Mircea Merca - cp's OEIS frontend

A238451 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an even number of distinct parts.

0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 2, 2, 2, 0, 1, 1, 1, 1, 0, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 0, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 0, 4, 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 0

Offset: 1

Mircea Merca, Feb 26 2014

			n/k | 1 2 3 4 5 6 7 8 9 10
   1: 0
   2: 0 0
   3: 1 1 0
   4: 1 0 1 0
   5: 1 1 1 1 0
   6: 1 1 0 1 1 0
   7: 1 1 1 1 1 1 0
   8: 1 1 1 0 1 1 1 0
   9: 1 1 1 1 1 1 1 1 0
  10: 2 2 2 2 0 1 1 1 1 0

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns k=1..6 are A238215, A238217, A238218, A238219, A238220, A238221.
Row sums are A238132.
Cf. A015716, A067659, A067661, A238450.

T(n,k) = {my(m=n-k); if(m>0, polcoef(prod(j=1, m, 1+x^j + O(x*x^m))/(1+x^k) - prod(j=1, m, 1-x^j + O(x*x^m))/(1-x^k), m)/2, 0)} \\ Andrew Howroyd, Apr 29 2020

T(n,k) = Sum_{j=1..round(n/(2*k))} A067659(n-(2*j-1)*k) - Sum_{j=1..floor(n/(2*k))} A067661(n-2*j*k).
G.f. of column k: (1/2)*(q^k/(1+q^k))*(-q;q){inf} - (1/2)*(q^k/(1-q^k))*(q;q){inf}.
T(n,k) = A015716(n,k) - A238450(n,k). - Andrew Howroyd, Apr 29 2020

A238450 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an odd number of distinct parts.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 0, 0, 1, 2, 2, 2, 1, 1, 1, 0, 0, 1, 3, 2, 2, 1, 2, 1, 1, 0, 0, 1, 3, 3, 2, 2, 1, 2, 1, 1, 0, 0, 1, 4, 3, 3, 3, 2, 2, 2, 1, 1, 0, 0, 1, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 0, 0, 1

Offset: 1

Mircea Merca, Feb 26 2014

			n\k | 1 2 3 4 5 6 7 8 9 10
   1: 1
   2: 0 1
   3: 0 0 1
   4: 0 0 0 1
   5: 0 0 0 0 1
   6: 1 1 1 0 0 1
   7: 1 1 0 1 0 0 1
   8: 2 1 1 1 1 0 0 1
   9: 2 2 2 1 1 1 0 0 1
  10: 3 2 2 1 2 1 1 0 0 1

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns k=1..6 are A238208, A238209, A238210, A238211, A238212, A238213.
Row sums are A238131.
Cf. A015716, A067659, A067661, A238451.

T(n,k) = {my(m=n-k); if(m>0, polcoef(prod(j=1, m, 1+x^j + O(x*x^m))/(1+x^k) + prod(j=1, m, 1-x^j + O(x*x^m))/(1-x^k), m)/2, m==0)} \\ Andrew Howroyd, Apr 29 2020

T(n,k) = Sum_{j=1..round(n/(2*k))} A067661(n-(2*j-1)*k) - Sum_{j=1..floor(n/(2*k))} A067659(n-2*j*k).
G.f. of column k: (1/2)*(q^k/(1+q^k))*(-q;q){inf} + (1/2)*(q^k/(1-q^k))*(q;q){inf}.
T(n,k) = A015716(n,k) - A238451(n,k). - Andrew Howroyd, Apr 29 2020

Terms a(79) and beyond from Andrew Howroyd, Apr 29 2020

A238221 The total number of 6's in all partitions of n into an even number of distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 3, 3, 4, 4, 6, 7, 9, 11, 12, 14, 17, 20, 24, 28, 32, 37, 44, 51, 59, 69, 78, 90, 104, 119, 136, 156, 177, 202, 230, 261, 296, 336, 379, 428, 483, 544, 612, 689, 773, 867, 972, 1088, 1217, 1360, 1518, 1693, 1887

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(13) = 2 because the partitions in question are: 7+6, 6+4+2+1.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=6 of A238451.

