A304386 -id:A304386 - cp's OEIS frontend

A304387 a(n) = 27*2^n - 5.

22, 49, 103, 211, 427, 859, 1723, 3451, 6907, 13819, 27643, 55291, 110587, 221179, 442363, 884731, 1769467, 3538939, 7077883, 14155771, 28311547, 56623099, 113246203, 226492411, 452984827, 905969659, 1811939323, 3623878651, 7247757307, 14495514619, 28991029243, 57982058491

Offset: 0

Emeric Deutsch, May 13 2018

For n>0, a(n) is the number of edges in the dendrimer nanostar NS1[n] defined pictorially in the Ashrafi et al. reference (Ns1[3] is shown in Fig. 1) or in the Ahmadi et al. reference (Fig. 1).

Colin Barker, Table of n, a(n) for n = 0..1000
M. B. Ahmadi and M. Sadeghimehr, Atom bond connectivity index of an infinite class NS1[n] of dendrimer nanostars, Optoelectronics and Advanced Materials, 4(7):1040-1042 July 2010.
Ali Reza Ashrafi and Parisa Nikzad, Kekulé index and bounds of energy for nanostar dendrimers, Digest J. of Nanomaterials and Biostructures, 4, No. 2, 2009, 383-388.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A304386, A250653.

List([1..40],n->27*2^n-5); # Muniru A Asiru, May 13 2018

Maple
```
seq(27*2^n-5, n = 0 .. 40);
```

27*2^Range[0,40]-5 (* or *) LinearRecurrence[{3,-2},{22,49},40] (* Harvey P. Dale, Jan 12 2019 *)

a(n) = 27*2^n - 5; \\ Altug Alkan, May 13 2018

Vec((22 - 17*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, May 18 2018

From Colin Barker, May 18 2018: (Start)
G.f.: (22 - 17*x) / ((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1) - 2*a(n-2) for n>1.
(End)

Offset changed by N. J. A. Sloane, May 13 2018

A304968 Number of labeled hypertrees spanning some subset of {1,...,n}, with singleton edges allowed.

Original entry on oeis.org

1, 2, 7, 48, 621, 12638, 351987, 12426060, 531225945, 26674100154, 1538781595999, 100292956964456, 7288903575373509, 584454485844541718, 51256293341752583499, 4880654469385955209092, 501471626403154217825457, 55300894427785157597436786

Offset: 0

Gus Wiseman, May 22 2018

nonn

			The a(2) = 7 hypertrees are the following:
{}
{{1}}
{{2}}
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A030019, A035053, A134954, A134955, A134956, A134957, A134958, A134959, A144959, A304386, A304867, A304911, A304912, A304918, A304968, A304970.

\\ here b(n) is A134958 with b(1)=1.
b(n)=if(n<2, n>=0, 2^n*sum(i=0, n, stirling(n-1, i, 2)*n^(i-1)));
a(n)=sum(k=0, n, binomial(n, k)*b(k)); \\ Andrew Howroyd, Aug 27 2018

Binomial transform of b(1) = 1, b(n) = A134958(n) otherwise.

A304970 Number of unlabeled hypertrees with up to n vertices and without singleton edges.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 39, 98, 263, 759, 2299, 7259, 23649, 79057, 269629, 935328, 3290260, 11714285, 42139053, 152963037, 559697097, 2062574000, 7649550572, 28534096988, 106994891146, 403119433266, 1525466082179, 5795853930652, 22102635416716, 84579153865570

Offset: 0

Gus Wiseman, May 22 2018

nonn

			Non-isomorphic representatives of the a(4) = 8 hypertrees are the following:
{}
{{1,2}}
{{1,2,3}}
{{1,2,3,4}}
{{1,3},{2,3}}
{{1,4},{2,3,4}}
{{1,3},{2,4},{3,4}}
{{1,4},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A030019, A035053, A134954, A134955, A134956, A134957, A134958, A134959, A144959, A304386, A304867, A304911, A304912, A304918, A304968, A304970.

\\ here b(n) is A007563 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(v)))); v}
seq(n)={my(u=b(n)); Vec(1 + (x*Ser(EulerT(u))*(1-x*Ser(u)))/(1-x))} \\ Andrew Howroyd, Aug 27 2018

Partial sums of A035053 if we assume A035053(1) = 0.

a(n) = A304937(n) + 1 for n > 0.

A304388 a(n) = 144*2^n - 20 (n>=1).

Original entry on oeis.org

268, 556, 1132, 2284, 4588, 9196, 18412, 36844, 73708, 147436, 294892, 589804, 1179628, 2359276, 4718572, 9437164, 18874348, 37748716, 75497452, 150994924, 301989868, 603979756, 1207959532, 2415919084, 4831838188, 9663676396, 19327352812, 38654705644, 77309411308

Offset: 1

Emeric Deutsch, May 13 2018

a(n) is the second Zagreb index of the dendrimer nanostar NS1[n], defined pictorially in the Ashrafi et al. reference (Ns1[3] is shown in Fig. 1) or in the Ahmadi et al. reference (Fig. 1).
The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.
The M-polynomial of NS1[n] is M(NS1[n]; x,y) = xy^4 + (9*2^n +3)x^2*y^2 + (18*2^n - 12)x^2*y^3 + 3x^3*y^4.

