A134530 -id:A134530

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A134531

G.f.: Sum_{n>=0} a(n)*x^n/(n!*2^(n*(n-1)/2)) = log( Sum_{n>=0} x^n/(n!*2^(n*(n-1)/2)) ).

+10
11

0, 1, -1, 5, -79, 3377, -362431, 93473345, -56272471039, 77442176448257, -239804482525402111, 1650172344732021412865, -24981899010711376986398719, 825164608171793476724052668417, -59053816996641612758331731690504191, 9102696765174239045811746247171452452865

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..77

Kimmo Berg, Complexity of solution structures in nonlinear pricing, Ann. Oper. Res. 206, 23-37 (2013).

Sergei Chmutov, Maxim Kazarian and Sergey Lando, Polynomial graph invariants and the KP hierarchy, arXiv:1803.09800 [math.CO], 2018.

FORMULA

Equals column 0 of triangle A134530, which is the matrix log of triangle A111636, where A111636(n,k) = (2^k)^(n-k)*C(n,k).

From Peter Bala, Apr 01 2013: (Start)

Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)). Then a generating function for this sequence (but with a different offset) is E(x)/E(2*x) = Sum_{n >= 0} a(n-1)*x^n/(n!*2^C(n,2)) = 1 - x + 5*x^2/(2!*2) - 79*x^3/(3!*2^3) + 3377*x^4/(4!*2^6) - ....

Recurrence equation:

a(n) = 1 - Sum_{k = 1..n-1} 2^(k*(n-k))*C(n-1,k-1)*a(k) with a(1) = 1. (End)

a(n) = (-1)^(n-1)*A003025(n)/n. - Andrew Howroyd, Jan 07 2022

EXAMPLE

Let g.f. G(x) = Sum_{n>=0} a(n)*x^n/[ n! * 2^(n*(n-1)/2) ]

then exp(G(x)) = Sum_{n>=0} x^n/[ n! * 2^(n*(n-1)/2) ];

G.f.: G(x) = x - x^2/4 + 5x^3/48 - 79x^4/1536 + 3377x^5/122880 + ...

exp(G(x)) = 1 + x + x^2/4 + x^3/48 + x^4/1536 + x^5/122880 + ...

MATHEMATICA

a[0] = 0;

a[n_] := a[n] = 1 - Sum[2^(k(n-k)) Binomial[n-1, k-1] a[k], {k, 1, n-1}];

Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 26 2018 *)

PROG

(PARI) {a(n)=n!*2^(n*(n-1)/2)*polcoeff(log(sum(k=0, n, x^k/(k!*2^(k*(k-1)/2)))+x*O(x^n)), n)}

CROSSREFS

Cf. related triangles: A134530, A111636.

Cf. A003025, A011266, A118197 (variant).

KEYWORD

sign,easy

AUTHOR

Paul D. Hanna, Oct 30 2007

STATUS

approved

A111636

Triangle read by rows: T(n,k) (0<=k<=n) is the number of labeled graphs having k blue nodes and n-k green ones and only nodes of different colors can be joined by an edge.

+10
6

1, 1, 1, 1, 4, 1, 1, 12, 12, 1, 1, 32, 96, 32, 1, 1, 80, 640, 640, 80, 1, 1, 192, 3840, 10240, 3840, 192, 1, 1, 448, 21504, 143360, 143360, 21504, 448, 1, 1, 1024, 114688, 1835008, 4587520, 1835008, 114688, 1024, 1, 1, 2304, 589824, 22020096, 132120576, 132120576, 22020096, 589824, 2304, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Row sums yield A047863. T(2n,n) = A111637(n). T(n,1) = A001787(n).

REFERENCES

H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 88, Eq. 3.11.2.

LINKS

Table of n, a(n) for n=0..54.

