A135021 -id:A135021 - cp's OEIS frontend

A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

0, 1, 16, 70, 200, 455, 896, 1596, 2640, 4125, 6160, 8866, 12376, 16835, 22400, 29240, 37536, 47481, 59280, 73150, 89320, 108031, 129536, 154100, 182000, 213525, 248976, 288666, 332920, 382075, 436480, 496496, 562496, 634865, 714000, 800310, 894216, 996151

Offset: 0

Bruno Berselli, Dec 08 2012

Partial sums of A172073.

Apart from 0, all terms are in A135021: a(n) = A135021(A034856(n+1)) with n>0.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Index to sequences related to pyramidal numbers.

Cf. A135021, A172073.
Cf. convolution of the natural numbers (A000027) with the k-gonal numbers (* means "except 0"):
k= 2 (A000027 ): A000292;
k= 3 (A000217 ): A000332 (after the third term);
k= 4 (A000290 ): A002415 (after the first term);
k= 5 (A000326 ): A001296;
k= 6 (A000384*): A002417;
k= 7 (A000566 ): A002418;
k= 8 (A000567*): A002419;
k= 9 (A001106*): A051740;
k=10 (A001107*): A051797;
k=11 (A051682*): A051798;
k=12 (A051624*): A051799;
k=13 (A051865*): A055268.
Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.

A051866:=func; [&+[(n-k+1)*A051866(k): k in [0..n]]: n in [0..37]];

I:=[0,1,16,70,200]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

A051866[k_] := k (6 k - 5); Table[Sum[(n - k + 1) A051866[k], {k, 0, n}], {n, 0, 37}]
CoefficientList[Series[x (1 + 11 x) / (1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

G.f.: x*(1+11*x)/(1-x)^5.
a(n) = n*(n+1)*(n+2)*(3*n-2)/6.
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 3*(3*sqrt(3)*Pi + 27*log(3) - 17)/80.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(6*sqrt(3)*Pi - 64*log(2) + 37)/80. (End)

A370770 Triangle read by rows: T(n,k) is the number of k-trees with n unlabeled nodes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 6, 5, 2, 1, 1, 1, 1, 11, 12, 5, 2, 1, 1, 1, 1, 23, 39, 15, 5, 2, 1, 1, 1, 1, 47, 136, 58, 15, 5, 2, 1, 1, 1, 1, 106, 529, 275, 64, 15, 5, 2, 1, 1, 1, 1, 235, 2171, 1505, 331, 64, 15, 5, 2, 1, 1, 1

Offset: 0

Andrew Howroyd, Mar 01 2024

			Triangle begins:
  1;
  1,   1;
  1,   1,   1;
  1,   1,   1,   1;
  1,   2,   1,   1,  1;
  1,   3,   2,   1,  1,  1;
  1,   6,   5,   2,  1,  1, 1;
  1,  11,  12,   5,  2,  1, 1, 1;
  1,  23,  39,  15,  5,  2, 1, 1, 1;
  1,  47, 136,  58, 15,  5, 2, 1, 1, 1;
  1, 106, 529, 275, 64, 15, 5, 2, 1, 1, 1;
  ...

Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45.
I. M. Gessel and A. Gainer-Dewar, Counting unlabeled k-trees, arXiv:1309.1429 [math.CO], 2013-2014.
I. M. Gessel and A. Gainer-Dewar, Counting unlabeled k-trees, J. Combin. Theory Ser. A 126 (2014), 177-193.
Andrew Howroyd, SageMath Program code (from Andrew Gainer-Dewar reference).

Columns k=0..7 are A000012, A000055, A054581, A078792, A078793, A201702, A202037, A322754.

Cf. A135021 (labeled version), A370771, A370772, A370773.

T(n,k) = A370771(n,k) + A370772(n,k) - A370773(n,k).

A036362 Number of labeled 3-trees with n nodes.

Original entry on oeis.org

0, 0, 1, 1, 10, 200, 5915, 229376, 10946964, 618435840, 40283203125, 2968444272640, 243926836708126, 22100985366992896, 2187905889450121295, 234881024000000000000, 27172548942138551952680, 3369317755618569294053376, 445726953911853022186520169

Offset: 1

N. J. A. Sloane

F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 30, Problem 1.13(b) with k=3.

