login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A277504 Array T(n, k) giving the number of reversible strings with n beads of k colors, read by antidiagonals. 12
1, 2, 1, 3, 3, 1, 4, 6, 6, 1, 5, 10, 18, 10, 1, 6, 15, 40, 45, 20, 1, 7, 21, 75, 136, 135, 36, 1, 8, 28, 126, 325, 544, 378, 72, 1, 9, 36, 196, 666, 1625, 2080, 1134, 136, 1, 10, 45, 288, 1225, 3996, 7875, 8320, 3321, 272, 1, 11, 55, 405, 2080, 8575, 23436, 39375, 32896, 9963, 528, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

From Petros Hadjicostas, Jul 07 2018: (Start)

Column k of this array is the "BIK" (reversible, indistinct, unlabeled) transform of k,0,0,0,....

Consider the input sequence (c_k(n): n >= 1) with g.f. C_k(x) = Sum_{n>=1} c_k(n)*x^n. Let a_k(n) = BIK(c_k(n): n >= 1) be the output sequence under Bower's BIK transform. It can proved that the g.f. of BIK(c_k(n): n >= 1) is A_k(x) = (1/2)*(C_k(x)/(1-C_k(x)) + (C_k(x^2) + C_k(x))/(1-C_k(x^2))). (See the comments for sequence A001224.)

For column k of this two-dimensional array, the input sequence is defined by c_k(1) = k and c_k(n) = 0 for n >= 1. Thus, C_k(x) = k*x, and hence the g.f. of column k is (1/2)*(C_k(x)/(1-C_k(x)) + (C_k(x^2) + C_k(x))/(1-C_k(x^2))) = (1/2)*(k*x/(1-k*x) + (k*x^2 + k*x)/(1-k*x^2)) = (2 + (1-k)*x - 2*k*x^2)*k*x/(2*(1-k*x^2)*(1-k*x)).

Using the first form the g.f. above and the expansion 1/(1-y) = 1 + y + y^2 + ..., we can easily prove J.-F. Alcover's formula T(n,k) = (k^n + k^((n + mod(n,2))/2))/2.

(End)

REFERENCES

See A005418.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275

C. G. Bower, Transforms (2)

FORMULA

T(n,k) = (k^n + k^((n + mod(n,2))/2))/2.

G.f. for column k: (2 + (1-k)*x - 2*k*x^2)*k*x/(2*(1-k*x^2)*(1-k*x)). - Petros Hadjicostas, Jul 07 2018

EXAMPLE

Array begins:

1   2     3      4       5       6          7         8 ...

1   3     6     10      15      21         28        36 ...

1   6    18     40      75     126        196       288 ...

1  10    45    136     325     666       1225      2080 ...

1  20   135    544    1625    3996       8575     16640 ...

1  36   378   2080    7875   23436      58996    131328 ...

1  72  1134   8320   39375  140616     412972   1050624 ...

1 136  3321  32896  195625  840456    2883601   8390656 ...

1 272  9963 131584  978125  5042736  20185207  67125248 ...

1 528 29646 524800 4884375 30236976 141246028 536887296 ...

...

MATHEMATICA

T[n_, k_] := (n^k + n^((k + Mod[k, 2])/2))/2;

Table[T[n - k + 1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* updated Jul 10 2018 *)

CROSSREFS

Columns 2-6 are A005418, A032120, A032121, A032122, A056308.

Rows 2-6 are A000217 (triangular numbers), A002411 (pentagonal pyramidal numbers), A037270, A168178, A071232.

Transpose is A284979.

Cf. A284871, A284949.

Sequence in context: A192001 A122176 A159881 * A319539 A098546 A126277

Adjacent sequences:  A277501 A277502 A277503 * A277505 A277506 A277507

KEYWORD

nonn,tabl

AUTHOR

Jean-Fran├žois Alcover, Oct 18 2016

EXTENSIONS

Array transposed for greater consistency by Andrew Howroyd, Apr 04 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 18 11:56 EDT 2018. Contains 316321 sequences. (Running on oeis4.)