cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

Mircea Merca, Feb 26 2014

Keywords

Examples

			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

Links

Crossrefs

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

Programs

  • PARI
    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

Formula

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

Extensions

Terms a(79) and beyond from 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

Views

Author

Clark Kimberling

Keywords

Comments

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

Examples

			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

Links

Crossrefs

Cf. A015723, A015744, A015745, A015746, A015750, A025147, A027293, A066633.
Cf. A006128, A022629, A066189, A246867, A325504, A325506, A325513.

Programs

  • Maple
    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
  • Mathematica
    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 *)

Formula

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)

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

Views

Author

Mircea Merca, Feb 20 2014

Keywords

Comments

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).

Examples

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

Links

Crossrefs

Column k=1 of A238451.
Cf. A067659, A067661.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

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

Extensions

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

Views

Author

Mircea Merca, Feb 20 2014

Keywords

Comments

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).

Examples

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

Links

Crossrefs

Column k=2 of A238451.
Cf. A067659, A067661.

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

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

Extensions

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

Views

Author

Mircea Merca, Feb 20 2014

Keywords

Comments

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).

Examples

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

Links

Crossrefs

Column k=3 of A238451.
Cf. A067659, A067661.

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

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

Extensions

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

Views

Author

Mircea Merca, Feb 20 2014

Keywords

Comments

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).

Examples

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

Links

Crossrefs

Column k=4 of A238451.
Cf. A067659, A067661.

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

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

Views

Author

Mircea Merca, Feb 20 2014

Keywords

Comments

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).

Examples

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

Links

Crossrefs

Column k=5 of A238451.
Cf. A067659, A067661.

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

Terms a(51) 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

Views

Author

Mircea Merca, Feb 20 2014

Keywords

Comments

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).

Examples

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

Links

Crossrefs

Column k=6 of A238451.
Cf. A067659, A067661.

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

Terms a(51) and beyond from Andrew Howroyd, Apr 29 2020
Showing 1-8 of 8 results.

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