A194621 -id:A194621 - cp's OEIS frontend

A117142 Number of partitions of n in which any two parts differ by at most 2.

1, 2, 3, 5, 6, 9, 10, 14, 15, 20, 21, 27, 28, 35, 36, 44, 45, 54, 55, 65, 66, 77, 78, 90, 91, 104, 105, 119, 120, 135, 136, 152, 153, 170, 171, 189, 190, 209, 210, 230, 231, 252, 253, 275, 276, 299, 300, 324, 325, 350, 351, 377, 378, 405, 406, 434, 435, 464, 465

Offset: 1

Emeric Deutsch, Feb 27 2006

Equals row sums of triangle A177991. - Gary W. Adamson, May 16 2010

Positive numbers that are either triangular (A000217) or triangular minus 1 (A000096). - Jon E. Schoenfield, Jun 12 2010

			a(6) = 9 because we have
  1: [6],
  2: [4, 2],
  3: [3, 3],
  4: [3, 2, 1],
  5: [3, 1, 1, 1],
  6: [2, 2, 2],
  7: [2, 2, 1, 1],
  8: [2, 1, 1, 1, 1],
  9: [1, 1, 1, 1, 1, 1]
([5,1] and [4,1,1] do not qualify).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Jonathan Bloom and Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000096, A000217, A117143, A152271, A177991, A152919.

Column k=2 of A194621. - Alois P. Heinz, Oct 17 2012

List([1..60],n->(2*n^2+10*n+3+(-1)^n*(2*n-3))/16); # Muniru A Asiru, Dec 21 2018

[(2*n*(n+5) +3 +(-1)^n*(2*n-3))/16: n in [1..60]]; // G. C. Greubel, Jul 18 2023

g:=sum(x^k/(1-x^k)/(1-x^(k+1))/(1-x^(k+2)),k=1..75): gser:=series(g,x=0,70): seq(coeff(gser,x^n),n=1..65); with(combinat): for n from 1 to 7 do P:=partition(n): A:={}: for j from 1 to nops(P) do if P[j][nops(P[j])]-P[j][1]<3 then A:=A union {P[j]} else A:=A fi od: print(A); od: # this program yields the partitions

Table[Count[IntegerPartitions[n], ?(Max[#] - Min[#] <= 2 &)], {n, 30}] (* _Birkas Gyorgy, Feb 20 2011 *)
Table[(2*n^2 +10*n +3 +(-1)^n*(2*n-3))/16, {n,30}] (* Birkas Gyorgy, Feb 20 2011 *)
Table[Sum[If[EvenQ[k], 1, (k+1)/2], {k,0,n}], {n,0,60}] (* Jon Maiga, Dec 21 2018 *)

Vec(x*(x^2-x-1)/((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Mar 05 2015

[(2*n*(n+5) +3 +(-1)^n*(2*n-3))/16 for n in range(1,61)] # G. C. Greubel, Jul 18 2023

G.f.: Sum_{k>=1} x^k/((1 - x^k)*(1 - x^(k + 1))*(1 - x^(k + 2))). More generally, the g.f. of the number of partitions in which any two parts differ by at most b is Sum_{k>=1} (x^k/(Product_{j=k..k+b} 1 - x^j)).
a(n) = (2*n^2 + 10*n + 3 + (-1)^n * (2*n - 3))/16. - Birkas Gyorgy, Feb 20 2011
G.f.: (1 + x)/(1 - x)/(Q(0) - x^2 - x^3), where Q(k) = 1 + x^2 + x^3 + k*x*(1 + x^2) - x^2*(1 + x*(k + 2))*(1 + k*x)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Jan 05 2014
G.f.: x*(1 + x - x^2)/((1 - x)^3*(1 + x)^2). - Colin Barker, Mar 05 2015
a(n) = Sum_{k=0..n-1} A152271(k). - Jon Maiga, Dec 21 2018
E.g.f.: (1/16)*( (3 + 2*x)*exp(-x) + (3 + 12*x + 2*x^2)*exp(x) ). - G. C. Greubel, Jul 18 2023
a(n) = A152919(n+1)/2. - Ridouane Oudra, Oct 29 2024

A117143 Number of partitions of n in which any two parts differ by at most 3.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 13, 17, 22, 27, 33, 41, 48, 57, 68, 78, 90, 105, 118, 134, 153, 170, 190, 214, 235, 260, 289, 315, 345, 380, 411, 447, 488, 525, 567, 615, 658, 707, 762, 812, 868, 931, 988, 1052, 1123, 1188, 1260, 1340, 1413, 1494, 1583, 1665, 1755, 1854

Offset: 1

Emeric Deutsch, Feb 27 2006

			a(6) = 10 because we have [6], [4,2], [4,1,1], [3,3], [3,2,1], [3,1,1,1], [2,2,2], [2,2,1,1], [2,1,1,1,1] and [1,1,1,1,1,1] ([5,1] does not qualify).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Jonathan Bloom, Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,-2,-2,1,1,1,-1).

Cf. A117142.
Column k=3 of A194621.
Cf. A002717, A045947, A248851.

