8..7..7..7..7..7..7..7..7..7..7..7..7..7..7
.135..55..33.24.18.16.13.12.11.10..9..9..8..8..8..8..8..8..8..8..8..8..8..8..8
.176..66..39.27.21.17.15.13.12.11.10..9..9..8..8..8..8..8..8..8..8..8..8..8..8
.231..88..49.34.26.21.18.16.14.13.12.11.10.10..9..9..9..9..9..9..9..9..9..9..9
.297.105..60.39.30.24.20.17.16.14.13.12.11.10.10..9..9..9..9..9..9..9..9..9..9
.385.137..73.50.36.29.24.21.18.17.15.14.13.12.11.11.10.10.10.10.10.10.10.10.10
.490.165..88.57.42.32.27.23.20.18.17.15.14.13.12.11.11.10.10.10.10.10.10.10.10
.627.210.110.70.50.40.32.27.24.21.19.18.16.15.14.13.12.12.11.11.11.11.11.11.11

			All solutions for n=5, k=3: 3+3+3, 3+6, 4+5, 9.

R. H. Hardin and Alois P. Heinz, Table of n, a(n) for n = 1..10011

Column 1 is A000041
Column 2 is A002865(n+2)
Column 3 is A008483(n+4)
Column 4 is A008484(n+6)
Column 5 is A026798(n+13)
Column 6 is A026799(n+16)
Column 7 is A026800(n+19)
Column 8 is A026801(n+22)
Column 9 is A026802(n+25)
Column 10 is A026803(n+28)

b:= proc(n, i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i>n then 0
    else b(n-i, i) +b(n, i+1)
      fi
    end:
T:= (n, k)-> b(n+2*k-2, k):
seq(seq(T(n, d+1-n), n=1..d), d=1..20); # Alois P. Heinz, Nov 06 2011

b[n_, i_] := b[n, i] = Which[n < 0, 0, n == 0, 1, i > n, 0, True, b[n - i, i] + b[n, i + 1]]; T[n_, k_] := b[n + 2*k - 2, k]; Table[Table[T[n, d + 1 - n], {n, 1, d}], {d, 1, 20}] // Flatten (* Jean-François Alcover, Jan 23 2016, after Alois P. Heinz *)

G.f. of column k: x^(2-2*k) * Product_{j>=k} 1/(1-x^j). - Alois P. Heinz, Nov 06 2011

cp's OEIS Frontend

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

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