A255517 -id:A255517 - cp's OEIS frontend

A052797 Number of rooted identity trees with n nodes and 5-colored non-root nodes.

0, 1, 5, 35, 310, 2980, 30526, 325655, 3581200, 40301850, 461925625, 5373527605, 63281397830, 752962948360, 9038406399150, 109321688626100, 1331047556550240, 16300720361555725, 200658900798443135, 2481460407963908510, 30814094488256704650, 384069865485447909905

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

Alois P. Heinz, Table of n, a(n) for n = 0..500
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 754

Column k=5 of A255517.

spec := [S,{B=PowerSet(S),S=Prod(Z,B,B,B,B,B)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(5*a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n=1, 1, b((n-1)$2)):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 24 2015

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[5*a[i], j]*b[n - i*j, i-1], {j, 0, n/i}]]]; a[n_] := If[n==1, 1, b[n-1, n-1]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = 13.40708747253774757978704707270263863994591470583713..., c = 0.07868285364894808762720738672045999... . - Vaclav Kotesovec, Feb 24 2015
From Ilya Gutkovskiy, Apr 13 2019: (Start)
G.f. A(x) satisfies: A(x) = x*exp(5*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(5*a(n)). (End)

New name from Vaclav Kotesovec, Feb 24 2015

A255518 Number of rooted identity trees with n nodes and 6-colored non-root nodes.

Original entry on oeis.org

0, 1, 6, 51, 542, 6273, 77262, 991264, 13109394, 177428496, 2445799878, 34218909015, 484668543660, 6935979521095, 100136840127534, 1456726481340477, 21332234689262126, 314211310198197978, 4652073871301311128, 69194159284034697860, 1033441834436615884266

Offset: 0

Alois P. Heinz, Feb 24 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..800

Column k=6 of A255517.

with(numtheory):
a:= proc(n) option remember; `if`(n<2, n, -add(a(n-j)*add(
      6*a(d)*d*(-1)^(j/d), d=divisors(j)), j=1..n-1)/(n-1))
    end:
seq(a(n), n=0..30);

a(n) ~ c * d^n / n^(3/2), where d = 16.1255293604485586705050971466317639696978222052989685416919877643317..., c = 0.065726585606326795496161701400246926800764944... . - Vaclav Kotesovec, Feb 24 2015
From Ilya Gutkovskiy, Apr 13 2019: (Start)
G.f. A(x) satisfies: A(x) = x*exp(6*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(6*a(n)). (End)

A255519 Number of rooted identity trees with n nodes and 7-colored non-root nodes.

Original entry on oeis.org

0, 1, 7, 70, 868, 11746, 169022, 2533895, 39156244, 619261762, 9974917309, 163078496420, 2699109217420, 45136794135853, 761492594770649, 12944937176754422, 221518183089901621, 3812823547509811611, 65966582592220602621, 1146568019260356348978

Offset: 0

Alois P. Heinz, Feb 24 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..750

Column k=7 of A255517.

with(numtheory):
a:= proc(n) option remember; `if`(n<2, n, -add(a(n-j)*add(
      7*a(d)*d*(-1)^(j/d), d=divisors(j)), j=1..n-1)/(n-1))
    end:
seq(a(n), n=0..30);

a(n) ~ c * d^n / n^(3/2), where d = 18.843901825822305757579605844910623225182677164912157727486865738745..., c = 0.056432508184666421552433382560980970822794... . - Vaclav Kotesovec, Feb 24 2015
From Ilya Gutkovskiy, Apr 13 2019: (Start)
G.f. A(x) satisfies: A(x) = x*exp(7*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(7*a(n)). (End)

A255520 Number of rooted identity trees with n nodes and 8-colored non-root nodes.

Original entry on oeis.org

0, 1, 8, 92, 1304, 20198, 332520, 5703724, 100847976, 1824927697, 33634879304, 629201396744, 11915930584384, 228010216559592, 4401559021963488, 85616787777724400, 1676436841812675760, 33017479163392717192, 653643628799220208104, 12999812350464606307796

Offset: 0

Alois P. Heinz, Feb 24 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..750

Column k=8 of A255517.

with(numtheory):
a:= proc(n) option remember; `if`(n<2, n, -add(a(n-j)*add(
      8*a(d)*d*(-1)^(j/d), d=divisors(j)), j=1..n-1)/(n-1))
    end:
seq(a(n), n=0..30);

a(n) ~ c * d^n / n^(3/2), where d = 21.5622387024302370660187831154056800411286761376313324441779580180359..., c = 0.049440632575743414117260362085656158155861722... . - Vaclav Kotesovec, Feb 24 2015
From Ilya Gutkovskiy, Apr 14 2019: (Start)
G.f. A(x) satisfies: A(x) = x*exp(8*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(8*a(n)). (End)

A255521 Number of rooted identity trees with n nodes and 9-colored non-root nodes.

Original entry on oeis.org

0, 1, 9, 117, 1866, 32553, 603414, 11654634, 232034283, 4728048201, 98125181461, 2066983603704, 44079196497075, 949772378078829, 20645820782745363, 452215682045713701, 9970925646977589555, 221133330528834114000, 4929622717525248345174, 110400838255998014848137

Offset: 0

Alois P. Heinz, Feb 24 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..700

Column k=9 of A255517.

with(numtheory):
a:= proc(n) option remember; `if`(n<2, n, -add(a(n-j)*add(
      9*a(d)*d*(-1)^(j/d), d=divisors(j)), j=1..n-1)/(n-1))
    end:
seq(a(n), n=0..30);

a(n) ~ c * d^n / n^(3/2), where d = 24.2805556948066926165789325334976292249076194687965619357813839307368..., c = 0.04399000859622510673129847184312171422452194... . - Vaclav Kotesovec, Feb 24 2015
From Ilya Gutkovskiy, Apr 14 2019: (Start)
G.f. A(x) satisfies: A(x) = x*exp(9*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(9*a(n)). (End)

A255522 Number of rooted identity trees with n nodes and 10-colored non-root nodes.

Original entry on oeis.org

0, 1, 10, 145, 2570, 49860, 1027602, 22068705, 488541820, 11068982545, 255437694060, 5983042467096, 141873247900650, 3399140192819340, 82160878859739650, 2001070766525744725, 49061025740711233080, 1209873601374566796515, 29990547373994063764080

Offset: 0

Alois P. Heinz, Feb 24 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..700

Column k=10 of A255517.

with(numtheory):
a:= proc(n) option remember; `if`(n<2, n, -add(a(n-j)*add(
      10*a(d)*d*(-1)^(j/d), d=divisors(j)), j=1..n-1)/(n-1))
    end:
seq(a(n), n=0..30);

a(n) ~ c * d^n / n^(3/2), where d = 26.998860838916733933849490675388336975888308433826638445291076817..., c = 0.0396216952587990270999387393053224285... . - Vaclav Kotesovec, Feb 24 2015
From Ilya Gutkovskiy, Apr 14 2019: (Start)
G.f. A(x) satisfies: A(x) = x*exp(10*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(10*a(n)). (End)

cp's OEIS Frontend

A052797 Number of rooted identity trees with n nodes and 5-colored non-root nodes.

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A255518 Number of rooted identity trees with n nodes and 6-colored non-root nodes.

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A255519 Number of rooted identity trees with n nodes and 7-colored non-root nodes.

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A255520 Number of rooted identity trees with n nodes and 8-colored non-root nodes.

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A255521 Number of rooted identity trees with n nodes and 9-colored non-root nodes.

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Maple

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A255522 Number of rooted identity trees with n nodes and 10-colored non-root nodes.

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Maple

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