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.

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A351593 Number of odd-length integer partitions of n into parts that are alternately equal and strictly decreasing.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 3, 5, 4, 6, 4, 8, 6, 9, 6, 12, 7, 14, 10, 16, 11, 20, 13, 24, 16, 28, 18, 34, 21, 40, 26, 46, 30, 56, 34, 64, 41, 75, 48, 88, 54, 102, 64, 118, 73, 138, 84, 159, 98, 182, 112, 210, 128, 242, 148, 276, 168, 318
Offset: 0

Views

Author

Gus Wiseman, Feb 23 2022

Keywords

Comments

Also odd-length partitions whose run-lengths are all 2's, except for the last, which is 1.

Examples

			The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
  1  2  3  4  5    6  7    8    9    A    B      C    D      E    F
              221     331  332  441  442  443    552  553    554  663
                                          551         661    662  771
                                          33221       44221       44331
                                                                  55221

Crossrefs

The even-length ordered version is A003242, ranked by A351010.
The opposite version is A053251, even-length A351007, any length A351006.
This is the odd-length case of A351005, even-length A035457.
With only equalities we get:
- opposite any length: A351003
- opposite odd-length: A000009 (except at 0)
- opposite even-length: A351012
- any length: A351004
- odd-length: A351594
- even-length: A035363
Without equalities we get:
- opposite any length: A122129 (apparently)
- opposite odd-length: A122130 (apparently)
- opposite even-length: A351008
- any length: A122135 (apparently)
- odd-length: A351595
- even-length: A122134 (apparently)
Cf. A000070, A000984, A027383, A053738, A236559, A236914, A350842, A350844.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[If[EvenQ[i],#[[i]]!=#[[i+1]],#[[i]]==#[[i+1]]],{i,Length[#]-1}]&]],{n,0,30}]

A351594 Number of odd-length integer partitions y of n that are alternately constant, meaning y_i = y_{i+1} for all odd i.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 4, 2, 7, 3, 9, 4, 13, 6, 19, 6, 26, 10, 35, 12, 49, 16, 64, 20, 87, 27, 115, 32, 151, 44, 195, 53, 256, 69, 328, 84, 421, 108, 537, 130, 682, 167, 859, 202, 1085, 252, 1354, 305, 1694, 380, 2104, 456, 2609, 564, 3218, 676, 3968, 826, 4863
Offset: 0

Views

Author

Gus Wiseman, Feb 24 2022

Keywords

Comments

These are partitions with all even run-lengths except for the last, which is odd.

Examples

			The a(1) = 1 through a(9) = 7 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)    (7)        (8)    (9)
            (111)       (221)    (222)  (331)      (332)  (333)
                        (11111)         (22111)           (441)
                                        (1111111)         (22221)
                                                          (33111)
                                                          (2211111)
                                                          (111111111)

Crossrefs

The ordered version (compositions) is A016116 shifted right once.
All odd-length partitions are counted by A027193.
The opposite version is A117409, even-length A351012, any length A351003.
Replacing equal with unequal relations appears to give:
- any length: A122129
- odd length: A122130
- even length: A351008
- opposite any length: A122135
- opposite odd length: A351595
- opposite even length: A122134
This is the odd-length case of A351004, even-length A035363.
The case that is also strict at even indices is:
- any length: A351005
- odd length: A351593
- even length: A035457
- opposite any length: A351006
- opposite odd length: A053251
- opposite even length: A351007
A reverse version is A096441; see also A349060.
Cf. A000009, A000041, A000070, A000984, A003242, A027383, A053738, A236559, A236914, A350842.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]

A100823 G.f.: Product_{k>0} (1+x^k)/((1-x^k)*(1+x^(3k))*(1+x^(5k))).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 46, 69, 101, 146, 208, 293, 408, 563, 768, 1040, 1397, 1864, 2470, 3254, 4261, 5550, 7192, 9277, 11911, 15229, 19391, 24597, 31085, 39150, 49142, 61489, 76702, 95401, 118324, 146362, 180573, 222226, 272826, 334173, 408394, 498022
Offset: 0

Views

Author

Noureddine Chair, Jan 06 2005

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 30*x^6 + 46*x^7 + ...
G.f. = q^-1 + 2*q^2 + 4*q^5 + 7*q^8 + 12*q^11 + 19*q^14 + 30*q^17 + 46*q^20 + ...