[(2*Floor((n+2)/3)*(14*Floor((n+2)/3)^2-(10*n+21)*Floor((n+2)/3)+2*(n^2+5*n+7))-(1-(-1)^Floor((n+2)/3))*(-1)^(n+2-Floor((n+2)/3)))/16: n in [1..60]]; // Vincenzo Librandi, May 12 2015

g:=sum(x^k/(1-x^k)/(1-x^(k+1))/(1-x^(k+2))/(1-x^(k+3)),k=1..85): gser:=series(g,x=0,65): seq(coeff(gser,x^n),n=1..59); with(combinat): for n from 1 to 7 do P:=partition(n): A:={}: for j from 1 to nops(P) do if P[j][nops(P[j])]-P[j][1]<4 then A:=A union {P[j]} else A:=A fi od: print(A); od: # this program yields the partitions

Table[Count[IntegerPartitions[n], ?(Max[#] - Min[#] <= 3 &)], {n, 30}] (* _Birkas Gyorgy, Feb 20 2011 *)

Vec(x*(x^5-x^4-x^3+x+1)/((x-1)^4*(x+1)*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Mar 05 2015

G.f.: sum(x^k/[(1-x^k)(1-x^(k+1))(1-x^(k+2))(1-x^(k+3))], k=1..infinity). More generally, the g.f. of the number of partitions in which any two parts differ by at most b is sum(x^k/product(1-x^j, j=k..k+b), k=1..infinity).
G.f.: x*(x^5-x^4-x^3+x+1) / ((x-1)^4*(x+1)*(x^2+x+1)^2). - Colin Barker, Mar 05 2015
a(n)=(2*floor((n+2)/3)*(14*floor((n+2)/3)^2-(10*n+21)*floor((n+2)/3)+2*(n^2+5*n+7))-(1-(-1)^floor((n+2)/3))*(-1)^(n+2-floor((n+2)/3)))/16. - Luce ETIENNE, May 12 2015

A218506 Number of partitions of n in which any two parts differ by at most 4.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 20, 25, 34, 41, 54, 64, 81, 95, 118, 136, 165, 189, 226, 256, 301, 339, 395, 441, 507, 564, 644, 711, 804, 885, 995, 1089, 1215, 1326, 1473, 1600, 1766, 1914, 2105, 2272, 2486, 2678, 2921, 3136, 3406, 3650, 3954, 4225, 4560, 4865

Offset: 0

Alois P. Heinz, Oct 31 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,0,1,-3,-1,0,1,3,-1,0,-1,-1,1).

Column k=4 of A194621.

b:= proc(n, i, k) option remember; `if`(n<0 or k<0, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, k-1) +b(n-i, i, k))))
    end:
a:= n-> `if`(n=0, 1, 0) +add(b(n-i, i, 4), i=1..n):
seq(a(n), n=0..80);

b[n_, i_, k_] := b[n, i, k] = If[n < 0 || k < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k - 1] + b[n - i, i, k]]]];
a[n_] := If[n == 0, 1, 0] + Sum[b[n - i, i, 4], {i, 1, n}];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, May 20 2018, after Alois P. Heinz *)

Vec((x^14-x^13-x^12+2*x^9-x^6-x^5+x^4-1)/((x-1)^5*(x+1)^3*(x^2+1)^2*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Mar 05 2015

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

G.f.: (x^14-x^13-x^12+2*x^9-x^6-x^5+x^4-1) / ((x-1)^5*(x+1)^3*(x^2+1)^2*(x^2+x+1)). - Colin Barker, Mar 05 2015

A218507 Number of partitions of n in which any two parts differ by at most 5.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 37, 48, 62, 78, 98, 121, 149, 181, 219, 262, 313, 370, 436, 510, 595, 690, 797, 916, 1050, 1198, 1364, 1545, 1747, 1968, 2212, 2479, 2771, 3089, 3437, 3814, 4226, 4669, 5151, 5670, 6232, 6837, 7487, 8185, 8936, 9739, 10602

Offset: 0

Alois P. Heinz, Oct 31 2012

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,0,-2,-2,1,1,2,1,1,-2,-2,0,0,0,1,1,-1).

Column k=5 of A194621.

b:= proc(n, i, k) option remember; `if`(n<0 or k<0, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, k-1) +b(n-i, i, k))))
    end:
a:= n-> `if`(n=0, 1, 0) +add(b(n-i, i, 5), i=1..n):
seq(a(n), n=0..80);

LinearRecurrence[{1,1,0,0,0,-2,-2,1,1,2,1,1,-2,-2,0,0,0,1,1,-1},{1,1,2,3,5,7,11,15,21,28,37,48,62,78,98,121,149,181,219,262,313},60] (* Harvey P. Dale, Jan 18 2016 *)

Vec((x^20-x^19-x^18+x^15+x^14+x^13-x^12-x^11-x^10+x^7+x^6-x^5+1)/((x-1)^6*(x+1)^2*(x^2+1)*(x^2+x+1)*(x^4+x^3+x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Mar 05 2015

