A006720 -id:A006720 - cp's OEIS frontend

A161307 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 2 times.

0, 1, 1, 2, 1, 4, 2, 5, 5, 7, 6, 11, 9, 14, 14, 18, 18, 25, 23, 31, 32, 38, 40, 49, 49, 59, 63, 71, 76, 88, 91, 104, 111, 123, 131, 147, 154, 171, 182, 198, 211, 231, 243, 265, 281, 303, 320, 347, 364, 393, 414, 444, 466, 501, 525, 561, 589, 628, 657, 701, 732, 778, 814, 863

Offset: 1

R. H. Hardin Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

A161319 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 14 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 2, 4, 4, 5, 4, 7, 5, 7, 7, 8, 7, 10, 8, 11, 11, 12, 12, 15, 13, 16, 16, 17, 17, 20, 18, 21, 21, 23, 22, 26, 24, 27, 27, 29, 28, 32, 30, 33, 33, 35, 34, 39, 37, 41, 42, 45, 45, 51, 50, 55, 57, 61, 62, 69, 69, 76

Offset: 1

R. H. Hardin Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

A254314 Hankel transform of a(n) is A006720(n). Hankel transform of a(n+1) is A006720(n+2).

Original entry on oeis.org

1, 1, 2, 5, 14, 43, 143, 507, 1887, 7279, 28828, 116455, 477709, 1983779, 8321474, 35203777, 150014157, 643302743, 2773997104, 12020733635, 52319374842, 228616865437, 1002544803949, 4410700121313, 19462407890220, 86111960348939, 381956399941011

Offset: 0

Michael Somos, Jan 28 2015

nonn

a(n+1) is the number of rooted plane trees with integer compositions labeling the leaves (empty labels are allowed), with total size n. The total size is the number of edges in the tree plus the sum of the sizes of the integer compositions labeling the leaves.

Example: a(3)=5 because there are 5 elements of size 2: two trees that consist of the root and no descendants, hence the root is itself a leaf and it can be labeled by either 2=2 or by 1=1+1, then a tree with the root and one descendant that is a leaf labeled with 1=1, then a tree with the root and two descendants with no labels on the leaves, and finally a tree with the root with one descendant with a descendant that is a leaf with no label. - Ricardo Gómez Aíza, Feb 29 2024

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 43*x^5 + 143*x^6 + 507*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 17-18.

Cf. A006720.

m:=60; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((3-6*x+x^2 - Sqrt((1-4*x+x^2)^2 -4*x^3))/(2*(1 - 2*x)))); // G. C. Greubel, Aug 10 2018

CoefficientList[Series[(3-6*x+x^2 - Sqrt[(1-4*x+x^2)^2 -4*x^3])/(2*(1 - 2*x)), {x, 0, 60}], x] (* G. C. Greubel, Aug 10 2018 *)

{a(n) = if( n<0, 0, polcoeff( (3 - 6*x + x^2 - sqrt( (1-4*x+x^2)^2 - 4*x^3 + x^2 * O(x^n))) / (2*(1 - 2*x)), n))};

G.f. A(x) satisfies 0 = (2*x-1)*A(x)^2 + (x^2-6*x+3)*A(x) + (3*x-2).
G.f.: (3 - 6*x + x^2 - sqrt( (1-4*x+x^2)^2 - 4*x^3 )) / (2*(1 - 2*x)).
Conjecture: n*a(n) +2*(-5*n+6)*a(n-1) +2*(17*n-39)*a(n-2) +6*(-8*n+27)*a(n-3) +(25*n-114)*a(n-4) +2*(-n+6)*a(n-5)=0. - R. J. Mathar, Jun 07 2016
a(n) ~ sqrt(b*(5-32*b+46*b^2))/(2*sqrt((1-2*b)^3*Pi*n^3))*(1/b)^n where b = (11-c-100/c)/3 and c = (-998+6*sqrt(111)*i)^(1/3). - Ricardo Gómez Aíza, Feb 29 2024

A161306 Number of partitions of n into Somos-4 sequence numbers A006720.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 14, 17, 20, 24, 28, 33, 38, 44, 50, 57, 64, 72, 81, 90, 101, 112, 124, 137, 151, 166, 182, 200, 218, 238, 259, 281, 305, 330, 357, 385, 415, 446, 479, 514, 550, 589, 629, 672, 716, 763, 812, 863, 917, 973, 1032, 1093, 1157, 1224, 1293, 1366, 1442

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

			a(3)=3: [3], [2,1], [1,1,1].

