A066897 -id:A066897 - cp's OEIS frontend

A305105 G.f.: Sum_{k>=1} x^(2k-1)/(1-x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

0, 1, 3, 8, 17, 34, 64, 114, 195, 325, 526, 832, 1292, 1970, 2958, 4384, 6413, 9276, 13283, 18836, 26478, 36924, 51096, 70210, 95844, 130019, 175350, 235192, 313802, 416618, 550540, 724250, 948719, 1237732, 1608508, 2082600, 2686857, 3454590, 4427144, 5655652

Offset: 0

Vaclav Kotesovec, May 25 2018

Convolution of A066897 and A000009.
Convolution of A067588 and A000041.
Let A(x) = Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) * Product_{k >= 1} (1 + x^k)/(1 - x^k). Then A(x) = Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) * Product_{k >= 1} (1 + x^k)/(1 + x^k - 2*x^k) == Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) (mod 2). It follows from the comment in A001227 by Juri-Stepan Gerasimov, dated Jul 17 2016, that a(n) is odd iff n is a square or twice a square. - Peter Bala, Jan 10 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A001227, A015128, A066897, A067588, A305102, A305104.

nmax = 50; CoefficientList[Series[Sum[x^(2*k-1)/(1-x^(2*k-1)), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (2*gamma + log(16*n/Pi^2)) * exp(Pi*sqrt(n)) / (16*Pi*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620.

A325771 Rectangular array: row n shows the number of parts in all partitions of n that are == k (mod 2), for k = 0, 1.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 4, 8, 5, 15, 11, 24, 15, 39, 28, 58, 38, 90, 62, 130, 85, 190, 131, 268, 177, 379, 258, 522, 346, 722, 489, 974, 648, 1317, 890, 1754, 1168, 2330, 1572, 3058, 2042, 4010, 2699, 5200, 3475, 6731, 4532, 8642, 5783, 11068, 7446, 14076, 9430

Offset: 1

Clark Kimberling, Jun 05 2019

Row n partitions A006128 into 2 parts, r(n,0) + r(n,1) = p(n) = A006128(n). What is the limiting behavior of r(n,0)/p(n)?

			First 15 rows:
    0    1
    1    2
    1    5
    4    8
    5   15
   11   24
   15   39
   28   58
   38   90
   62  130
   85  190
  131  268
  177  379
  258  522
  346  722

Clark Kimberling, Table of n, a(n) for n = 1..100

Cf. A006128, A066898, A066897, A325772, A325773, A325774.

f[n_] := Mod[Flatten[IntegerPartitions[n]], 2];
Table[Count[f[n], k], {n, 1, 40}, {k, 0, 1}]  (* A325771 array *)
Flatten[%] (* A325771 sequence *)

(row n) = (A066898(n), A066897(n)).

A090868 Number of partitions of n such that the set of odd parts has only one element.

Original entry on oeis.org

1, 1, 3, 2, 6, 5, 11, 8, 20, 15, 32, 24, 51, 39, 80, 58, 119, 90, 175, 130, 255, 190, 361, 268, 508, 379, 706, 522, 967, 722, 1313, 974, 1771, 1317, 2363, 1754, 3131, 2330, 4123, 3058, 5388, 4010, 7001, 5200, 9053, 6731, 11631, 8642, 14878, 11068, 18944, 14076

Offset: 1

Vladeta Jovovic, Feb 12 2004

Vaclav Kotesovec, Table of n, a(n) for n = 1..14572 (terms 1..5000 from Alois P. Heinz)

Cf. A066897.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t, 1, 0),
      `if`(i<1, 0, add(b(n-i*j, i-1, t or j>0 and i::odd),
       j=0..`if`(t and i::odd, 0, n/i))))
    end:
a:= n-> b(n$2, false):
seq(a(n), n=1..60);  # Alois P. Heinz, Jun 30 2016

first Needs["DiscreteMath`Combinatorica`"], then f[n_] := Count[ Plus @@@ Mod[ Union /@ Partitions[n], 2], 1]; Table[ f[n], {n, 1, 51}] (* Robert G. Wilson v, Feb 16 2004 *)

G.f.: Sum_{m>0} x^(2*m-1)/(1-x^(2*m-1))/Product_{m>0} (1-x^(2*m)).

More terms from Robert G. Wilson v, Feb 16 2004

A102289 Total number of odd lists in all sets of lists, cf. A000262.