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

G.f.: (x^20 -x^19 -x^18 +x^15 +x^14 +x^13 -x^12 -x^11 -x^10 +x^7 +x^6 -x^5 +1) / ((x -1)^6*(x +1)^2*(x^2 +1)*(x^2 +x +1)*(x^4 +x^3 +x^2 +x +1)^2). - Colin Barker, Mar 05 2015

A218508 Number of partitions of n in which any two parts differ by at most 6.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 40, 51, 69, 86, 112, 139, 176, 214, 268, 321, 394, 470, 567, 668, 800, 933, 1103, 1281, 1498, 1725, 2006, 2293, 2643, 3010, 3443, 3897, 4438, 4995, 5652, 6341, 7135, 7967, 8932, 9930, 11079, 12283, 13645, 15071, 16692, 18372

Offset: 0

Alois P. Heinz, Oct 31 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=6 of A194621.

b:= proc(n, i, k) option remember; `if`(n<0 or k<0, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, k-1) +b(n-i, i, k))))
    end:
a:= n-> `if`(n=0, 1, 0) +add(b(n-i, i, 6), i=1..n):
seq(a(n), n=0..80);

b[n_, i_, k_] := b[n, i, k] = If[n < 0 || k < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k - 1] + b[n - i, i, k]]]];
a[n_] := If[n == 0, 1, 0] + Sum[b[n - i, i, 6], {i, 1, n}];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, May 20 2018, after Alois P. Heinz *)

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

A218509 Number of partitions of n in which any two parts differ by at most 7.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 54, 72, 93, 120, 153, 194, 242, 302, 372, 457, 556, 675, 812, 975, 1162, 1381, 1632, 1923, 2254, 2636, 3068, 3562, 4119, 4752, 5462, 6265, 7162, 8170, 9293, 10549, 11942, 13495, 15211, 17115, 19214, 21534, 24083, 26892

Offset: 0

Alois P. Heinz, Oct 31 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=7 of A194621.

b:= proc(n, i, k) option remember; `if`(n<0 or k<0, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, k-1) +b(n-i, i, k))))
    end:
a:= n-> `if`(n=0, 1, 0) +add(b(n-i, i, 7), i=1..n):
seq(a(n), n=0..80);

b[n_, i_, k_] := b[n, i, k] = If[n < 0 || k < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k - 1] + b[n - i, i, k]]]];
a[n_] := If[n == 0, 1, 0] + Sum[b[n - i, i, 7], {i, 1, n}];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, May 20 2018, after Alois P. Heinz *)

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

A218510 Number of partitions of n in which any two parts differ by at most 8.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 55, 75, 96, 127, 161, 208, 260, 330, 407, 509, 621, 765, 925, 1127, 1350, 1627, 1934, 2310, 2725, 3227, 3782, 4446, 5178, 6044, 7000, 8122, 9355, 10791, 12370, 14195, 16196, 18494, 21012, 23887, 27029, 30596, 34492, 38894

Offset: 0

Alois P. Heinz, Oct 31 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=8 of A194621.

b:= proc(n, i, k) option remember; `if`(n<0 or k<0, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, k-1) +b(n-i, i, k))))
    end:
a:= n-> `if`(n=0, 1, 0) +add(b(n-i, i, 8), i=1..n):
seq(a(n), n=0..80);

b[n_, i_, k_] := b[n, i, k] = If[n < 0 || k < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k - 1] + b[n - i, i, k]]]];
a[n_] := If[n == 0, 1, 0] + Sum[b[n - i, i, 8], {i, 1, n}];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, May 20 2018, after Alois P. Heinz *)

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

A218511 Number of partitions of n in which any two parts differ by at most 9.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 99, 130, 168, 216, 274, 348, 435, 544, 674, 831, 1017, 1244, 1507, 1823, 2193, 2629, 3136, 3734, 4420, 5223, 6148, 7215, 8438, 9850, 11453, 13292, 15382, 17758, 20447, 23502, 26935, 30818, 35181, 40082, 45570

Offset: 0

Alois P. Heinz, Oct 31 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=9 of A194621.

b:= proc(n, i, k) option remember; `if`(n<0 or k<0, 0,
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, k-1) +b(n-i, i, k))))
    end:
a:= n-> `if`(n=0, 1, 0) +add(b(n-i, i, 9), i=1..n):
seq(a(n), n=0..80);

b[n_, i_, k_] := b[n, i, k] = If[n < 0 || k < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k - 1] + b[n - i, i, k]]]];
a[n_] := If[n == 0, 1, 0] + Sum[b[n - i, i, 9], {i, 1, n}];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, May 20 2018, after Alois P. Heinz *)

G.f.: 1 + Sum_{j>0} x^j / Product_{i=0..9} (1-x^(i+j)).

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).

cp's OEIS Frontend

A117142 Number of partitions of n in which any two parts differ by at most 2.

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A117143 Number of partitions of n in which any two parts differ by at most 3.

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A218506 Number of partitions of n in which any two parts differ by at most 4.

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A218507 Number of partitions of n in which any two parts differ by at most 5.

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A218508 Number of partitions of n in which any two parts differ by at most 6.

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A218509 Number of partitions of n in which any two parts differ by at most 7.

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A218510 Number of partitions of n in which any two parts differ by at most 8.

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