A097364 -id:A097364 - cp's OEIS frontend

A008805 Triangular numbers repeated.

1, 1, 3, 3, 6, 6, 10, 10, 15, 15, 21, 21, 28, 28, 36, 36, 45, 45, 55, 55, 66, 66, 78, 78, 91, 91, 105, 105, 120, 120, 136, 136, 153, 153, 171, 171, 190, 190, 210, 210, 231, 231, 253, 253, 276, 276, 300, 300, 325, 325, 351, 351, 378, 378, 406, 406, 435, 435

Offset: 0

N. J. A. Sloane

Number of choices for nonnegative integers x,y,z such that x and y are even and x + y + z = n.
Diagonal sums of A002260, when arranged as a number triangle. - Paul Barry, Feb 28 2003
a(n) = number of partitions of n+4 such that the differences between greatest and smallest parts are 2: a(n-4) = A097364(n,2) for n>3. - Reinhard Zumkeller, Aug 09 2004
For n >= i, i=4,5, a(n-i) is the number of incongruent two-color bracelets of n beads, i from them are black (cf. A005232, A032279), having a diameter of symmetry. - Vladimir Shevelev, May 03 2011
Prefixing A008805 by 0,0,0,0 gives the sequence c(0), c(1), ... defined by c(n)=number of (w,x,y) such that w = 2x+2y, where w,x,y are all in {1,...,n}; see A211422. - Clark Kimberling, Apr 15 2012
Partial sums of positive terms of A142150. - Reinhard Zumkeller, Jul 07 2012
The sum of the first parts of the nondecreasing partitions of n+2 into exactly two parts, n >= 0. - Wesley Ivan Hurt, Jun 08 2013
Number of the distinct symmetric pentagons in a regular n-gon, see illustration for some small n in links. - Kival Ngaokrajang, Jun 25 2013
a(n) is the number of nonnegative integer solutions to the equation x + y + z = n such that x + y <= z. For example, a(4) = 6 because we have 0+0+4 = 0+1+3 = 0+2+2 = 1+0+3 = 1+1+2 = 2+0+2. - Geoffrey Critzer, Jul 09 2013
a(n) is the number of distinct opening moves in n X n tic-tac-toe. - I. J. Kennedy, Sep 04 2013
a(n) is the number of symmetry-allowed, linearly-independent terms at n-th order in the series expansion of the T2 X t2 vibronic perturbation matrix, H(Q) (cf. Opalka & Domcke). - Bradley Klee, Jul 20 2015
a(n-1) also gives the number of D_4 (dihedral group of order 4) orbits of an n X n square grid with squares coming in either of two colors and only one square has one of the colors. - Wolfdieter Lang, Oct 03 2016
Also, this sequence is the third column in the triangle of the coefficients of the sum of two consecutive Fibonacci polynomials F(n+1, x) and F(n, x) (n>=0) in ascending powers of x. - Mohammad K. Azarian, Jul 18 2018
In an n-person symmetric matching pennies game (a zero-sum normal-form game) with n > 2 symmetric and indistinguishable players, each with two strategies (viz. heads or tails), a(n-3) is the number of distinct subsets of players that must play the same strategy to avoid incurring losses (single pure Nash equilibrium in the reduced game). The total number of distinct partitions is A000217(n-1). - Ambrosio Valencia-Romero, Apr 17 2022
a(n) is the number of connected bipartite graphs with n+1 edges and a stable set of cardinality 2. - Christian Barrientos, Jun 15 2022
a(n) is the number of 132-avoiding odd Grassmannian permutations of size n+2. - Juan B. Gil, Mar 10 2023
Consider a regular n-gon with all diagonals drawn. Define a "layer" to be the set of all regions sharing an edge with the exterior. Removing a layer creates another layer. Count the layers, removing them until none remain. The number of layers is a(n-2). See illustration. - Christopher Scussel, Nov 07 2023

			a(5) = 6, since (5) + 2 = 7 has three nondecreasing partitions with exactly 2 parts: (1,6),(2,5),(3,4). The sum of the first parts of these partitions = 1 + 2 + 3 = 6. - _Wesley Ivan Hurt_, Jun 08 2013

