A000684 -id:A000684 - cp's OEIS frontend

A001828 Related to graded partially ordered sets.

1, 5, 33, 287, 3309, 50975, 1058493, 29885567, 1156711869, 61815727295, 4589058616413, 475576073939807, 69061902766811229, 14093318360697120095, 4049931601653596366013, 1641314561238334948886207, 939097032426474389539281789

Offset: 0

N. J. A. Sloane

nonn

Corresponds to the numbers c(5,n) in the Klarner paper. - Sean A. Irvine, Sep 24 2015

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Cerkan, Table of n, a(n) for n = 0..112
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
Index entries for sequences related to posets

Column k=5 of A361950.

Cf. A000684, A001827, A001829, A001830, A047863.

a(n) = Sum_{p+q+r+s+t=n} (n!/p!q!r!s!t!) 2^(pq+qr+rs+st) where (p,q,r,s,t) is any nonnegative composition of n. - Sean A. Irvine, Sep 24 2015

More terms from Sean A. Irvine, Sep 24 2015

A001829 Related to graded partially ordered sets.

Original entry on oeis.org

1, 6, 46, 450, 5650, 91866, 1957066, 55363650, 2109599650, 109773407466, 7894945079386, 792252362302770, 111671194813402930, 22202849561274787866, 6241728810901739517226, 2484011055161613143144610, 1400187830319472451472442690

Offset: 0

N. J. A. Sloane

nonn

Corresponds to the numbers c(6,n) in the Klarner paper. - Sean A. Irvine, Sep 24 2015

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Cerkan, Table of n, a(n) for n = 0..112
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
Index entries for sequences related to posets

Column k=6 of A361950.

Cf. A000684, A001827, A001828, A001830, A047863.

a(n) = Sum_{p+q+r+s+t+u=n} (n!/p!q!r!s!t!u!) 2^(pq+qr+rs+st+tu) where (p,q,r,s,t,u) is any nonnegative composition of n. - Sean A. Irvine, Sep 24 2015

A001830 Related to graded partially ordered sets.

Original entry on oeis.org

1, 7, 61, 661, 8953, 152917, 3334921, 94354981, 3528929353, 177999003157, 12340001650921, 1194005625114661, 162936187792764073, 31536761103831315157, 8677703806537883683081, 3395880602480076153665701, 1889190751946097573211698313

Offset: 0

N. J. A. Sloane

nonn

Corresponds to the numbers c(7,n) in the Klarner paper. - Sean A. Irvine, Sep 24 2015

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Cerkan, Table of n, a(n) for n = 0..112
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
Index entries for sequences related to posets

Column k=7 of A361950.

Cf. A000684, A001827, A001828, A001829, A047863.

a(n) = Sum_{p+q+r+s+t+u+v=n} (n!/p!q!r!s!t!u!v!) 2^(pq+qr+rs+st+tu+uv) where (p,q,r,s,t,u,v) is any nonnegative composition of n. - Sean A. Irvine, Sep 24 2015

More terms from Sean A. Irvine, Sep 24 2015

A006201 Number of colorings of labeled graphs on n nodes using exactly 3 colors, divided by 48.

Original entry on oeis.org

0, 0, 1, 24, 640, 24000, 1367296, 122056704, 17282252800, 3897054412800, 1400795928395776, 802530102499344384, 732523556206878392320, 1064849635418836398243840, 2464403435614136308036796416

Offset: 1

N. J. A. Sloane

Equals 1/48*A213442. - Peter Bala, Apr 12 2013

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 18, table 1.5.1, column 3 (divided by 8).
R. C. Read, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..75
R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976

Cf. A000683. A diagonal of A058843. A213442.

