A103457 -id:A103457 - cp's OEIS frontend

A045741 Number of edges in all noncrossing connected graphs on n nodes on a circle.

1, 9, 82, 765, 7266, 69930, 679764, 6659037, 65635570, 650194974, 6467730204, 64562259762, 646399361076, 6488447895540, 65276186864232, 657998685456093, 6644370824416530, 67198463606576790, 680568874690989900

Offset: 2

Emeric Deutsch

nonn

			a(3)=9; indeed, with vertices u, v, w, the noncrossing connected graphs are {uv,uw}, {vu, vw}, {wu, wv}, and {uv, vw, wu} with a total of 9 edges.

G. C. Greubel, Table of n, a(n) for n = 2..980
Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh, On the congruences of some combinatorial numbers, Stud. Appl. Math. vol. 116 (2006) pp. 135-144
I. M. Gessel, A short proof of the Deutsch-Sagan congruence for connected non crossing graphs, arXiv preprint arXiv:1403.7656, 2014

Cf. A007297, A244038, A244039.

A045741 := proc(n) local k ; add(binomial(3*n-3,n+k)*binomial(k,n-1),k=0..2*n-3) ; end: seq(A045741(n),n=2..20) ; # R. J. Mathar, Feb 27 2008

Table[Sum[k*Binomial[3*n - 3, n + k]*Binomial[k - 1, k - n + 1], {k, n - 1, 2*n}]/(n - 1), {n,2,50}] (* G. C. Greubel, Jan 30 2017 *)

for(n=2,50, print1(sum(k=n-1,2*n, k*binomial(3*n-3,n+k)* binomial(k-1,k-n+1))/(n-1), ", ")) \\ G. C. Greubel, Jan 30 2017

a(n) = Sum_{k = n-1 .. 2*n} (k*binomial(3*n-3, n+k)*binomial(k-1, k-n+1))/(n-1).
a(n) = 1 mod 3 if n in A103457; a(n) = 0 mod 3 otherwise [Eu et al.]. - R. J. Mathar, Feb 27 2008
Recurrence: (n-2)*(n-1)*(6*n-17)*a(n) = 18*(n-2)*a(n-1) + 12*(3*n-8)*(3*n-7)*(6*n-11)*a(n-2). - Vaclav Kotesovec, Dec 29 2012
a(n) ~ (sqrt(3)-1)/sqrt(Pi) * (2^(n-5/2)*3^(3*n/2-3/2))/sqrt(n). - Vaclav Kotesovec, Dec 29 2012
A244038(n) = a(n) + A244039(n) [Gessel]. - N. J. A. Sloane, Jun 28 2014

A103462 A triangle with palindromic cubes, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 5, 4, 1, 1, 2, 9, 10, 5, 1, 1, 2, 17, 28, 17, 6, 1, 1, 2, 33, 82, 65, 26, 7, 1, 1, 2, 65, 244, 257, 126, 37, 8, 1, 1, 2, 129, 730, 1025, 626, 217, 50, 9, 1, 1, 2, 257, 2188, 4097, 3126, 1297, 344, 65, 10, 1, 1, 2, 513, 6562, 16385, 15626, 7777

Offset: 0

Paul Barry, Feb 07 2005

			Rows start {1}, {1,1}, {1,2,1}, {1,2,3,1}, {1,2,5,4,1},..

P. De Geest, World!Of Numbers

Columns include A040000, A083318, A103457, A046231, A046233, A103458, A103459, A000533. Cubes of column k are palindromic to base k, k>3 (start with column 0). Row sums are A103480. Diagonal sums are A103481.

Number triangle T(n, k)=if(k<=n, k^(n-k)+1-0^(n-k), 0); Column k has g.f. x^k(1-kx^2)/((1-x)(1-kx)).

A176666 A triangle of polynomial coefficients:p(x,n)=Sum[(2k + 1)^nk!*Binomial[x, k], {k, 0, n}].

Original entry on oeis.org

1, 1, 3, 1, -16, 25, 1, 588, -904, 343, 1, -35108, 65593, -36965, 6561, 1, 3541662, -7450307, 5299298, -1551461, 161051, 1, -539667860, 1239476145, -1027098387, 393094596, -70630574, 4826809, 1, 115929493398, -285126982237, 264011385389

Offset: 0

Roger L. Bagula, Apr 23 2010

Row sums are:A103457;

{1, 4, 10, 28, 82, 244, 730, 2188, 6562, 19684, 59050,...}.

			{1},
{1, 3},
{1, -16, 25},
{1, 588, -904, 343},
{1, -35108, 65593, -36965, 6561},
{1, 3541662, -7450307, 5299298, -1551461, 161051},
{ 1, -539667860, 1239476145, -1027098387, 393094596, -70630574, 4826809},
{1, 115929493398, -285126982237, 264011385389, -120438105421, 28978650041, -3525298358, 170859375},
{1, -33405526460804, 86851508060145, -87619801707127, 45402414077950, -13236000326919, 2193188923598, -192758317723, 6975757441},
{1, 12439546100725062, -33876724511327305, 36619991865553925, -20936375400104384, 7018154767854372, -1426322806941012, 172905465804793, -11498169243547, 322687697779},
{1, -5815351979718349460, 16476663041157314889, -18861838035155184791, 11671607490973992658, -4358525114199083475, 1028212770824559839, -154333184246062051, 14292794059654483, -744463577761244, 16679880978201}

Cf. A103457

Clear[p, x, n]
p[x_, n_] := Sum[(2*k + 1)^n*k!*Binomial[x, k], {k, 0, n}];
Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}];
Flatten[%]

p(x,n)=Sum[(2*k + 1)^n*k!*Binomial[x, k], {k, 0, n}];

t(n,m)=coefficients(p(x,n))

cp's OEIS Frontend

A045741 Number of edges in all noncrossing connected graphs on n nodes on a circle.

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A103462 A triangle with palindromic cubes, read by rows.

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A176666 A triangle of polynomial coefficients:p(x,n)=Sum[(2k + 1)^nk!*Binomial[x, k], {k, 0, n}].

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Mathematica

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cp's OEIS Frontend

A045741 Number of edges in all noncrossing connected graphs on n nodes on a circle.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A103462 A triangle with palindromic cubes, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A176666 A triangle of polynomial coefficients:p(x,n)=Sum[(2*k + 1)^n*k!*Binomial[x, k], {k, 0, n}].

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A176666 A triangle of polynomial coefficients:p(x,n)=Sum[(2k + 1)^nk!*Binomial[x, k], {k, 0, n}].