A238450 -id:A238450 - 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

A015716 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, one of which is k (1<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 1, 1, 1, 3, 3, 3, 2, 2, 2, 1, 1, 1, 5, 4, 4, 3, 2, 2, 2, 1, 1, 1, 5, 5, 4, 3, 3, 3, 2, 2, 1, 1, 1, 7, 6, 5, 5, 4, 3, 3, 2, 2, 1, 1, 1, 8, 7, 6, 6, 4, 4, 4, 3, 2, 2, 1, 1

Offset: 1

Clark Kimberling

Row sums yield A015723. T(n,1)=A025147(n-1); T(n,2)=A015744(n-2); T(n,3)=A015745(n-3); T(n,4)=A015746(n-4); T(n,5)=A015750(n-5). - Emeric Deutsch, Mar 29 2006

Number of parts of size k in all partitions of n into distinct parts. Number of partitions of n-k into distinct parts not including a part of size k. - Franklin T. Adams-Watters, Jan 24 2012

			T(8,3)=2 because we have [5,3] and [4,3,1].
Triangle begins:
n/k 1 2 3 4 5 6 7 8 9 10
01: 1
02: 0 1
03: 1 1 1
04: 1 0 1 1
05: 1 1 1 1 1
06: 2 2 1 1 1 1
07: 2 2 1 2 1 1 1
08: 3 2 2 1 2 1 1 1
09: 3 3 3 2 2 2 1 1 1
10: 5 4 4 3 2 2 2 1 1 1
...
The strict integer partitions of 6 are {(6), (5,1), (4,2), (3,2,1)}, with multiset union {1,1,2,2,3,4,5,6}, with multiplicities (2,2,1,1,1,1), which is row n = 6. - _Gus Wiseman_, May 07 2019

Mircea Merca, Table of n, a(n) for n = 1..7260

Cf. A015723, A015744, A015745, A015746, A015750, A025147, A027293, A066633.

Cf. A006128, A022629, A066189, A246867, A325504, A325506, A325513.

g:=product(1+x^j,j=1..50)*sum(t^i*x^i/(1+x^i),i=1..50): gser:=simplify(series(g,x=0,18)): for n from 1 to 14 do P[n]:=sort(coeff(gser,x^n)) od: for n from 1 to 14 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form - Emeric Deutsch, Mar 29 2006
seq(seq(coeff(x^k*(product(1+x^j, j=1..n))/(1+x^k), x, n), k=1..n), n=1..13); # Mircea Merca, Feb 28 2014

z = 15; d[n_] := d[n] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &]; p[n_, k_] := p[n, k] = d[n][[k]]; s[n_] := s[n] = Flatten[Table[p[n, k], {k, 1, PartitionsQ[n]}]]; t[n_, k_] := Count[s[n], k]; u = Table[t[n, k], {n, 1, z}, {k, 1, n}]; TableForm[u] (* A015716 as a triangle *)
v = Flatten[u] (* A015716 as a sequence *)
(* Clark Kimberling, Mar 14 2014 *)

G.f.: G(t,x) = Product_{j>=1} (1+x^j) * Sum_{i>=1} t^i*x^i/(1+x^i). - Emeric Deutsch, Mar 29 2006
From Mircea Merca, Feb 28 2014: (Start)
a(n) = A238450(n) + A238451(n).
T(n,k) = Sum_{j=1..floor(n/k)} (-1)^(j-1)*A000009(n-j*k).
G.f.: for column k: q^k/(1+q^k)*(-q;q)_{inf}. (End)

A238208 The total number of 1's in all partitions of n into an odd number of distinct parts.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 12, 14, 17, 20, 24, 28, 33, 38, 45, 52, 60, 69, 80, 91, 105, 120, 137, 156, 178, 202, 230, 261, 295, 334, 378, 426, 481, 542, 609, 685, 769, 862, 966, 1082, 1209, 1351, 1508, 1681, 1873, 2086, 2319, 2578

Offset: 0

Mircea Merca, Feb 20 2014

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).

Or: the number of partitions of n-1 into an even number of distinct parts >=2. - R. J. Mathar, May 11 2016

			a(10) = 3 because the partitions in question are: 7+2+1, 6+3+1, 5+4+1.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Andrew Howroyd)

Column k=1 of A238450.

Cf. A067659, A067661, A133280.

