A094262 -id:A094262 - cp's OEIS frontend

A005174 Number of rooted trees with 4 nodes of disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.

0, 0, 10, 124, 890, 5060, 25410, 118524, 527530, 2276020, 9613010, 40001324, 164698170, 672961380, 2734531810, 11066546524, 44652164810, 179768037140, 722553165810, 2900661482124, 11634003919450, 46630112719300, 186802788139010, 748058256616124

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..500
F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10.
F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for sequences related to trees
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Column 4 of A094262.

A005174:=2*z**2*(5+12*z)/(z-1)/(3*z-1)/(2*z-1)/(4*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

The terms a(1)-a(18) are given by a(n) = (8/3)*(4^n - 4) - 9*3^n + 11*2^n + 5. - John W. Layman, Jul 20 1999
Formula of Layman matches the proven formula in McMorris and Zaslavsky. - Sean A. Irvine, Apr 12 2016
E.g.f.: (1/3)*(-17*exp(x) + 66*exp(2*x) - 81*exp(3*x) + 32*exp(4*x)). - Ilya Gutkovskiy, Apr 12 2016
G.f.: 2*x^3*(5 + 12*x)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)). - Andrew Howroyd, Mar 28 2025

Name clarified by Andrew Howroyd, Mar 28 2025

A005175 Number of rooted trees with 5 nodes of disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.

Original entry on oeis.org

0, 0, 3, 131, 1830, 16990, 127953, 851361, 5231460, 30459980, 170761503, 931484191, 4979773890, 26223530970, 136522672653, 704553794621, 3611494269120, 18415268221960, 93516225653403, 473366777478651, 2390054857197150, 12043393363764950, 60590148885015753

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..500
F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10.
F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for sequences related to trees
Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120).

Column 5 of A094262.

A005175:=-z**2*(3+86*z+120*z**2)/(z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

Table[(125/24) 5^n - (64/3) 4^n + (135/4) 3^n - (76/3) 2^n + 209/24, {n, 20}] (* Michael De Vlieger, Apr 12 2016 *)

a(n+1) = 3*(3^n - 2*2^n + 1)/2 + 113*(4^n - 3*3^n + 3*2^n - 1)/6 + 625*(5^n - 4*4^n + 6*3^n - 4*2^n + 1)/24. - formula fitted by John W. Layman
a(n) = (125/24) * 5^n - (64/3) * 4^n + (135/4)*3^n - (76/3) * 2^n + 209/24 proven in McMorris and Zaslavsky, matches Layman's formula with an offset of 1. - Sean A. Irvine, Apr 12 2016
E.g.f.: (1/24)*exp(x)*(-1 + exp(x))^2*(209 - 798*exp(x) + 625*exp(2*x)). - Ilya Gutkovskiy, Apr 12 2016
G.f.: x^3*(3 + 86*x + 120*x^2)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)). - Andrew Howroyd, Mar 28 2025

Name clarified by Andrew Howroyd, Mar 28 2025

A102735 Table read by rows giving the coefficients of general sum formulas of n-th sums of Bell numbers (A005001). The k-th row (k>=1) contains T(i,k) for i=1 to 2k and k=1 to n-3, where T(i,k) satisfies Sum_{q=1..n} Bell(q) = 1 + C(n,2) + Sum_{k=1..n-3} Sum_{i=1..2k} T(i,k) * C(n-k-2,1).

Original entry on oeis.org

2, 1, 8, 13, 10, 3, 22, 74, 134, 134, 70, 15, 52, 314, 1024, 1964, 2296, 1615, 630, 105, 114, 1155, 6084, 18954, 37512, 48677, 41426, 22330, 6930, 945, 240, 3927, 31494, 146907, 438948, 885653, 1237958, 1204525, 802648, 349965, 90090, 10395, 494

Offset: 1

André F. Labossière, Feb 07 2005

The coefficients T(i,k) along the i-th columns of the triangle are the consecutive partial sums of those found in table A094262.

			Sum_Bell(7) = 1 + C(7,2) + 2*C(7-3,1) + C(7-3,2) + ... + 74*C(7-5,2) + 52*C(7-6,1)
= 1 + 21 + 8 + 6 + 8*C(3,1) + 13*C(3,2) + 10*C(3,3) + 22*C(2,1) + 74 + 52 = 1 + 21 + 8 + 6
+ 24 + 39 + 10 + 44 + 74 + 52 = 279.

Cf. A094262, A005001, A000110, A008277, A003422, A000166, A000204, A000045, A000108.

A094261 a(n) = n(n-1)(n-3)(n-6)...(n-t), where t is the largest triangular number less than n; number of factors in the product is ceiling((sqrt(1+8*n)-1)/2).

Original entry on oeis.org

1, 2, 6, 12, 40, 90, 168, 560, 1296, 2520, 4400, 14256, 32760, 64064, 113400, 187200, 586432, 1321920, 2560896, 4522000, 7484400, 11797632, 35784320, 78871968, 150480000, 263120000, 433060992, 681080400, 1033305728, 3044304000

Offset: 1

Amarnath Murthy, Apr 26 2004

nonn

			a(8) = 8*(8-1)*(8-3)*(8-6) = 8*7*5*2 = 560.

Cf. A057944, A094262.

a:=n->product(n-k*(k+1)/2,k=0..ceil((sqrt(1+8*n)-1)/2)-1): seq(a(n),n=1..35); # Emeric Deutsch, Feb 03 2006

Corrected and extended by Emeric Deutsch, Feb 03 2006

cp's OEIS Frontend

A005174 Number of rooted trees with 4 nodes of disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Formula

Extensions

A005175 Number of rooted trees with 5 nodes of disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A102735 Table read by rows giving the coefficients of general sum formulas of n-th sums of Bell numbers (A005001). The k-th row (k>=1) contains T(i,k) for i=1 to 2k and k=1 to n-3, where T(i,k) satisfies Sum_{q=1..n} Bell(q) = 1 + C(n,2) + Sum_{k=1..n-3} Sum_{i=1..2k} T(i,k) * C(n-k-2,1).

Views

Author

Keywords

Comments

Examples

Crossrefs

A094261 a(n) = n(n-1)(n-3)(n-6)...(n-t), where t is the largest triangular number less than n; number of factors in the product is ceiling((sqrt(1+8*n)-1)/2).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Extensions

cp's OEIS Frontend

A005174 Number of rooted trees with 4 nodes of disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Formula

Extensions

A005175 Number of rooted trees with 5 nodes of disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A102735 Table read by rows giving the coefficients of general sum formulas of n-th sums of Bell numbers (A005001). The k-th row (k>=1) contains T(i,k) for i=1 to 2*k and k=1 to n-3, where T(i,k) satisfies Sum_{q=1..n} Bell(q) = 1 + C(n,2) + Sum_{k=1..n-3} Sum_{i=1..2*k} T(i,k) * C(n-k-2,1).

Views

Author

Keywords

Comments

Examples

Crossrefs

A094261 a(n) = n(n-1)(n-3)(n-6)...(n-t), where t is the largest triangular number less than n; number of factors in the product is ceiling((sqrt(1+8*n)-1)/2).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Extensions

A102735 Table read by rows giving the coefficients of general sum formulas of n-th sums of Bell numbers (A005001). The k-th row (k>=1) contains T(i,k) for i=1 to 2k and k=1 to n-3, where T(i,k) satisfies Sum_{q=1..n} Bell(q) = 1 + C(n,2) + Sum_{k=1..n-3} Sum_{i=1..2k} T(i,k) * C(n-k-2,1).