A242498 -id:A242498

A098123

Number of compositions of n with equal number of even and odd parts.

+10
16

1, 0, 0, 2, 0, 4, 6, 6, 24, 28, 60, 130, 190, 432, 770, 1386, 2856, 5056, 9828, 18918, 34908, 68132, 128502, 244090, 470646, 890628, 1709136, 3271866, 6238986, 11986288, 22925630, 43932906, 84349336, 161625288, 310404768, 596009494

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = Sum_{k=floor(n/3)..floor(n/2)} C(2*n-4*k,n-2*k)*C(n-1-k,2*n-4*k-1).

Recurrence: n*(2*n-7)*a(n) = 2*(n-2)*(2*n-5)*a(n-2) + 2*(2*n-7)*(2*n-3)*a(n-3) - (n-4)*(2*n-3)*a(n-4). - Vaclav Kotesovec, May 01 2014

a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 1.94696532812840456026081823863... is the root of the equation 1-4*d-2*d^2+d^4 = 0, c = 0.225563290820392765554898545739... is the root of the equation 43*c^4-18*c^2+8*c-1=0. - Vaclav Kotesovec, May 01 2014

EXAMPLE

From Gus Wiseman, Jun 26 2022: (Start)

The a(0) = 1 through a(7) = 6 compositions (empty columns indicated by dots):

() . . (12) . (14) (1122) (16)

(21) (23) (1212) (25)

(32) (1221) (34)

(41) (2112) (43)

(2121) (52)

(2211) (61)

(End)

MATHEMATICA

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Count[#, _?EvenQ]==Count[#, _?OddQ]&]], {n, 0, 15}] (* Gus Wiseman, Jun 26 2022 *)

CROSSREFS

For partitions: A045931, ranked by A325698, strict A239241 (conj A352129).

Column k=0 of A242498.

Without multiplicity: A242821, for partitions A241638 (ranked by A325700).

These compositions are ranked by A355321.

A047993 counts balanced partitions, ranked by A106529.

A108950/A108949 count partitions with more odd/even parts.

A130780/A171966 count partitions with more or as many odd/even parts.

Cf. A000700, A000712, A001405, A026424, A027193, A028260, A097613, A240009, A277579 (ranked by A349157).

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Sep 24 2004

STATUS

approved

A242626

Number T(n,k) of compositions of n, where k is the difference between the number of odd parts and the number of even parts, both counted without multiplicity; triangle T(n,k), n>=0, read by rows.

+10
13

1, 1, 1, 0, 1, 2, 2, 2, 3, 1, 2, 11, 2, 3, 2, 2, 14, 8, 6, 6, 33, 14, 11, 5, 15, 43, 45, 20, 44, 82, 99, 25, 6, 14, 74, 141, 230, 41, 12, 202, 260, 451, 85, 26, 6, 22, 351, 514, 953, 148, 54, 24, 766, 1049, 1798, 355, 104, 18, 104, 1301, 2321, 3503, 751, 194

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

T(n^2,n) = T(n^2+n,-n) = n! = A000142(n) for n>=0.

LINKS

Alois P. Heinz, Rows n = 0..500, flattened

EXAMPLE

T(8,-1) = 15: [2,2,2,2], [1,1,2,4], [1,1,4,2], [1,2,1,4], [1,2,4,1], [1,4,1,2], [1,4,2,1], [2,1,1,4], [2,1,4,1], [2,4,1,1], [4,1,1,2], [4,1,2,1], [4,2,1,1], [4,4], [8].

Triangle T(n,k) begins:

: n\k : -3 -2 -1 0 1 2 3 ...

+-----+------------------------------------

: 0 : 1;

: 1 : 1;

: 2 : 1, 0, 1;

: 3 : 2, 2;

: 4 : 2, 3, 1, 2;

: 5 : 11, 2, 3;

: 6 : 2, 2, 14, 8, 6;

: 7 : 6, 33, 14, 11;

: 8 : 5, 15, 43, 45, 20;

: 9 : 44, 82, 99, 25, 6;

: 10 : 14, 74, 141, 230, 41, 12;

: 11 : 202, 260, 451, 85, 26;

: 12 : 6, 22, 351, 514, 953, 148, 54;

: 13 : 24, 766, 1049, 1798, 355, 104;

: 14 : 18, 104, 1301, 2321, 3503, 751, 194;

MAPLE

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,

expand(add(`if`(j=0, 1, x^(2*irem(i, 2)-1))*

b(n-i*j, i-1, p+j)/j!, j=0..n/i))))

end:

T:= n->(p->seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n$2, 0)):

seq(T(n), n=0..20);

MATHEMATICA

b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, Expand[Sum[If[j==0, 1, x^(2*Mod[i, 2]-1)]*b[n-i*j, i-1, p+j]/j!, {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, Exponent[p, x, Min], Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 17 2017, translated from Maple *)

CROSSREFS

Columns k=(-5)-5 give: A242836, A242837, A242838, A242839, A242840, A242821, A242841, A242842, A242843, A242844, A242845.