F2[n_] := Sum[Binomial[n, r]*2^(r*(n-r)), {r, 1, n-1}]; F3[n_] := Sum[Binomial[n, r]*2^(r*(n-r))*F2[r], {r, 1, n-1}]; a[n_] := F3[n]/48; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Mar 06 2014, after Maple code in A213442 *)

seq(n)={Vec(serconvol(sum(j=1, n, x^j*j!*2^binomial(j,2)) + O(x*x^n), (sum(j=1, n, x^j/(j!*2^binomial(j,2))) + O(x*x^n))^3)/48, -n)} \\ Andrew Howroyd, Nov 30 2018

Let E(x) = sum {n >= 0} x^n/(n!*2^C(n,2)) = 1 + x + x^2/(2!*2) + x^3/(3!*2^3) + x^4/(4!*2^6) + .... Then a generating function is 1/48*(E(x) - 1)^3 = x^3/(3!*2^3) + 24*x^4/(4!*2^6) + 640*x^6/(5!*2^10) + ... (see Read). - Peter Bala, Apr 12 2013

More terms from Vladeta Jovovic, Feb 03 2000

A004100 Number of labeled nonseparable bipartite graphs on n nodes.

Original entry on oeis.org

0, 1, 0, 3, 10, 355, 6986, 297619, 15077658, 1120452771, 111765799882, 15350524923547, 2875055248515242, 738416821509929731, 260316039943139322858, 126430202628042630866787, 84814075550928212558332858, 78847417416749666369637926851

Offset: 1

N. J. A. Sloane

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 406.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
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..100 (terms 1..32 from R. W. Robinson)
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68.
F. Harary and R. W. Robinson, Labeled bipartite blocks, Canad. J. Math., 31 (1979), 60-68. (Annotated scanned copy)
A. Nymeyer and R. W. Robinson, Tabulation of the Numbers of Labeled Bipartite Blocks and Related Classes of Bicolored Graphs, 1982 [Annotated scanned copy of unpublished MS and letter from R.W.R.]

Cf. A001832, A013922.

b[n_] := Log[Sum[Exp[2^k*x + O[x]^n]*x^k/k!, {k, 0, n}]/2];
seq[n_] := CoefficientList[-Log[2] + Log[x/InverseSeries[x*D[b[n], x]]], x]*Table[(2k)!!, {k, 0, n-2}];
seq[19] (* Jean-François Alcover, Sep 04 2019, after Andrew Howroyd *)

\\ here b(n) is A001832 as e.g.f.
b(n)={log(sum(k=0, n, exp(2^k*x + O(x*x^n))*x^k/k!))/2}
seq(n)={Vec(serlaplace(log(x/serreverse(x*deriv(b(n))))), -n)} \\ Andrew Howroyd, Sep 26 2018

a(16) onwards added by N. J. A. Sloane, Oct 19 2006 from the Robinson reference

A193199 G.f.: A(x) = Sum_{n>=0} x^n/(1 - 4^n*x)^n.

Original entry on oeis.org

1, 1, 5, 49, 1025, 42241, 3610625, 609251329, 210923290625, 144320565411841, 201501092228890625, 556475188311619534849, 3125896980250691972890625, 34751531654955460673195212801, 784223845648499469575195012890625

Offset: 0

Paul D. Hanna, Jul 17 2011

nonn

			G.f.: A(x) = 1 + x + 5*x^2 + 49*x^3 + 1025*x^4 + 42241*x^5 +...
where:
A(x) = 1 + x/(1-4*x) + x^2/(1-16*x)^2 + x^3/(1-64*x)^3 + x^4/(1-256*x)^4 +...

Seiichi Manyama, Table of n, a(n) for n = 0..80

Cf. A000684, A193198.

{a(n)=local(A=1);A=1+sum(m=1,n,x^m/(1-4^m*x +x*O(x^n))^m);polcoeff(A,n)}

{a(n)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*4^(k*(n-k))))}

a(n) = Sum_{k=0..n-1} binomial(n-1,k)*4^(k*(n-k)) for n>0 with a(0)=1.

A000426 Coefficients of ménage hit polynomials.

Original entry on oeis.org

0, 1, 1, 1, 8, 35, 211, 1459, 11584, 103605, 1030805, 11291237, 135015896, 1749915271, 24435107047, 365696282855, 5839492221440, 99096354764009, 1780930394412009, 33789956266629001, 674939337282352360, 14157377139256183723, 311135096550816014651

Offset: 1

N. J. A. Sloane and Simon Plouffe

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 198.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. M. Taylor, A problem on arrangements, Mess. Math., 32 (1902), 60ff.

David W. Wilson, Table of n, a(n) for n = 1..100
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
H. M. Taylor, A problem on arrangements, Mess. Math., 32 (1902), 60ff. [Annotated scanned copy]

Cf. A000179, A000271. A diagonal of A058057.

[0] cat [&+[(-1)^k*Factorial(2*n-k-1)*Factorial(n-k) / (Factorial(2*n-2*k)*Factorial(k-2)): k in [2..n]]: n in [2..25]]; // Vincenzo Librandi, Jun 11 2019

Table[Sum[(-1)^k*(2*n-k-1)!*(n-k)!/((2*n-2*k)!*(k-2)!),{k,2,n}],{n,1,20}] (* Vaclav Kotesovec, Oct 26 2012 *)

a(n) = Sum_{k=2..n} (-1)^k*(2n-k-1)!*(n-k)!/((2n-2k)!*(k-2)!).
a(n) = A000033(n)/n.
a(n) = ((2*n-5)*a(n-1) + (5*n-11)*a(n-2) + (5*n-14)*a(n-3) + (2*n-5)*a(n-4) + 2*a(n-5))/2 for n >= 6.
Shorter recurrence: (14*n-67)*a(n) = (14*n^2-95*n+137)*a(n-1) + (14*n^2-105*n+180)*a(n-2) - 24*a(n-4) + (57-10*n)*a(n-3). - Vaclav Kotesovec, Oct 26 2012
a(n) ~ 2/e^2*(n-1)!. - Vaclav Kotesovec, Oct 26 2012
a(n) = round((exp(-2)*(8*BesselK(n,2) - (4*n-10)*BesselK(n-1,2)))) for n > 6. - Mark van Hoeij, Jun 09 2019
a(n)+2*a(n+p)+a(n+2*p) is divisible by p for any prime p. - Mark van Hoeij, Jun 13 2019

Edited by David W. Wilson, Dec 27 2007

A000673 Number of bicentered 3-valent (or boron, or binary) trees with n nodes.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 2, 6, 8, 18, 30, 67, 127, 275, 551, 1192, 2507, 5475, 11820, 26007, 57077, 126686, 281625, 630660, 1416116, 3195784, 7232624, 16430563, 37429146, 85528079, 195940960, 450074270, 1036226173, 2391193488, 5529420585

Offset: 0

N. J. A. Sloane

A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 451).
R. C. Read, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Nicolas Broutin and Philippe Flajolet, The distribution of height and diameter in random non-plane binary trees, Random Struct. Algorithms 41, No. 2, 215-252 (2012).
E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
Index entries for sequences related to trees

A000672 = A000673 + A000675. Cf. A000022, A000200, A000602.

n = 50; (* algorithm from Rains and Sloane *)
S2[f_,h_,x_] := f[h,x]^2/2 + f[h,x^2]/2;
T[-1,z_] := 1;  T[h_,z_] := T[h,z] = Table[z^k, {k,0,n}].Take[CoefficientList[z^(n+1) + 1 + S2[T,h-1,z]z, z], n+1];
Sum[Take[CoefficientList[z^(n+1) + (T[h,z] - T[h-1,z])^2/2 + (T[h,z^2] - T[h-1,z^2])/2, z],n+1], {h,0,n/2}] (* Robert A. Russell, Sep 15 2018 *)

A000675 Number of centered 3-valent (or boron, or binary) trees with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 4, 5, 10, 19, 36, 68, 138, 277, 581, 1218, 2591, 5545, 12026, 26226, 57719, 127685, 284109, 634919, 1425516, 3212890, 7269605, 16504439, 37592604, 85876345, 196717882, 451768247, 1039990913, 2399476030, 5547849750

Offset: 0

N. J. A. Sloane

A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 451).
R. C. Read, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
Index entries for sequences related to trees