S. R. Finch, Bipartite, k-colorable and k-colored graphs

S. R. Finch, Bipartite, k-colorable and k-colored graphs, June 5, 2003. [Cached copy, with permission of the author]

W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

FORMULA

T(n, k)=2^[k(n-k)]*C(n, k).

Matrix log yields triangle A134530, where A134530(n,k) = A134531(n-k)*(2^k)^(n-k)*C(n,k). - Paul D. Hanna, Nov 11 2007

From Peter Bala, Aug 13 2012: (Start)

Let f(n) = n!*2^binomial(n,2) = A011266(n). Then T(n,k) = f(n)/(f(k)*f(n-k)).

E.g.f.: sum {n >= 0} exp(2^n*t*x)*x^n/n! = 1 + (1+t)*x + (1+4*t+t^2)*x^2/2! + ....

O.g.f.: sum {n >= 0} x^n/(1-2^n*t*x)^(n+1) = 1 + (1+t)*x + (1+4*t+t^2)*x^2 + .... O.g.f. for column k: 1/(1-2^k*x)^(k+1).

Recurrence equation: T(n,k) = 2^k*T(n-1,k) + 2^(n-k)*T(n-1,k-1).

Column k = 2: A038845. Column k = 3: A140802. Sum {k = 0..n} k*T(n,k) = n*A000684(n).

(End)

From Peter Bala, Apr 09 2013: (Start)

Let E(x) = Sum_{n >= 0} x^n/(n!*2^C(n,2)) = 1 + x + x^2/(2!*2) + x^3/(3!*2^3) + .... Then a generating function for this sequence is E(z)*E(x*z) = 1 + (1 + x)*z + (1 + 4*x + x^2)*z^2/(2!*2) + (1 + 12*x + 12*x^2 + x^3)*z^3/(3!*2^3) + .... Cf. Pascal's triangle A007318 with an e.g.f. of exp(z)*exp(x*z).

This is a generalized Riordan array (E(x), x) with respect to the sequence n!*2^C(n,2), as defined by Wang and Wang.

The n-th power of this triangle has a generating function E(z)^n*E(x*z). See A224069 for the inverse array (n = -1).

The n-th row is a log-concave sequence and hence unimodal.

The row polynomials satisfy the recurrence equation R(n+1,x) = 2^n*x*R(n,x/2) + R(n,2*x) with R(0,x) = 1, as well as R'(n,2*x) = n*2^(n-1)*R(n-1,x) (the ' denotes differentiation w.r.t. x). The row polynomials appear to have only real zeros.

Sum_{k = 0..n} (-1)^k*T(2*n+1,k) = 0;

Sum_{k = 0..n} (-1)^k*2^k*T(2*n,k) = 0;

Sum_{k = 0..n} 2^k*T(n,k) = A000684(n). (End)

T(n,k+1) = Product_{i=0..k} (T(n-i,1)/T(i+1,1)) for 0 <= k < n. - Werner Schulte, Nov 13 2018

EXAMPLE

T(2,1)=4 because we have B G, B--G, G B and G--B, where B (G) stands for a blue (green) node and -- denotes an edge.

Triangle starts:

1, 1;

1, 4, 1;

1, 12, 12, 1;

1, 32, 96, 32, 1;

...

MAPLE

T:=(n, k)->binomial(n, k)*2^(k*(n-k)): for n from 0 to 9 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

MATHEMATICA

nn=6; f[x_, y_]:=Sum[Exp[x 2^n](y x)^n/n!, {n, 0, nn}]; Range[0, nn]!CoefficientList[Series[f[x, y], {x, 0, nn}], {x, y}]//Grid (* Geoffrey Critzer, Sep 07 2013 *).

CROSSREFS

Cf. A047863, A111637, A001787.

Cf. A134530 (matrix log), A134531.

A000684, A011266, A038845, A140802. A224069 (matrix inverse).

KEYWORD

nonn,tabl,easy

AUTHOR

Emeric Deutsch, Aug 09 2005

STATUS

approved

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