Row sums give A011782.

Cf. A242498 (compositions with multiplicity), A242618 (partitions without multiplicity).

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, May 19 2014

STATUS

approved

A242499

Number of compositions of n, where the difference between the number of odd parts and the number of even parts is 1.

+10
2

1, 0, 1, 3, 1, 9, 11, 18, 51, 65, 151, 290, 477, 1043, 1835, 3486, 6931, 12540, 24607, 46797, 87979, 171072, 323269, 619245, 1190619, 2264925, 4357211, 8343322, 15973309, 30711853, 58846191, 113027716, 217192103, 416964202, 801880039, 1541412015, 2963997227

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

With offset 2 number of compositions of n, where the difference between the number of odd parts and the number of even parts is -1.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

FORMULA

Recurrence (for n>=5): (n+2)*(16*n^4 - 128*n^3 + 344*n^2 - 352*n + 89)*a(n) = -32*(n+1)*(2*n-5)*a(n-1) + 2*(16*n^5 - 112*n^4 + 264*n^3 - 320*n^2 + 301*n - 89)*a(n-2) + 2*(2*n-5)*(16*n^4 - 80*n^3 + 80*n^2 + 36*n - 53)*a(n-3) - (n-4)*(16*n^4 - 64*n^3 + 56*n^2 + 16*n - 31)*a(n-4). - Vaclav Kotesovec, May 20 2014

MAPLE

a:= proc(n) option remember;

`if`(n<6, [0, 1, 0, 1, 3, 1][n+1],

((3*n-2)*a(n-2) +(4*n+2)*a(n-3) -(3*n-10)*a(n-4)

-(4*n-22)*a(n-5) +(n-6)*a(n-6))/(n+2))

end:

seq(a(n), n=1..50);

MATHEMATICA

a[n_] := a[n] = If[n<6, {0, 1, 0, 1, 3, 1}[[n+1]], ((3n-2)a[n-2] + (4n+2)a[n-3] - (3n-10)a[n-4] - (4n-22)a[n-5] + (n-6)a[n-6])/(n+2)];

Array[a, 50] (* Jean-François Alcover, Dec 28 2020, after Alois P. Heinz *)

CROSSREFS

Column k=1 of A242498.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 16 2014

STATUS

approved

A242500

Number of compositions of n, where the difference between the number of odd parts and the number of even parts is 2.

+10
2

1, 0, 2, 4, 3, 16, 19, 40, 95, 136, 321, 588, 1057, 2240, 3998, 7848, 15339, 28464, 56143, 106788, 204083, 396704, 755052, 1457456, 2806531, 5377112, 10382243, 19947252, 38382957, 73996576, 142311198, 274283168, 528438319, 1017784016, 1962451118, 3781912684

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,3

COMMENTS

With offset 4 number of compositions of n, where the difference between the number of odd parts and the number of even parts is -2.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 2..1000

FORMULA

Recurrence (for n>=6): (n+4)*(2*n-5)*(2*n-3)*(n^4 - 6*n^3 + 11*n^2 - 6*n - 16)*a(n) = -16*(n-3)*(n+3)*(2*n-5)*(2*n-1)*a(n-1) + 2*(n-2)*(2*n-3)*(2*n^5 - 7*n^4 + 8*n^3 - 51*n^2 + 28*n + 32)*a(n-2) + 2*(n-3)*(2*n-5)*(2*n-1)*(2*n^4 - 3*n^3 - 2*n^2 + 11*n - 24)*a(n-3) - (n-4)*(2*n-3)*(2*n-1)*(n^4 - 2*n^3 - n^2 + 2*n - 16)*a(n-4). - Vaclav Kotesovec, May 20 2014

CROSSREFS

Column k=2 of A242498.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 16 2014

STATUS

approved

A242501

Number of compositions of n, where the difference between the number of odd parts and the number of even parts is 3.