Cf. A067659, A067661.

endpQ[n_]:=Module[{len=Length[n]},EvenQ[len]&&len==Length[Union[n]]]; Table[ Count[Flatten[Select[IntegerPartitions[i],endpQ]],6],{i,0,50}] (* Harvey P. Dale, Mar 03 2014 *)
nmax = 100; With[{k=6}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] - x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

a(n) = Sum_{j=1..round(n/12)} A067659(n-(2*j-1)*6) - Sum_{j=1..floor(n/12)} A067661(n-12*j).
G.f.: (1/2)*(x^6/(1+x^6))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^6/(1-x^6))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Terms a(51) and beyond from Andrew Howroyd, Apr 29 2020

A238220 The total number of 5's in all partitions of n into an even number of distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 2, 2, 2, 3, 3, 5, 6, 7, 8, 9, 12, 14, 16, 19, 22, 27, 31, 36, 42, 48, 56, 65, 75, 86, 99, 114, 130, 149, 170, 193, 220, 250, 283, 321, 364, 410, 463, 522, 587, 661, 742, 832, 933, 1045, 1169, 1306, 1459, 1627, 1814, 2021

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(13) = 2 because the partitions in question are: 8+5, 5+4+3+1.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=5 of A238451.

Cf. A067659, A067661.

nmax = 100; With[{k=5}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] - x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

a(n) = Sum_{j=1..round(n/10)} A067659(n-(2*j-1)*5) - Sum_{j=1..floor(n/10)} A067661(n-10*j).
G.f.: (1/2)*(x^5/(1+x^5))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^5/(1-x^5))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Terms a(51) and beyond from Andrew Howroyd, Apr 29 2020

A238219 The total number of 4's in all partitions of n into an even number of distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 2, 3, 4, 4, 5, 6, 8, 9, 11, 13, 16, 18, 21, 25, 29, 34, 40, 46, 53, 62, 71, 82, 94, 108, 124, 142, 161, 185, 210, 238, 270, 307, 347, 392, 442, 499, 562, 632, 709, 797, 894, 1000, 1119, 1252, 1398, 1560, 1739, 1937, 2157

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(13) = 3 because the partitions in question are: 9+4, 6+4+2+1, 5+4+3+1.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=4 of A238451.

Cf. A067659, A067661.

nmax = 100; With[{k=4}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] - x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

a(n) = Sum_{j=1..round(n/8)} A067659(n-(2*j-1)*4) - Sum_{j=1..floor(n/8)} A067661(n-8*j).
G.f.: (1/2)*(x^4/(1+x^4))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^4/(1-x^4))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Terms a(51) and beyond from Andrew Howroyd, Apr 29 2020

A238218 The total number of 3's in all partitions of n into an even number of distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 2, 2, 2, 3, 4, 5, 6, 7, 9, 10, 12, 15, 17, 20, 24, 27, 32, 38, 43, 50, 59, 67, 77, 90, 102, 117, 135, 153, 175, 200, 226, 257, 292, 330, 373, 422, 475, 535, 603, 677, 760, 853, 955, 1069, 1196, 1336, 1491, 1663, 1853, 2063, 2295

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(13) = 3 because the partitions in question are: 10+3, 7+3+2+1, 5+4+3+1.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=3 of A238451.

Cf. A067659, A067661.

nmax = 100; With[{k=3}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] - x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

seq(n)={my(A=O(x^(n-2))); Vec(x*(eta(x^2 + A)/(eta(x + A)*(1+x^3)) - eta(x + A)/(1-x^3))/2, -(n+1))} \\ Andrew Howroyd, May 01 2020

a(n) = Sum_{j=1..round(n/6)} A067659(n-(2*j-1)*3) - Sum_{j=1..floor(n/6)} A067661(n-6*j).
G.f.: (1/2)*(x^3/(1+x^3))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^3/(1-x^3))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Terms a(51) and beyond from Andrew Howroyd, May 01 2020

A238217 The total number of 2's in all partitions of n into an even number of distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 10, 11, 14, 16, 19, 22, 26, 30, 35, 41, 47, 55, 63, 73, 84, 97, 110, 127, 145, 166, 188, 215, 243, 277, 313, 354, 400, 452, 508, 573, 644, 723, 811, 910, 1018, 1139, 1273, 1421, 1586, 1768, 1968, 2190, 2436

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(12) = 3 because the partitions in question are: 10+2, 6+3+2+1, 5+4+2+1.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=2 of A238451.

Cf. A067659, A067661.

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Length[#] == Length[ Union[#]]&&MemberQ[#,2]&]],{n,0,50}] (* Harvey P. Dale, Dec 09 2014 *)
nmax = 100; With[{k=2}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] - x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

seq(n)={my(A=O(x^(n-1))); Vec(x*(eta(x^2 + A)/(eta(x + A)*(1+x^2)) - eta(x + A)/(1-x^2))/2, -(n+1))} \\ Andrew Howroyd, May 01 2020

a(n) = Sum_{j=1..round(n/4)} A067659(n-(2*j-1)*2) - Sum_{j=1..floor(n/4)} A067661(n-4*j).
G.f.: (1/2)*(x^2/(1+x^2))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^2/(1-x^2))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Terms a(51) and beyond from Andrew Howroyd, May 01 2020

A238215 The total number of 1's in all partitions of n into an even number of distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 18, 21, 24, 28, 33, 38, 44, 51, 59, 68, 79, 90, 104, 119, 136, 156, 178, 202, 230, 261, 296, 335, 379, 427, 482, 543, 610, 686, 770, 863, 967, 1082, 1209, 1351, 1508, 1681, 1873, 2085, 2318, 2577

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(12) = 3 because the partitions in question are: 11+1, 6+3+2+1, 5+4+2+1.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=1 of A238451.

Cf. A067659, A067661.

b:= proc(n, i, t) option remember; `if`(n=0, t,
     `if`(i>n, 0, b(n, i+1, t)+b(n-i, i+1, 1-t)))
    end:
a:= n-> b(n-1, 2, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, May 01 2020

b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i > n, 0, b[n, i + 1, t] + b[n - i, i + 1, 1 - t]]];
a[n_] := b[n - 1, 2, 0];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

seq(n)={my(A=O(x^n)); Vec(x*(eta(x^2 + A)/(eta(x + A)*(1+x)) - eta(x + A)/(1-x))/2, -(n+1))} \\ Andrew Howroyd, May 01 2020

a(n) = Sum_{j=1..round(n/2)} A067659(n-(2*j-1)) - Sum_{j=1..floor(n/2)} A067661(n-2*j).
G.f.: (1/2)*(x/(1+x))*(Product_{n>=1} 1 + x^n) - (1/2)*(x/(1-x))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Nov 07 2024

Terms a(51) and beyond from Andrew Howroyd, May 01 2020

A238213 The total number of 6's in all partitions of n into an odd number of distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 2, 2, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 17, 20, 23, 27, 33, 38, 44, 51, 59, 68, 79, 91, 104, 119, 136, 155, 178, 202, 230, 261, 296, 335, 379, 428, 483, 544, 612, 688, 773, 867, 972, 1088, 1217, 1360, 1518, 1693, 1887

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(12) = 2 because the partitions in question are: 6+5+1, 6+4+2.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=6 of A238450.

Cf. A067659, A067661.

nmax = 100; With[{k=6}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] + x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

a(n) = Sum_{j=1..round(n/12)} A067661(n-(2*j-1)*6) - Sum_{j=1..floor(n/12)} A067659(n-12*j).
G.f.: (1/2)*(x^6/(1+x^6))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^6/(1-x^6))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Terms a(51) and beyond from Andrew Howroyd, Apr 29 2020

A238212 The total number of 5's in all partitions of n into an odd number of distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 1, 2, 2, 3, 5, 4, 5, 7, 8, 10, 11, 13, 16, 19, 23, 26, 31, 36, 42, 49, 56, 65, 75, 86, 100, 114, 130, 149, 170, 193, 220, 250, 283, 321, 363, 410, 463, 522, 587, 660, 742, 832, 933, 1045, 1168, 1307, 1459, 1627, 1814, 2020

Offset: 0

Mircea Merca, Feb 20 2014

nonn

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

			a(12) = 2 because the partitions in question are: 6+5+1, 5+4+3.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=5 of A238450.

Cf. A067659, A067661.

tn5[n_]:=Module[{op=IntegerPartitions[n],m},m=Flatten[Select[op,OddQ[ Length[#]] && Length[#]==Length[Union[#]]&]];Count[m,5]]; Array[tn5,60,0] (* Harvey P. Dale, Feb 06 2015 *)
nmax = 100; With[{k=5}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] + x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

a(n) = Sum_{j=1..round(n/10)} A067661(n-(2*j-1)*5) - Sum_{j=1..floor(n/10)} A067659(n-10*j).
G.f.: (1/2)*(x^5/(1+x^5))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^5/(1-x^5))*(Product_{n>=1} 1 - x^n).
a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Terms a(51) and beyond from Andrew Howroyd, Apr 29 2020

cp's OEIS Frontend

User: Mircea Merca

A238451 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an even number of distinct parts.

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A238450 Triangle read by rows: T(n,k) is the number of k’s in all partitions of n into an odd number of distinct parts.

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A238221 The total number of 6's in all partitions of n into an even number of distinct parts.

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A238220 The total number of 5's in all partitions of n into an even number of distinct parts.

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A238219 The total number of 4's in all partitions of n into an even number of distinct parts.

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A238218 The total number of 3's in all partitions of n into an even number of distinct parts.

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A238217 The total number of 2's in all partitions of n into an even number of distinct parts.

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