H. D. Brunk, An Introduction to Mathematical Statistics, Ginn, Boston, 1960; p. 360.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
G. E. Andrews, M. Beck, and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv preprint arXiv:1406.3374 [math.NT], 2014.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 46.
Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 19.
Kival Ngaokrajang, The distinct symmetric 5-gons in a regular n-gon for n = 6..13
D. Opalka and W. Domcke, High-order expansion of T2xt2 Jahn-Teller potential energy surfaces in tetrahedral molecules, J. Chem. Phys., 132, 154108 (2010).
Christopher Scussel, Illustration of layers in regular n-gons with all diagonals drawn
Vladimir Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
Index entries for Molien series

Cf. A000217, A002260, A002620, A006918 (partial sums), A054252, A135276, A142150, A158920 (binomial trans.).

List([0..60], n-> (2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32); # G. C. Greubel, Sep 12 2019

import Data.List (transpose)
a008805 = a000217 . (`div` 2) . (+ 1)
a008805_list = drop 2 $ concat $ transpose [a000217_list, a000217_list]
-- Reinhard Zumkeller, Feb 01 2013

[(2*n+3+(-1)^n)*(2*n+7+(-1)^n)/32 : n in [0..50]]; // Wesley Ivan Hurt, Apr 22 2015

A008805:=n->(2*n+3+(-1)^n)*(2*n+7+(-1)^n)/32: seq(A008805(n), n=0..50); # Wesley Ivan Hurt, Apr 22 2015

CoefficientList[Series[1/(1-x^2)^2/(1-x), {x, 0, 50}], x]
Table[Binomial[Floor[n/2] + 2, 2], {n, 0, 57}] (* Michael De Vlieger, Oct 03 2016 *)

PARI
```
a(n)=(n\2+2)*(n\2+1)/2
```

def A008805(n): return (m:=(n>>1)+1)*(m+1)>>1 # Chai Wah Wu, Oct 20 2023

[(2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32 for n in (0..60)] # G. C. Greubel, Sep 12 2019

G.f.: 1/((1-x)*(1-x^2)^2) = 1/((1+x)^2*(1-x)^3).
E.g.f.: (exp(x)*(2*x^2 +12*x+ 11) - exp(-x)*(2*x -5))/16.
a(-n) = a(-5+n).
a(n) = binomial(floor(n/2)+2, 2). - Vladimir Shevelev, May 03 2011
From Paul Barry, May 31 2003: (Start)
a(n) = ((2*n +5)*(-1)^n + (2*n^2 +10*n +11))/16.
a(n) = Sum_{k=0..n} ((k+2)*(1+(-1)^k))/4. (End)
From Paul Barry, Apr 16 2005: (Start)
a(n) = Sum_{k=0..n} floor((k+2)/2)*(1-(-1)^(n+k-1))/2.
a(n) = Sum_{k=0..floor(n/2)} floor((n-2k+2)/2). (End)
A signed version is given by Sum_{k=0..n} (-1)^k*floor(k^2/4). - Paul Barry, Aug 19 2003
a(n) = A108299(n-2,n)*(-1)^floor((n+1)/2) for n>1. - Reinhard Zumkeller, Jun 01 2005
a(n) = A004125(n+3) - A049798(n+2). - Carl Najafi, Jan 31 2013
a(n) = Sum_{i=1..floor((n+2)/2)} i. - Wesley Ivan Hurt, Jun 08 2013
a(n) = (1/2)*floor((n+2)/2)*(floor((n+2)/2)+1). - Wesley Ivan Hurt, Jun 08 2013
From Wesley Ivan Hurt, Apr 22 2015: (Start)
a(n) = a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).
a(n) = (2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32. (End)
a(n-1) = A054252(n,1) = A054252(n^2-1), n >= 1. See a Oct 03 2016 comment above. - Wolfdieter Lang, Oct 03 2016
a(n) = A000217(A008619(n)). - Guenther Schrack, Sep 12 2018
From Ambrosio Valencia-Romero, Apr 17 2022: (Start)
a(n) = a(n-1) if n odd, a(n) = a(n-1) + (n+2)/2 if n is even, for n > 0, a(0) = 1.
a(n) = (n+1)*(n+3)/8 if n odd, a(n) = (n+2)*(n+4)/8 if n is even, for n >= 0.
a(n) = A002620(n+2) - a(n-1), for n > 0, a(0) = 1.
a(n) = A142150(n+2) + a(n-1), for n > 0, a(0) = 1.
a(n) = A000217(n+3)/2 - A135276(n+3)/2. (End)