+10
2

1, 0, 3, 5, 6, 25, 31, 75, 162, 259, 609, 1106, 2122, 4410, 8076, 16197, 31527, 59961, 118844, 227700, 441507, 860860, 1654731, 3218501, 6226818, 12027405, 23337471, 45082050, 87258876, 168935018, 326536646, 632132760, 1222716653, 2364969824, 4576680195

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,3

COMMENTS

With offset 6 number of compositions of n, where the difference between the number of odd parts and the number of even parts is -3.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 3..1000

FORMULA

Recurrence (for n>=7): (n-3)*(n-2)*(n-1)*(n+6)*(16*n^4 - 64*n^3 + 56*n^2 + 16*n - 1311)*a(n) = -288*(n-4)*(n-2)*n*(n+5)*(2*n-3)*a(n-1) + 2*(n-1)*(16*n^7 - 64*n^6 + 136*n^5 - 1048*n^4 + 1621*n^3 + 1202*n^2 - 9162*n + 7866)*a(n-2) + 2*(n-2)*n*(2*n-3)*(16*n^5 - 32*n^4 - 48*n^3 + 212*n^2 - 1429*n + 2145)*a(n-3) - (n-4)*(n-1)^2*n*(16*n^4 - 40*n^2 - 1287)*a(n-4). - Vaclav Kotesovec, May 20 2014

CROSSREFS

Column k=3 of A242498.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 16 2014

STATUS

approved

A242502

Number of compositions of n, where the difference between the number of odd parts and the number of even parts is 4.

+10
2

1, 0, 4, 6, 10, 36, 48, 126, 259, 456, 1064, 1956, 3939, 8112, 15300, 31174, 60951, 118580, 236456, 458172, 900185, 1765556, 3431792, 6728410, 13107393, 25538448, 49856392, 96966572, 188914574, 367741688, 715053048, 1391512424, 2705016795, 5258241032

(list; graph; refs; listen; history; text; internal format)

OFFSET

4,3

COMMENTS

With offset 8 number of compositions of n, where the difference between the number of odd parts and the number of even parts is -4.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 4..1000

FORMULA

Recurrence (for n>=8): (n-4)*(n+8)*(2*n-3)*(2*n-1)*(n^4 - 2*n^3 - n^2 + 2*n - 256)*a(n) = -64*(n-5)*(n-1)*(n+7)*(2*n-3)*(2*n+1)*a(n-1) + 2*(2*n-1)*(2*n^7 - n^6 + 14*n^5 - 199*n^4 - 288*n^3 + 600*n^2 - 5360*n + 2928)*a(n-2) + 2*(n-1)*(2*n-3)*(2*n+1)*(2*n^5 + n^4 - 9*n^3 + 28*n^2 - 508*n + 608)*a(n-3) - (n-4)*n*(2*n-1)*(2*n+1)*(n^4 + 2*n^3 - n^2 - 2*n - 256)*a(n-4). - Vaclav Kotesovec, May 20 2014

CROSSREFS

Column k=4 of A242498.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 16 2014

STATUS

approved

A242503

Number of compositions of n, where the difference between the number of odd parts and the number of even parts is 5.

+10
2

1, 0, 5, 7, 15, 49, 71, 196, 394, 753, 1746, 3285, 6865, 14124, 27445, 56661, 111892, 222222, 446524, 876876, 1744353, 3448783, 6782633, 13411528, 26346074, 51799306, 101840098, 199601828, 391637976, 767247094, 1501758784, 2939789022, 5747749147, 11235696151

(list; graph; refs; listen; history; text; internal format)

OFFSET

5,3

COMMENTS

With offset 10 number of compositions of n, where the difference between the number of odd parts and the number of even parts is -5.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 5..1000

FORMULA

Recurrence (for n>=9): (n-5)*(n-1)*n*(n+10)*(16*n^4 - 40*n^2 - 9991)*a(n) = -800*(n-6)*(n-1)*(n+1)*(n+9)*(2*n-1)*a(n-1) + 2*n*(16*n^7 + 48*n^6 + 280*n^5 - 1920*n^4 - 11691*n^3 - 5023*n^2 - 167795*n + 7975)*a(n-2) + 2*(n-1)*(n+1)*(2*n-1)*(16*n^5 + 48*n^4 - 16*n^3 + 292*n^2 - 9645*n + 7200)*a(n-3) - (n-4)*n*(n+1)^2*(16*n^4 + 64*n^3 + 56*n^2 - 16*n - 10015)*a(n-4). - Vaclav Kotesovec, May 20 2014

CROSSREFS

Column k=5 of A242498.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 16 2014

STATUS

approved

A242504

Number of compositions of n, where the difference between the number of odd parts and the number of even parts is 6.