T. D. Noe, Table of n, a(n) for n=1..100
Index entries for sequences related to trees

Column 4 of A135021.

Cf. A000272 (labeled trees), A036361 (labeled 2-trees), this sequence (labeled 3-trees), A036506 (labeled 4-trees), A000055 (unlabeled trees), A054581 (unlabeled 2-trees).

[ seq(binomial(n,3)*(3*n-8)^(n-5), n=1..20) ];

Table[Binomial[n,3](3n-8)^(n-5),{n,20}] (* Harvey P. Dale, Dec 31 2023 *)

def A036362(n): return int(n*(n - 2)*(n - 1)*(3*n - 8)**(n - 5)//6) # Chai Wah Wu, Feb 03 2022

a(n) = binomial(n, 3)*(3*n-8)^(n-5).

Number of labeled k-trees on n nodes is binomial(n, k) * (k(n-k)+1)^(n-k-2).

A036506 Number of labeled 4-trees with n nodes.

Original entry on oeis.org

0, 0, 0, 1, 1, 15, 455, 20230, 1166886, 82031250, 6768679170, 639276644655, 67876292150095, 7992910154350121, 1032869077119140625, 145221924661653841820, 22060305511905816000860, 3599313659344525384083060, 627583654087024080928783956

Offset: 1

N. J. A. Sloane

nonn

F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 30, Problem 1.13(b) with k=4.

T. D. Noe, Table of n, a(n) for n=1..100
Index entries for sequences related to trees

Column 5 of A135021.

Cf. A000272 (labeled trees), A036361 (labeled 2-trees), A036362 (labeled 3-trees), A078793 (unlabeled 4-trees), A000055 (unlabeled trees), A054581 (unlabeled 2-trees).

def A036506(n): return int(n*(n - 3)*(n - 2)*(n - 1)*(4*n - 15)**(n - 6)//24) # Chai Wah Wu, Feb 03 2022

a(n) = C(n,4)*(4*n-15)^(n-6).

Number of labeled k-trees on n nodes is binomial(n, k) * (k(n-k)+1)^(n-k-2).

A036361 Number of labeled 2-trees with n nodes.

Original entry on oeis.org

0, 1, 1, 6, 70, 1215, 27951, 799708, 27337500, 1086190605, 49162945645, 2496308717826, 140489907594114, 8678436279296875, 583701359488329915, 42457773984656284920, 3320786296452525792376, 277898747312921495246937, 24775177557380767822265625

Offset: 1

N. J. A. Sloane

F. Harary and E. Palmer, Graphical Enumeration, (1973), p. 30.

T. D. Noe, Table of n, a(n) for n = 1..100
L. W. Beineke and R. E. Pipert, The number of labeled k-dimensional trees, J. Comb. Theory 6 (2) (1969) 200-205. Math. Rev. 38 #3182.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
T. Fowler, I. Gessel, G. Labelle and P. Leroux, The specification of 2-trees, Adv. Appl. Math. 28 (2) (2002) 145-168, eq. (18).
Index entries for sequences related to trees

Column 3 of A135021.

Cf. A000272 (labeled trees), this sequence (labeled 2-trees), A036362 (labeled 3-trees), A036506 (labeled 4-trees), A000055 (unlabeled trees), A054581 (unlabeled 2-trees).

A036361:=n->binomial(n, 2)*(2*n-3)^(n-4): seq(A036361(n), n=1..30);

Table[Binomial[n,2](2n-3)^(n-4),{n,20}] (* Harvey P. Dale, Nov 24 2011 *)

def A036361(n): return int(n*(n - 1)*(2*n - 3)**(n - 4)//2) # Chai Wah Wu, Feb 03 2022

Number of labeled k-trees on n nodes is binomial(n, k) * (k(n-k)+1)^(n-k-2).

cp's OEIS Frontend

A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Mathematica

Formula

A370770 Triangle read by rows: T(n,k) is the number of k-trees with n unlabeled nodes.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A036362 Number of labeled 3-trees with n nodes.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A036506 Number of labeled 4-trees with n nodes.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Python

Formula

A036361 Number of labeled 2-trees with n nodes.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula