A194621 -id:A194621 - cp's OEIS frontend

A218512 Number of partitions of n in which any two parts differ by at most 10.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 100, 133, 171, 223, 282, 362, 453, 572, 709, 884, 1084, 1337, 1626, 1984, 2394, 2896, 3468, 4162, 4951, 5897, 6972, 8249, 9696, 11402, 13330, 15586, 18131, 21090, 24417, 28264, 32580, 37541, 43097, 49449, 56544

Offset: 0

Alois P. Heinz, Oct 31 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Column k=10 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, 10), 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, 10], {i, 1, n}];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

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

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).

A218572 Number of partitions p of n such that max(p)-min(p) = 9.

Original entry on oeis.org

1, 1, 3, 3, 7, 8, 14, 18, 28, 35, 53, 66, 92, 117, 157, 196, 259, 319, 411, 507, 638, 777, 970, 1171, 1438, 1728, 2098, 2501, 3012, 3563, 4251, 5008, 5923, 6931, 8152, 9486, 11078, 12835, 14900, 17177, 19844, 22768, 26169, 29916, 34219, 38954, 44387, 50338

Offset: 11

Alois P. Heinz, Nov 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 11..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 = 11; 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[9], offset] (* Jean-François Alcover, Sep 11 2017, after Alois P. Heinz *)

G.f.: Sum_{k>0} x^(2*k+9)/Product_{j=0..9} (1-x^(k+j)).

a(n) = A097364(n,9) = A116685(n,9) = A194621(n,9) - A194621(n,8) = A218511(n) - A218510(n).

cp's OEIS Frontend

A218512 Number of partitions of n in which any two parts differ by at most 10.

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

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

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

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

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A218572 Number of partitions p of n such that max(p)-min(p) = 9.

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