Colin Barker, Table of n, a(n) for n = 1..1000
M. B. Ahmadi and M. Sadeghimehr, Atom bond connectivity index of an infinite class NS1[n] of dendrimer nanostars, Optoelectronics and Advanced Materials, 4(7):1040-1042 July 2010.
Ali Reza Ashrafi and Parisa Nikzad, Kekulé index and bounds of energy for nanostar dendrimers, Digest J. of Nanomaterials and Biostructures, 4, No. 2, 2009, 383-388.
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A304386, A304387, A304389.

List([1..40],n->144*2^n-20); # Muniru A Asiru, May 13 2018

Maple
```
seq(144*2^n-20, n = 1 .. 40);
```

LinearRecurrence[{3,-2},{268,556},30] (* Harvey P. Dale, Nov 02 2021 *)

a(n) = 144*2^n - 20; \\ Altug Alkan, May 13 2018

Vec(4*x*(67 - 62*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, May 18 2018

From Colin Barker, May 18 2018: (Start)
G.f.: 4*x*(67 - 62*x) / ((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1) - 2*a(n-2) for n>2.
(End)

A304389 a(n) = 126*2^n - 22 (n>=1).

Original entry on oeis.org

230, 482, 986, 1994, 4010, 8042, 16106, 32234, 64490, 129002, 258026, 516074, 1032170, 2064362, 4128746, 8257514, 16515050, 33030122, 66060266, 132120554, 264241130, 528482282, 1056964586, 2113929194, 4227858410, 8455716842, 16911433706, 33822867434, 67645734890, 135291469802, 270582939626, 541165879274

Offset: 1

Emeric Deutsch, May 13 2018

a(n) is the first Zagreb index of the dendrimer nanostar NS1[n], defined pictorially in the Ashrafi et al. reference (Ns1[3] is shown in Fig. 1) or in the Ahmadi et al. reference (Fig. 1).
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of NS1[n] is M(NS1[n]; x,y) = xy^4 + (9*2^n + 3)x^2*y^2 + (18*2^n - 12)x^2*y^3 + 3x^3*y^4 .

Colin Barker, Table of n, a(n) for n = 1..1000
M. B. Ahmadi and M. Sadeghimehr, Atom bond connectivity index of an infinite class NS1[n] of dendrimer nanostars, Optoelectronics and Advanced Materials, 4(7):1040-1042 July 2010.
Ali Reza Ashrafi and Parisa Nikzad, Kekulé index and bounds of energy for nanostar dendrimers, Digest J. of Nanomaterials and Biostructures, 4, No. 2, 2009, 383-388.
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A304386, A304387, A304388.

List([1..40],n->126*2^n-22); # Muniru A Asiru, May 13 2018

Maple
```
seq(126*2^n-22, n = 1 .. 40);
```

a(n) = 126*2^n - 22; \\ Altug Alkan, May 13 2018

Vec(2*x*(115 - 104*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, May 18 2018

From Colin Barker, May 18 2018: (Start)
G.f.: 2*x*(115 - 104*x) / ((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1) - 2*a(n-2) for n>2.
(End)

A305004 Number of labeled hypertrees (connected acyclic antichains) spanning some subset of {1,...,n} without singleton edges.

Original entry on oeis.org

1, 1, 2, 8, 52, 507, 6844, 118582, 2504856, 62370530, 1788082154, 57997339633, 2099638691440, 83922479506504, 3670657248913386, 174387350448735878, 8942472292255441104, 492294103555090048459, 28958704109012732921524

Offset: 0

Gus Wiseman, May 23 2018

nonn

			The a(3) = 8 hypertrees:
{}
{{1,2}}
{{1,3}}
{{2,3}}
{{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A030019, A035053, A134958, A134959, A304386, A304939, A304968, A304970.

\\ here b(n) is A030019 with b(1)=0.
b(n)=if(n<2, n==0, sum(i=0, n, stirling(n-1, i, 2)*n^(i-1)));
a(n)=sum(k=0, n, binomial(n, k)*b(k)); \\ Andrew Howroyd, Aug 27 2018

a(n > 0) = A304939(n) + 1.

Binomial transform of A030019 if we assume A030019(1) = 0.

cp's OEIS Frontend

A304387 a(n) = 27*2^n - 5.

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A304968 Number of labeled hypertrees spanning some subset of {1,...,n}, with singleton edges allowed.

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A304970 Number of unlabeled hypertrees with up to n vertices and without singleton edges.

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A304388 a(n) = 144*2^n - 20 (n>=1).

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A304389 a(n) = 126*2^n - 22 (n>=1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

PARI

PARI

Formula

A305004 Number of labeled hypertrees (connected acyclic antichains) spanning some subset of {1,...,n} without singleton edges.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula