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Revision History for A111636

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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.
(history; published version)

#33 by Alois P. Heinz at Mon Apr 03 09:24:58 EDT 2023

STATUS

editing

approved

#32 by Alois P. Heinz at Mon Apr 03 09:24:09 EDT 2023

COMMENTS

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

#31 by Alois P. Heinz at Mon Apr 03 09:22:31 EDT 2023

EXAMPLE

1;

1, 1;

1, 4, 1;

1, 12, 12, 1;

1, 32, 96, 32, 1;

...

STATUS

approved

editing

#30 by N. J. A. Sloane at Sun Apr 14 13:11:54 EDT 2019

STATUS

editing

approved

#29 by N. J. A. Sloane at Sun Apr 14 13:11:52 EDT 2019

LINKS

S. R. Finch, <a href="/A191371/a191371.pdf">Bipartite, k-colorable and k-colored graphs</a>, June 5, 2003. [Cached copy, with permission of the author]

STATUS

approved

editing

#28 by Alois P. Heinz at Tue Nov 13 16:12:15 EST 2018

STATUS

editing

approved

#27 by Alois P. Heinz at Tue Nov 13 16:07:02 EST 2018

DATA

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

Discussion

Tue Nov 13

16:07

Alois P. Heinz: filled last row for the tabl display.

16:09

Alois P. Heinz: Thank you for finding the missing term!

#26 by Alois P. Heinz at Tue Nov 13 16:05:33 EST 2018

DATA

OFFSET

0,45

STATUS

proposed

editing

#25 by Michel Marcus at Tue Nov 13 14:49:12 EST 2018

STATUS

editing

proposed

#24 by Michel Marcus at Tue Nov 13 14:49:07 EST 2018

LINKS

W. Wang and T. Wang, <a href="httphttps://www.sciencedirectdoi.com/science/articleorg/pii10.1016/S0012365X07010321j.disc.2007.12.037">Generalized Riordan array</a>, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

FORMULA

Let E(x) = sum 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).

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

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

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

STATUS

proposed

editing