A117989 Number of partitions of n such that the least part occurs at least twice.

Original entry on oeis.org

0, 1, 1, 3, 3, 7, 8, 14, 18, 28, 35, 53, 67, 94, 121, 165, 209, 280, 353, 462, 582, 749, 935, 1192, 1480, 1862, 2302, 2871, 3526, 4366, 5335, 6555, 7976, 9737, 11789, 14317, 17259, 20845, 25032, 30093, 35992, 43087, 51347, 61216, 72710, 86362, 102235

Offset: 1

Emeric Deutsch, Apr 08 2006

nonn

More generally, the g.f. for the number of partitions of n such that the least part occurs at least m times is sum(x^(mk)/product(1-x^j, j=k..infinity), k=1..infinity). Also, the number of partitions of n such that if k is the largest part, then k>=2 and k-1 does not occur. Example: a(5)=3 because we have [5],[4,1] and [3,1,1].

Also, the number of partitions of 2n such that the difference between greatest part and smallest part is n. - Vladeta Jovovic, May 09 2008

			a(5) = 3 because we have [3,1,1], [2,1,1,1] and [1,1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Aritram Dhar, Proofs of Two Formulas of Vladeta Jovovic, arXiv:2112.07762 [math.CO], 2021.

Cf. A096373, A000041, A056823.

Cf. A097364, A323111.

a117989 n = a117989_list !! (n-1)
a117989_list = tail $ zipWith (-)
                      (map (* 2) a000041_list) $ tail a000041_list
-- Reinhard Zumkeller, Nov 12 2015

g:=sum(x^k*(1-x^(k-1))/product(1-x^j,j=1..k),k=2..70): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..50);
A117989 := proc(n)
    2*combinat[numbpart](n)-combinat[numbpart](n+1) ;
end proc: # R. J. Mathar, May 19 2016

Table[Length[Select[IntegerPartitions[n],Count[#,Min[#]]>1&]], {n,50}]  (* Harvey P. Dale, Apr 23 2011 *)
max = 48; Sum[x^(2*k)/Product[1 - x^j, {j, k, Infinity}], {k, 1, Ceiling[ max/2]}] + O[x]^max // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Sep 11 2017 *)

G.f.: sum(k>=1, x^(2*k)/prod(j>=k, 1-x^j ) ).
G.f.: sum(k>=1, x^k*(1-x^(k-1))/prod(j=1..k, 1-x^j ) ).
a(n) = 2*A000041(n) - A000041(n+1). - Vladeta Jovovic, Jul 21 2006
a(n) = A056823(n+1) - 2*A056823(n). - Bob Selcoe, Apr 11 2014
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 25*Pi/(24*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Nov 03 2020

A116685 Triangle read by rows: T(n,k) is number of partitions of n that have k parts smaller than the largest part (n>=1, k>=0).

Original entry on oeis.org

1, 2, 2, 1, 3, 1, 1, 2, 3, 1, 1, 4, 2, 3, 1, 1, 2, 5, 3, 3, 1, 1, 4, 4, 6, 3, 3, 1, 1, 3, 6, 6, 7, 3, 3, 1, 1, 4, 6, 10, 7, 7, 3, 3, 1, 1, 2, 9, 10, 12, 8, 7, 3, 3, 1, 1, 6, 6, 15, 14, 13, 8, 7, 3, 3, 1, 1, 2, 11, 15, 20, 16, 14, 8, 7, 3, 3, 1, 1, 4, 10, 21, 22, 24, 17, 14, 8, 7, 3, 3, 1, 1, 4, 11, 21