R. H. Hardin, Table of n, a(n) for n=1..1000

Cf. A006720, A161307-A161319.

Definition shortened by Georg Fischer, Jul 06 2022

A161308 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 3 times.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 2, 3, 3, 3, 6, 5, 6, 9, 8, 9, 13, 12, 14, 19, 18, 20, 26, 26, 29, 35, 36, 39, 46, 48, 52, 61, 63, 68, 78, 80, 87, 98, 103, 110, 123, 129, 137, 152, 160, 171, 187, 197, 210, 227, 240, 255, 276, 290, 308, 331, 347, 368, 394, 414, 437, 466, 489, 514, 547, 573, 603

Offset: 1

R. H. Hardin Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

A161309 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 4 times.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 4, 2, 4, 4, 6, 5, 8, 7, 10, 9, 12, 11, 15, 14, 18, 18, 23, 22, 27, 27, 33, 34, 39, 41, 47, 47, 54, 56, 65, 65, 75, 77, 85, 88, 98, 103, 112, 117, 128, 132, 144, 150, 165, 170, 185, 193, 207, 216, 233, 245, 261, 274, 293, 305, 326, 341, 365, 380, 405, 423

Offset: 1

R. H. Hardin Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

A161310 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 5 times.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 3, 3, 5, 4, 6, 7, 7, 8, 10, 10, 11, 13, 13, 14, 17, 17, 19, 22, 23, 26, 29, 30, 33, 37, 40, 43, 49, 50, 54, 60, 62, 68, 73, 78, 83, 89, 94, 100, 108, 113, 122, 129, 135, 144, 153, 160, 170, 181, 188, 199, 211, 220, 233, 246, 258, 271, 286, 299

Offset: 1

R. H. Hardin, Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

Cf. A006720.

A161311 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 6 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 2, 4, 4, 5, 4, 8, 6, 8, 9, 10, 9, 14, 11, 14, 15, 16, 15, 21, 18, 22, 24, 26, 26, 34, 31, 36, 39, 42, 43, 52, 52, 57, 61, 65, 67, 77, 77, 86, 89, 95, 97, 110, 110, 120, 128, 133, 137, 151, 154, 164, 174, 184, 187, 204, 207, 221, 231, 243, 251

Offset: 1

R. H. Hardin Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

A161312 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 7 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 3, 3, 3, 5, 4, 5, 6, 7, 7, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 24, 26, 28, 30, 33, 36, 38, 41, 44, 47, 51, 54, 60, 63, 67, 71, 76, 80, 85, 93, 96, 102, 107, 113, 118, 125, 135, 139, 147, 154, 161, 168, 177, 189, 195, 205

Offset: 1

R. H. Hardin Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

A161313 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 8 times.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 2, 4, 4, 5, 4, 7, 5, 8, 8, 9, 9, 12, 10, 13, 13, 15, 14, 18, 16, 19, 19, 21, 20, 25, 23, 27, 28, 31, 31, 37, 36, 42, 43, 47, 48, 55, 55, 61, 65, 70, 72, 80, 81, 88, 92, 99, 102, 112, 113, 122, 126, 134, 138, 149, 152, 163, 168, 177

Offset: 1

R. H. Hardin Jun 06 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..1000

cp's OEIS Frontend

A161307 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 2 times.

Views

Author

Keywords

Links

A161319 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 14 times.

Views

Author

Keywords

Links

A254314 Hankel transform of a(n) is A006720(n). Hankel transform of a(n+1) is A006720(n+2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A161306 Number of partitions of n into Somos-4 sequence numbers A006720.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A161308 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 3 times.

Views

Author

Keywords

Links

A161309 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 4 times.

Views

Author

Keywords

Links

A161310 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 5 times.

Views

Author

Keywords

Links

Crossrefs

A161311 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 6 times.

Views

Author

Keywords

Links

A161312 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 7 times.

Views

Author

Keywords

Links

A161313 Number of partitions of n into Somos-4 sequence numbers A006720 where every part appears at least 8 times.

Views

Author

Keywords

Links