A000672 = A000673 + A000675. Cf. A000022, A000200, A000602.

n = 50; (* algorithm from Rains and Sloane *)
S2[f_,h_,x_] := f[h,x]^2/2 + f[h,x^2]/2;
S3[f_,h_,x_] := f[h,x]^3/6 + f[h,x] f[h,x^2]/2 + f[h,x^3]/3;
T[-1,z_] := 1;  T[h_,z_] := T[h,z] = Table[z^k, {k,0,n}].Take[CoefficientList[z^(n+1) + 1 + S2[T,h-1,z]z, z], n+1];
Sum[Take[CoefficientList[z^(n+1) + S3[T,h-1,z]z - S3[T,h-2,z]z - (T[h-1,z] - T[h-2,z]) (T[h-1,z]-1),z], n+1], {h,1,n/2}] + PadRight[{1,1}, n+1] (* Robert A. Russell, Sep 15 2018 *)

A000685 Number of 3-colored labeled graphs on n nodes, divided by 3.

Original entry on oeis.org

1, 5, 41, 545, 11681, 402305, 22207361, 1961396225, 276825510401, 62368881977345, 22413909724518401, 12840603873823473665, 11720394922432296755201, 17037597932370037286600705

Offset: 1

N. J. A. Sloane

Sequence represents 1/3 of the number of 3-colored labeled graphs on n nodes. Indeed, on p. 413 of the Read paper, column 3 is 3, 15, 123, 1635, ...; or see A047863. - Emeric Deutsch, May 06 2004

R. C. Read, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..50
S. R. Finch, Bipartite, k-colorable and k-colored graphs (3*A000685)
S. R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author]
R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410-414.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
R. P. Stanley, Acyclic orientation of graphs Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.
Eric Weisstein's World of Mathematics, k-Colorable Graph

Cf. A000683, A047863, A191371, A223887.

c[0]:=1: for n from 1 to 30 do c[n]:=sum(binomial(n,i)*2^(i*(n-i)),i=0..n) od: a:=n->(1/3)*sum(binomial(n,j)*2^(j*(n-j))*c[j],j=0..n): seq(a(n),n=1..19);

a[n_] := 1/3*Sum[ 2^((i-j)*j + i*(n-i))*Binomial[n, i]*Binomial[i, j], {i, 0, n}, {j, 0, i}]; Table[ a[n], {n, 1, 14}] (* Jean-François Alcover, Dec 07 2011, after Emeric Deutsch *)

a(n) = (1/3)Sum_{j=0..n} binomial(n, j)*2^(j(n-j))*c(j) where c(n) = Sum_{i=0..n} binomial(n, i)*2^(i(n-i)) = A047863(n). - Emeric Deutsch, May 06 2004
From Peter Bala, Apr 12 2013: (Start)
a(n) = 1/3*A191371(n). Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence is 1/3*E(x)^3 - 1/3 = Sum_{n >= 1} a(n)*x^n/(n!*2^C(n,2)) = x + 5*x^2/(2!*2) + 41*x^3/(3!*2^3) + .... In general, E(x)^k, k = 1, 2, ..., is a generating function for labeled k-colored graphs (see Read). For examples see A047863 (k = 2), A191371 (k = 3) and A223887 (k = 4). (End)

More terms from Pab Ter (pabrlos(AT)yahoo.com) and Emeric Deutsch, May 05 2004

cp's OEIS Frontend

A001828 Related to graded partially ordered sets.

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A001829 Related to graded partially ordered sets.

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A001830 Related to graded partially ordered sets.

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A006201 Number of colorings of labeled graphs on n nodes using exactly 3 colors, divided by 48.

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A004100 Number of labeled nonseparable bipartite graphs on n nodes.

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A193199 G.f.: A(x) = Sum_{n>=0} x^n/(1 - 4^n*x)^n.

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PARI

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A000426 Coefficients of ménage hit polynomials.

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Magma

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A000673 Number of bicentered 3-valent (or boron, or binary) trees with n nodes.

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