Links

Crossrefs

Cf. A035939, A098151, A102346, A122129.

Programs

  • Maple
    series(product((1+x^k)/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))),k=1..100),x=0,100);
  • Mathematica
    CoefficientList[ Series[ Product[(1 + x^k)/((1 - x^k)*(1 + x^(3k))*(1 + x^(5k))), {k, 100}], {x, 0, 45}], x] (* Robert G. Wilson v, Jan 12 2005 *)
nmax = 50; CoefficientList[Series[Product[(1+x^(5*k-1))*(1+x^(5*k-2))*(1+x^(5*k-3))*(1+x^(5*k-4)) / ((1-x^(6*k))*(1-x^(3*k-1))*(1-x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^6] QPochhammer[ x^5, x^10] / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Mar 07 2016 *)
  • PARI
    q='q+O('q^33); E(k)=eta(q^k);
Vec( (E(2)*E(3)*E(5)) / (E(1)^2*E(6)*E(10)) ) \\ Joerg Arndt, Sep 01 2015

Formula

a(n) ~ exp(Pi*sqrt(37*n/5)/3) * sqrt(37) / (12*sqrt(5)*n). - Vaclav Kotesovec, Sep 01 2015
G.f.: (E(2)*E(3)*E(5)) / (E(1)^2*E(6)*E(10)) where E(k) = prod(n>=1, 1-q^k ). - Joerg Arndt, Sep 01 2015
Euler transform of period 30 sequence [ 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 0, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, ...]. - Michael Somos, Mar 07 2016
Expansion of chi(-x^3) * chi(-x^5) / phi(-x) in powers of x where phi(), chi() are Ramanujan theta functions. - Michael Somos, Mar 07 2016
a(n) - A035939(2*n + 1) = A122129(2*n + 1). - Michael Somos, Mar 07 2016

Extensions

More terms from Robert G. Wilson v, Jan 12 2005
Offset corrected by Vaclav Kotesovec, Sep 01 2015
a(14) = 563 <- 562 corrected by Vaclav Kotesovec, Sep 01 2015

A137676 Expansion of f(-x^2, -x^3) / f(-x, -x^3) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 3, 3, 0, 0, 4, 4, 0, 0, 5, 6, 0, 0, 7, 7, 0, 0, 9, 10, 0, 0, 12, 12, 0, 0, 15, 16, 0, 0, 19, 20, 0, 0, 24, 26, 0, 0, 30, 31, 0, 0, 37, 40, 0, 0, 46, 48, 0, 0, 57, 60, 0, 0, 69, 72, 0, 0, 84, 89, 0, 0, 102, 106, 0
Offset: 0

Views

Author

Michael Somos, Feb 04 2008

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x + x^4 + x^5 + x^8 + 2*x^9 + 2*x^12 + 2*x^13 + 3*x^16 + 3*x^17 + ...
G.f. = 1/q + q^9 + q^39 + q^49 + q^79 + 2*q^89 + 2*q^119 + 2*q^129 + 3*q^159 + ...