+10
2

1, 0, 6, 8, 21, 64, 101, 288, 576, 1180, 2727, 5280, 11363, 23496, 46981, 98176, 196482, 397644, 806351, 1606488, 3234335, 6456048, 12849330, 25637632, 50835950, 100883304, 199903578, 395067760, 781029504, 1540973568, 3037666097, 5984978112, 11775884581

(list; graph; refs; listen; history; text; internal format)

OFFSET

6,3

COMMENTS

With offset 12 number of compositions of n, where the difference between the number of odd parts and the number of even parts is -6.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 6..1000

FORMULA

Recurrence (for n>=10): (n-6)*(n+12)*(2*n-1)*(2*n+1)*(n^4 + 2*n^3 - n^2 - 2*n - 1296)*a(n) = -144*(n-7)*n*(n+11)*(2*n-1)*(2*n+3)*a(n-1) + 2*(2*n+1)*(2*n^7 + 13*n^6 + 80*n^5 - 179*n^4 - 3424*n^3 - 6476*n^2 - 69072*n - 31104)*a(n-2) + 2*n*(2*n-1)*(2*n+3)*(2*n^5 + 11*n^4 + 15*n^3 + 67*n^2 - 2465*n + 642)*a(n-3) - (n-4)*(n+2)*(2*n+1)*(2*n+3)*(n^4 + 6*n^3 + 11*n^2 + 6*n - 1296)*a(n-4). - Vaclav Kotesovec, May 20 2014

CROSSREFS

Column k=6 of A242498.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 16 2014

STATUS

approved

A242505

Number of compositions of n, where the difference between the number of odd parts and the number of even parts is 7.

+10
2

1, 0, 7, 9, 28, 81, 139, 405, 815, 1771, 4092, 8173, 18019, 37609, 77246, 163345, 331968, 683631, 1400777, 2832362, 5770056, 11640546, 23446366, 47227530, 94582628, 189487950, 378658714, 754877809, 1504215522, 2990469337, 5939101301, 11782590061, 23340439078

(list; graph; refs; listen; history; text; internal format)

OFFSET

7,3

COMMENTS

With offset 14 number of compositions of n, where the difference between the number of odd parts and the number of even parts is -7.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 7..1000

FORMULA

Recurrence (for n>=11): (n-7)*n*(n+1)*(n+14)*(16*n^4 + 64*n^3 + 56*n^2 - 16*n - 38431)*a(n) = -1568*(n-8)*n*(n+2)*(n+13)*(2*n+1)*a(n-1) + 2*(n+1)*(16*n^7 + 160*n^6 + 1192*n^5 + 472*n^4 - 49083*n^3 - 168912*n^2 - 1534048*n - 1379196)*a(n-2) + 2*n*(n+2)*(2*n+1)*(16*n^5 + 128*n^4 + 336*n^3 + 1076*n^2 - 36101*n - 8729)*a(n-3) - (n-4)*(n+1)*(n+2)*(n+3)*(16*n^4 + 128*n^3 + 344*n^2 + 352*n - 38311)*a(n-4). - Vaclav Kotesovec, May 20 2014

CROSSREFS

Column k=7 of A242498.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 16 2014

STATUS

approved

A242506

Number of compositions of n, where the difference between the number of odd parts and the number of even parts is 8.

+10
2

1, 0, 8, 10, 36, 100, 186, 550, 1122, 2564, 5940, 12246, 27560, 58240, 122642, 262458, 542243, 1134944, 2352136, 4826980, 9949352, 20300312, 41377116, 84172508, 170322099, 344527304, 694617960, 1397219682, 2807142612, 5625453196, 11258808682, 22498804286

(list; graph; refs; listen; history; text; internal format)

OFFSET

8,3

COMMENTS

With offset 16 number of compositions of n, where the difference between the number of odd parts and the number of even parts is -8.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 8..1000

FORMULA

Recurrence (for n>=12): (n-8)*(n+16)*(2*n+1)*(2*n+3)*(n^4 + 6*n^3 + 11*n^2 + 6*n - 4096)*a(n) = -256*(n-9)*(n+1)*(n+15)*(2*n+1)*(2*n+5)*a(n-1) + 2*(2*n+3)*(2*n^7 + 27*n^6 + 242*n^5 + 549*n^4 - 9408*n^3 - 49916*n^2 - 462064*n - 606208)*a(n-2) + 2*(n+1)*(2*n+1)*(2*n+5)*(2*n^5 + 21*n^4 + 79*n^3 + 254*n^2 - 7608*n - 5760)*a(n-3) - (n-4)*(n+4)*(2*n+3)*(2*n+5)*(n^4 + 10*n^3 + 35*n^2 + 50*n - 4072)*a(n-4). - Vaclav Kotesovec, May 20 2014

CROSSREFS

Column k=8 of A242498.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, May 16 2014

STATUS

approved