Original entry on oeis.org

0, 1, 2, 15, 76, 665, 5286, 56287, 597080, 7601841, 99702730, 1484554511, 23049638052, 393702612745, 7036703742446, 135702811542495, 2737989749177776, 58848546456947297, 1321063959370833810, 31310238786268648591, 773291778432688011260, 20031956775840631151481

Offset: 0

Vladeta Jovovic, Feb 19 2005

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A052852, A066897, A066898, A059570, A059570, A081358, A092691.

G:=(x/(1-x^2))*exp(x/(1-x)): Gser:=series(G,x=0,25): seq(n!*coeff(Gser,x^n),n=1..22); # Emeric Deutsch
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+`if`(j::odd, [0, p[1]], 0))(b(n-j)*
        binomial(n-1, j-1)*j!), j=1..n))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

Rest[CoefficientList[Series[x/(1-x^2)*E^(x/(1-x)), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 29 2013 *)
nxt[{n_,a_,b_,c_}]:={n+1,b,c,(n+1)*c+(n+1)^2*b-(n-1)^2 (n+1)*a}; NestList[ nxt,{2,0,1,2},30][[All,2]] (* Harvey P. Dale, Jan 13 2019 *)

E.g.f.: x/(1-x^2)*exp(x/(1-x)).
a(n) = n*a(n-1) + n^2*a(n-2) - (n-2)^2*n*a(n-3). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ sqrt(2)/4 * n^(n+1/4)*exp(2*sqrt(n)-n-1/2) * (1 + 7/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013

More terms from Emeric Deutsch, Jun 24 2005

a(0)=0 pepended by Alois P. Heinz, May 10 2016

A102290 Total number of even lists in all sets of lists, cf. A000262.

Original entry on oeis.org

0, 0, 2, 6, 60, 380, 3990, 37002, 450296, 5373720, 76018410, 1096730030, 17814654132, 299645294676, 5511836578430, 105550556136690, 2171244984679920, 46545825736022192, 1059273836225051346, 25100215228045842390, 626204775725372971820, 16239127347086448236460

Offset: 0

Vladeta Jovovic, Feb 19 2005

Alois P. Heinz, Table of n, a(n) for n = 0..444
Index entries for sequences related to Laguerre polynomials

Cf. A052852, A059570, A066897, A066898, A081358, A092691.

l:= func< n,b | Evaluate(LaguerrePolynomial(n), b) >;
[0,0]cat[Factorial(n)*(&+[(-1)^(n+j)*l(j,-1): j in [0..n-2]]): n in [2..30]]; // G. C. Greubel, Mar 09 2021

Gser:=series(x^2*exp(x/(1-x))/(1-x^2),x=0,22):seq(n!*coeff(Gser,x^n),n=1..21); # Emeric Deutsch
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+`if`(j::even, [0, p[1]], 0))(b(n-j)*
        binomial(n-1, j-1)*j!), j=1..n))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

Rest[CoefficientList[Series[x^2/(1-x^2)*E^(x/(1-x)), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 29 2013 *)
Table[If[n<2, 0, n!*Sum[(-1)^(n-j)*LaguerreL[j, -1], {j,0,n-2}]], {n,0,30}] (* G. C. Greubel, Mar 09 2021 *)

[0,0]+[factorial(n)*sum((-1)^(n+j)*gen_laguerre(j,0,-1) for j in (0..n-2)) for n in (2..30)] # G. C. Greubel, Mar 09 2021

E.g.f.: x^2/(1-x^2)*exp(x/(1-x)).
Recurrence: (n-2)*a(n) = (n-2)*n*a(n-1) + (n-1)^2*n*a(n-2) - (n-3)*(n-2)*(n-1)*n*a(n-3). - Vaclav Kotesovec, Sep 29 2013
a(n) ~ sqrt(2)/4 * n^(n+1/4)*exp(2*sqrt(n)-n-1/2) * (1 - 41/(48*sqrt(n))). - Vaclav Kotesovec, Sep 29 2013
a(n) = n! * Sum_{j=0..n-2} (-1)^(n+j)*LaguerreL(j, -1) for n>1 with a(0)=a(1)=0. - G. C. Greubel, Mar 09 2021

More terms from Emeric Deutsch, Mar 27 2005

a(0)=0 prepended by Alois P. Heinz, May 10 2016

A308230 Irregular triangle: row n shows the alternating sums of partitions of n when the parts are arranged in nonincreasing order and the partitions are arranged lexicographically from [n] to [1,1,1,...,1].