A238208 := proc(n)
    local a,L,Lset;
    a := 0 ;
    L := combinat[firstpart](n) ;
    while true do
        # check that parts are distinct
        Lset := convert(L,set) ;
        if nops(L) = nops(Lset) then
            # check that number is odd
            if type(nops(L),'odd') then
                if 1 in Lset then
                    a := a+1 ;
                end if;
            end if;
        end if;
        L := combinat[nextpart](L) ;
        if L = FAIL then
            return a;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, May 11 2016
# second Maple program:
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, 1):
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, 1];
a /@ Range[0, 100] (* Jean-François Alcover, May 17 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)} A067661(n-(2*j-1)) - Sum_{j=1..floor(n/2)} A067659(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, May 17 2020
From Peter Bala, Feb 02 2021: (Start)
a(n+1) = d(n) - ( d(n-1) + d(n-3) ) + ( d(n-4) + d(n-6) + d(n-8) ) - ( d(n-9) + d(n-11) + d(n-13) + d(n-15) ) + ( d(n-16) + d(n-18) + d(n-20) + d(n-22) + d(n-24) ) - ( d(n-25) + d(n-27) + d(n-29) + d(n-31) + d(n-33) + d(n-35) ) + ..., where d(n) = A000009(n) is the number of partitions of n into distinct parts, with the convention that d(n) = 0 for n < 0.
G.f.: x/(1 - x^2)*Sum_{n >= 0} (-1)^n*x^((n^2+n+1-(-1)^n)/2)/Product_{k = 1..n} 1 - x^k.
Alternative g.f.: ( Product_{k >= 1} 1 + x^k ) * x*Sum_{n >= 0} (-1)^n*x^(n^2)*(1 - x^(2*n+2))/(1 - x^2).
Faster converging g.f. (conjecture): Sum_{n >= 0} x^((n+1)*(2*n+1))/ Product_{k = 1..2*n} 1 - x^k. (End)

a(51)-a(60) from R. J. Mathar, May 11 2016

A238209 The total number of 2's in all partitions of n into an odd number of distinct parts.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 13, 16, 18, 22, 26, 30, 35, 41, 48, 55, 64, 73, 85, 97, 111, 127, 146, 165, 189, 214, 244, 276, 313, 353, 400, 451, 508, 572, 644, 722, 811, 909, 1018, 1139, 1273, 1421, 1586, 1768, 1968, 2191, 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(11) = 3 because the partitions in question are: 8+2+1, 6+3+2, 5+4+2.

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

Column k=2 of A238450.

Cf. A067659, A067661.

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^2*(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)} A067661(n-(2*j-1)*2) - Sum_{j=1..floor(n/4)} A067659(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

A238210 The total number of 3's in all partitions of n into an odd number of distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 12, 14, 17, 20, 23, 28, 32, 37, 44, 51, 58, 68, 78, 89, 103, 118, 134, 154, 175, 199, 227, 257, 291, 330, 373, 421, 475, 535, 602, 677, 760, 852, 955, 1069, 1196, 1336, 1491, 1663, 1853, 2063, 2296

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: 8+3+1, 7+3+2, 5+4+3.

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

Column k=3 of A238450.

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^3*(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)} A067661(n-(2*j-1)*3) - Sum_{j=1..floor(n/6)} A067659(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

A238211 The total number of 4's in all partitions of n into an odd number of distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 5, 5, 6, 7, 9, 11, 13, 15, 18, 21, 25, 29, 34, 40, 46, 54, 62, 71, 82, 95, 108, 124, 142, 162, 184, 210, 238, 271, 306, 346, 392, 443, 498, 561, 632, 710, 796, 893, 1000, 1120, 1252, 1397, 1560, 1740, 1937, 2156

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: 7+4+1, 6+4+2, 5+4+3.

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

Column k=4 of A238450.

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)} A067661(n-(2*j-1)*4) - Sum_{j=1..floor(n/8)} A067659(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

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

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

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.

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A015716 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts, one of which is k (1<=k<=n).

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A238208 The total number of 1's in all partitions of n into an odd number of distinct parts.

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A238209 The total number of 2's in all partitions of n into an odd number of distinct parts.

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A238210 The total number of 3's in all partitions of n into an odd number of distinct parts.

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A238211 The total number of 4's in all partitions of n into an odd number of distinct parts.

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

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