Offset: 1

Emeric Deutsch, Feb 23 2006

Same as A097364 without the 0's.
Also number of partitions of n such that the difference between the largest and smallest parts is k (see A097364). Example: T(6,2)=3 because we have [4,2],[3,2,1] and [3,1,1,1].
Row 1 has one term; row n (n>=2) has n-1 terms.
Row sums yield the partition numbers (A000041).
T(n,0)=A000005(n) (number of divisors of n).
T(n,1)=A049820(n) (n minus number of divisors of n).
T(n,2)=A008805(n-4) for n>=4.
Sum(k*T(n,k),k=0..n-2)=A116686

			Triangle starts:
01:  1
02:  2
03:  2  1
04:  3  1  1
05:  2  3  1  1
06:  4  2  3  1  1
07:  2  5  3  3  1  1
08:  4  4  6  3  3  1 1
09:  3  6  6  7  3  3 1 1
10:  4  6 10  7  7  3 3 1 1
11:  2  9 10 12  8  7 3 3 1 1
12:  6  6 15 14 13  8 7 3 3 1 1
13:  2 11 15 20 16 14 8 7 3 3 1 1
14:  4 10 21 22 24 17 ...
T(6,2)=3 because we have [4,1,1],[3,2,1] and [2,2,1,1].

Alois P. Heinz, Rows n = 1..142, flattened
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014.
Bernard L. S. Lin, Saisai Zheng, k-regular partitions and overpartitions with bounded part differences, The Raman. J. 56 (2021) 685-695

Cf. A000041, A000005, A049820, A008805, A116686, A097364.

Columns k=3-10 give: A128508, A218567, A218568, A218569, A218570, A218571, A218572, A218573. T(2*n,n) = A117989(n). - Alois P. Heinz, Nov 02 2012

g:=sum(x^i/(1-x^i)/product(1-t*x^j,j=1..i-1),i=1..50): gser:=simplify(series(g,x=0,18)): for n from 1 to 15 do P[n]:=coeff(gser,x^n) od: 1; for n from 2 to 15 do seq(coeff(P[n],t,j),j=0..n-2) od;
# yields sequence in triangular form

rows = 15; max = rows + 2; col[k0_ /; k0 > 0] := col[k0] = Sum[x^(2*k + k0)/Product[ (1 - x^(k + j)), {j, 0, k0}], {k, 1, Ceiling[max/2]}] + O[x]^max // CoefficientList[#, x] &; col[0] := Table[Switch[n, 1, 0, 2, 1, , n - 1 - col[1][[n]]], {n, 1, Length[col[1]]}]; Join[{1}, Table[ col[k][[n+2]], {n, 0, rows-1}, {k, 0, n-1}] // Flatten] (* _Jean-François Alcover, Sep 11 2017, after Alois P. Heinz *)

G.f.: sum(i>=1, x^i/(1-x^i)/prod(j=1..i-1, 1-t*x^j) ).

A128508 Number of partitions p of n such that max(p) - min(p) = 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 3, 7, 7, 12, 14, 20, 22, 32, 34, 45, 51, 63, 69, 87, 93, 112, 124, 144, 156, 184, 196, 225, 245, 275, 295, 335, 355, 396, 426, 468, 498, 552, 582, 637, 679, 735, 777, 847, 889, 960, 1016, 1088, 1144, 1232, 1288, 1377, 1449, 1539, 1611, 1719

Offset: 0

John W. Layman, May 07 2007

nonn

See A008805 and A049820 for the numbers of partitions p of n such that max(p)-min(p)=1 or 2, respectively.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014

Cf. A008805, A049820, A097364, A116685, A117142, A117143.

np[n_]:=Length[Select[IntegerPartitions[n],Max[#]-Min[#]==3&]]; Array[np,60] (* Harvey P. Dale, Jul 02 2012 *)