Links

Crossrefs

Cf. A122129, A122135.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] / (QPochhammer[ x^4] QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}]; (* Michael Somos, Oct 08 2015 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2) / QPochhammer[ x^4, x^4, k], {k, 0, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Oct 08 2015 *)
a[ n_] := SeriesCoefficient[ Sqrt[2] x^(1/8) QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5] QPochhammer[ x^5] / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}]; (* Michael Somos, Oct 08 2015 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n), x^k^2 / prod(i=1, k, 1 - x^(4*i), 1 + x * O(x^(n - k^2)))), n))};

Formula

Expansion of f(-x^2) * f(-x^5) / (f(-x^4) * f(-x, -x^4)) in powers of x where f(, ) is Ramanujan's general theta function.
Expansion of (f(-x^13, -x^17) + x * f(-x^7, -x^23)) / f(-x^4) in powers of x where f(, ) is Ramanujan's general theta function.
Euler transform of period 20 sequence [ 1, -1, 0, 1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 1, 0, -1, 1, 0, ...].
G.f.: Sum_{k>=0} x^k^2 / (Product_{j=1..k} 1 - x^(4*j)).
a(4*n) = A122129(n). a(4*n + 1) = A122135(n). a(4*n + 2) = a(4*n + 3) = 0.
G.f.: (Sum_{k in Z} (-1)^k^2 * x^(k * (5*k + 1) / 2)) / (Sum_{k in Z} (-1)^k^2 * x^(k * (2*k + 1))). - Michael Somos, Oct 08 2015

A239329 The number of NE partitions of n (see Comments).

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 14, 19, 27, 36, 51, 67, 90, 117, 157, 204, 266, 337, 436, 554, 708, 890, 1123, 1401, 1750, 2172, 2701, 3329, 4106, 5026, 6161, 7507, 9147, 11095, 13455, 16245, 19597, 23555, 28288, 33867, 40514, 48328, 57590, 68456, 81286, 96286, 113947
Offset: 1

Views

Author

Clark Kimberling, Mar 19 2014

Keywords

Comments

Directional partitions are defined at A237981, and NE partitions are shown at A237982. a(n) is also the number of SW partitions of n, as at A237982.

Examples

			See A237982.

Crossrefs

Cf. A003114, A122129, A237981, A237982.

Programs

  • Mathematica
    z = 9; ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; cornerPart[list_] := Module[{f = ferrersMatrix[list], u, l, ur, lr, nw, ne, se, sw}, {u, l} = {UpperTriangularize[#, 1], LowerTriangularize[#]} &[f]; {ur, lr} = {UpperTriangularize[#, 1], LowerTriangularize[#]} &[Reverse[f]]; {nw, ne, se, sw} = {Total[Transpose[u]] + Total[l], Total[ur] + Total[Transpose[lr]], Total[u] + Total[Transpose[l]], Total[Transpose[ur]] + Total[lr]}; Map[DeleteCases[Reverse[Sort[#]], 0] &, {nw, ne, se, sw}]]; cornerParts[n_] :=  Map[#[[Reverse[Ordering[PadRight[#]]]]] &, Map[DeleteDuplicates[#] &, Transpose[Map[cornerPart, IntegerPartitions[n]]]]]; cP = Map[cornerParts, Range[z]];
Flatten[Map[cP[[#, 1]] &, Range[Length[cP]]]];(*NW A237981*)
Flatten[Map[cP[[#, 2]] &, Range[Length[cP]]]];(*NE A237982*)
Flatten[Map[cP[[#, 3]] &, Range[Length[cP]]]];(*SE A237983*)
Flatten[Map[cP[[#, 4]] &, Range[Length[cP]]]];(*SW A237982*)
m1 = Map[cP[[#, 1]] &, Range[Length[cP]]];
Table[Length[m1[[k]]], {k, 1, z}] (* A003114, NW *)
m2 = Map[cP[[#, 2]] &, Range[Length[cP]]];
Table[Length[m2[[k]]], {k, 1, z}] (* A239329, NE *)
m3 = Map[cP[[#, 3]] &, Range[Length[cP]]];
Table[Length[m3[[k]]], {k, 1, z}] (* A122129, SE *)
m4 = Map[cP[[#, 4]] &, Range[Length[cP]]];
Table[Length[m4[[k]]], {k, 1, z}] (* A239329, SW *)
Previous Showing 21-25 of 25 results.

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