Original entry on oeis.org

1, 2, 0, 3, 1, 1, 4, 2, 0, 2, 0, 5, 3, 1, 3, 1, 1, 1, 6, 4, 2, 4, 0, 2, 2, 2, 0, 2, 0, 7, 5, 3, 5, 1, 3, 3, 1, 3, 1, 3, 1, 1, 1, 1, 8, 6, 4, 6, 2, 4, 4, 0, 2, 4, 2, 4, 2, 0, 2, 2, 2, 0, 2, 0, 2, 0, 9, 7, 5, 7, 3, 5, 5, 1, 3, 5, 3, 5, 1, 3, 1, 3, 3, 3, 3, 1

Offset: 1

Clark Kimberling, May 17 2019

Row n consists of A000041(n) numbers, for n >= 1. The numbers in row n have the parity of n. Regarding row sums, see Comments at A066897.

			First 8 rows:
  1
  2  0
  3  1  1
  4  2  0  2  0
  5  3  1  3  1  1  1
  6  4  2  4  0  2  2  2  0  2  0
  7  5  3  5  1  3  3  1  3  1  3  1  1  1  1
  8  6  4  6  2  4  4  0  2  4  2  4  2  0  2  2  2  0  2  0  2  0
Row 5 comes from arranging the 7 partitions of 5 is this order:
[5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1] and taking alternating sums: 5, 4-1, 3-1, 3-1+1, 2-2+1, 2-1+1-1, 1-1+1-1.

Cf. A000041, A066897.

r[n_] := Map[Total[Map[Total, {Take[##], Drop[##]} &[#, {1, -1, 2}] {1, -1}]] &, IntegerPartitions[n]]; Column[Table[r[n], {n, 10}]] (* Peter J. C. Moses, May 15 2019 *)

A188139 Triangle by rows, A027293 * A129372 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 4, 2, 1, 1, 8, 3, 2, 1, 1, 11, 6, 3, 2, 1, 1, 19, 8, 5, 3, 2, 1, 1, 26, 13, 7, 5, 3, 2, 1, 1, 41, 18, 12, 7, 5, 3, 2, 1, 1, 56, 28, 16, 11, 7, 5, 3, 2, 1, 1, 83, 38, 24, 15, 11, 7, 5, 3, 2, 1, 1, 112, 55, 33, 23, 15, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Gary W. Adamson, Mar 21 2011

Row sums = A066897: (1, 2, 5, 8, 15, 24, 39,...), total number of odd parts in all partitions of n.

Apparently T(n,k) is the number of (2*k)'s in all the partitions of (n+k), k>=1, e.g. T(7,3) = number of 6's in partitions of 10 = A024790(10). [David Scambler, May 24 2012]

			First few rows of the triangle =
.
1,
1, 1
3, 1, 1
4, 2, 1, 1
8, 3, 2, 1, 1
11, 6, 3, 2, 1, 1
19, 8, 5, 3, 2, 1, 1
26, 13, 7, 5, 3, 2, 1, 1
41, 18, 12, 7, 5, 3, 2, 1, 1
56, 28, 16, 11, 7, 5, 3, 2, 1, 1
83, 38, 24, 15, 11, 7, 5, 3, 2, 1, 1
112, 55, 33, 23, 15, 11, 7, 5, 3, 2, 1, 1
160, 74, 47, 31, 22, 15, 11, 7, 5, 3, 2, 1, 1,
...

Cf. A027293, A066897, A129372.

Table[Count[Flatten[IntegerPartitions[n+k]], 2*k], {n,1,15}, {k,1,n}] (* David Scambler, May 24 2012 *)

a(22) ff. corrected and more terms from Georg Fischer, Jun 10 2023

cp's OEIS Frontend

A305105 G.f.: Sum_{k>=1} x^(2*k-1)/(1-x^(2*k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

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A325771 Rectangular array: row n shows the number of parts in all partitions of n that are == k (mod 2), for k = 0, 1.

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A090868 Number of partitions of n such that the set of odd parts has only one element.

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Maple

Mathematica

Formula

Extensions

A102289 Total number of odd lists in all sets of lists, cf. A000262.

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Author

Keywords

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Programs

Maple

Mathematica

Formula

Extensions

A102290 Total number of even lists in all sets of lists, cf. A000262.

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Magma

Maple

Mathematica

Sage

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Extensions

A308230 Irregular triangle: row n shows the alternating sums of partitions of n when the parts are arranged in nonincreasing order and the partitions are arranged lexicographically from [n] to [1,1,1,...,1].

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Comments

Examples

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Mathematica

A188139 Triangle by rows, A027293 * A129372 as infinite lower triangular matrices.

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Mathematica

Extensions

A305105 G.f.: Sum_{k>=1} x^(2k-1)/(1-x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).