Conjecture. a(1)=0 and, for n>1, a(n+1)=a(n)+d(n), where d(n) is defined as follows: d=0,0,0,1,0 for n=1,...,5 and, for n>5, d(n)=d(n-2)+1 if n=6k or n=6k+4, d(n)=d(n-2) if n=6k+1 or n=6k+3, d(n)=d(n-2)+2Floor[n/6] if n=6k+2 and d(n)=d(n-5) if n=6k+5.
G.f. for number of partitions p of n such that max(p)-min(p) = m is Sum_{k>0} x^(2*k+m)/Product_{i=0..m} (1-x^(k+i)). - Vladeta Jovovic, Jul 04 2007
a(n) = A097364(n,3) = A116685(n,3) = A117143(n) - A117142(n). - Alois P. Heinz, Nov 02 2012

More terms from Vladeta Jovovic, Jul 04 2007

A218567 Number of partitions p of n such that max(p)-min(p) = 4.

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 13, 16, 24, 27, 40, 46, 60, 71, 92, 103, 131, 149, 181, 206, 247, 275, 329, 366, 424, 474, 548, 601, 690, 759, 858, 942, 1059, 1152, 1293, 1404, 1555, 1690, 1869, 2013, 2218, 2390, 2614, 2812, 3066, 3282, 3574, 3820, 4131, 4415, 4769, 5071

Offset: 6

Alois P. Heinz, Nov 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014.

terms = 52; offset = 6; max = terms + offset; s[k0_ /; k0>0] := Sum[x^(2*k + k0) / Product[ (1 - x^(k+j)), {j, 0, k0}], {k, 1, Ceiling[max/2]}] + O[x]^max // CoefficientList[#, x]&; Drop[s[4], offset] (* Jean-François Alcover, Sep 11 2017, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n],?(#[[1]]-#[[-1]]==4&)],{n,6,60}] (* _Harvey P. Dale, Jul 10 2021 *)

G.f.: Sum_{k>0} x^(2*k+4)/Product_{j=0..4} (1-x^(k+j)).

a(n) = A097364(n,4) = A116685(n,4) = A194621(n,4) - A194621(n,3) = A218506(n) - A117143(n).

A218573 Number of partitions p of n such that max(p) - min(p) = 10.

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 14, 18, 28, 35, 53, 67, 93, 119, 161, 201, 267, 332, 428, 531, 674, 824, 1034, 1258, 1552, 1877, 2294, 2749, 3332, 3970, 4762, 5645, 6723, 7916, 9367, 10974, 12894, 15036, 17571, 20381, 23696, 27370, 31652, 36416, 41926, 48029, 55071, 62860

Offset: 12

Alois P. Heinz, Nov 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 12..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014.

terms = 48; offset = 12; max = terms + offset; s[k0_ /; k0 > 0] := Sum[x^(2*k + k0)/Product[ (1 - x^(k + j)), {j, 0, k0}], {k, 1, Ceiling[max/2]}] + O[x]^max // CoefficientList[#, x] &; Drop[s[10], offset] (* Jean-François Alcover, Sep 11 2017, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n],?(#[[1]]-#[[-1]]==10&)],{n,12,60}] (* _Harvey P. Dale, Aug 14 2024 *)

G.f.: Sum_{k>0} x^(2*k+10)/Product_{j=0..10} (1-x^(k+j)).

a(n) = A097364(n,10) = A116685(n,10) = A194621(n,10) - A194621(n,9) = A218512(n) - A218511(n).

A218568 Number of partitions p of n such that max(p)-min(p) = 5.

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 14, 17, 26, 31, 45, 54, 73, 87, 114, 135, 171, 200, 249, 290, 352, 406, 487, 560, 660, 752, 879, 997, 1153, 1298, 1489, 1671, 1900, 2121, 2397, 2665, 2992, 3311, 3701, 4081, 4535, 4982, 5514, 6042, 6655, 7265, 7977, 8686, 9502, 10314, 11248

Offset: 7

Alois P. Heinz, Nov 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014

Table[Count[IntegerPartitions[n],?(Max[#]-Min[#]==5&)],{n,7,60}] (* _Harvey P. Dale, Sep 25 2015 *)

G.f.: Sum_{k>0} x^(2*k+5)/Product_{j=0..5} (1-x^(k+j)).

a(n) = A097364(n,5) = A116685(n,5) = A194621(n,5) - A194621(n,4) = A218507(n) - A218506(n).

A218569 Number of partitions p of n such that max(p)-min(p) = 6.

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 14, 18, 27, 33, 49, 59, 81, 100, 131, 158, 205, 243, 306, 365, 448, 527, 642, 748, 896, 1042, 1231, 1418, 1667, 1906, 2215, 2527, 2909, 3298, 3781, 4260, 4847, 5446, 6158, 6886, 7756, 8633, 9669, 10738, 11970, 13239, 14713, 16212, 17943

Offset: 8

Alois P. Heinz, Nov 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014

Table[Count[IntegerPartitions[n],?(First[#]-Last[#]==6&)],{n,8,60}] (* _Harvey P. Dale, Feb 09 2015 *)

G.f.: Sum_{k>0} x^(2*k+6)/Product_{j=0..6} (1-x^(k+j)).

a(n) = A097364(n,6) = A116685(n,6) = A194621(n,6) - A194621(n,5) = A218508(n) - A218507(n).

A218570 Number of partitions p of n such that max(p)-min(p) = 7.

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 14, 18, 28, 34, 51, 63, 86, 108, 144, 175, 229, 278, 351, 425, 529, 630, 775, 919, 1109, 1309, 1565, 1827, 2167, 2518, 2952, 3414, 3975, 4563, 5281, 6036, 6931, 7889, 9012, 10200, 11598, 13078, 14785, 16613, 18704, 20925, 23470, 26174, 29229

Offset: 9

Alois P. Heinz, Nov 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014.

terms = 49; offset = 9; max = terms + offset; s[k0_ /; k0 > 0] := Sum[x^(2*k + k0)/Product[ (1 - x^(k + j)), {j, 0, k0}], {k, 1, Ceiling[max/2]}] + O[x]^max // CoefficientList[#, x] &; Drop[s[7], offset] (* Jean-François Alcover, Sep 11 2017, after Alois P. Heinz *)

G.f.: Sum_{k>0} x^(2*k+7)/Product_{j=0..7} (1-x^(k+j)).

a(n) = A097364(n,7) = A116685(n,7) = A194621(n,7) - A194621(n,6) = A218509(n) - A218508(n).

A218571 Number of partitions p of n such that max(p)-min(p) = 8.

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 14, 18, 28, 35, 52, 65, 90, 113, 152, 188, 246, 302, 387, 471, 591, 714, 884, 1059, 1292, 1538, 1857, 2193, 2621, 3077, 3646, 4254, 4999, 5801, 6772, 7815, 9062, 10409, 12002, 13719, 15733, 17909, 20438, 23169, 26318, 29722, 33623, 37833

Offset: 10

Alois P. Heinz, Nov 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014.

terms = 48; offset = 10; max = terms + offset; s[k0_ /; k0 > 0] := Sum[x^(2*k + k0)/Product[ (1 - x^(k + j)), {j, 0, k0}], {k, 1, Ceiling[max/2]}] + O[x]^max // CoefficientList[#, x] &; Drop[s[8], offset] (* Jean-François Alcover, Sep 11 2017, after Alois P. Heinz *)

G.f.: Sum_{k>0} x^(2*k+8)/Product_{j=0..8} (1-x^(k+j)).

a(n) = A097364(n,8) = A116685(n,8) = A194621(n,8) - A194621(n,7) = A218510(n) - A218509(n).

cp's OEIS Frontend

A008805 Triangular numbers repeated.

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A117989 Number of partitions of n such that the least part occurs at least twice.

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A116685 Triangle read by rows: T(n,k) is number of partitions of n that have k parts smaller than the largest part (n>=1, k>=0).

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A128508 Number of partitions p of n such that max(p) - min(p) = 3.

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A218567 Number of partitions p of n such that max(p)-min(p) = 4.

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A218573 Number of partitions p of n such that max(p) - min(p) = 10.

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A218568 Number of partitions p of n such that max(p)-min(p) = 5.

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