A284871 -id:A284871 - cp's OEIS frontend

A277504 Array read by descending antidiagonals: T(n,k) is the number of unoriented strings with n beads of k or fewer colors.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 6, 1, 0, 1, 5, 10, 18, 10, 1, 0, 1, 6, 15, 40, 45, 20, 1, 0, 1, 7, 21, 75, 136, 135, 36, 1, 0, 1, 8, 28, 126, 325, 544, 378, 72, 1, 0, 1, 9, 36, 196, 666, 1625, 2080, 1134, 136, 1, 0, 1, 10, 45, 288, 1225, 3996, 7875, 8320, 3321, 272, 1, 0

Offset: 0

Jean-François Alcover, Oct 18 2016

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)

			Array begins with T(0,0):
1 1   1     1      1       1        1         1         1          1 ...
0 1   2     3      4       5        6         7         8          9 ...
0 1   3     6     10      15       21        28        36         45 ...
0 1   6    18     40      75      126       196       288        405 ...
0 1  10    45    136     325      666      1225      2080       3321 ...
0 1  20   135    544    1625     3996      8575     16640      29889 ...
0 1  36   378   2080    7875    23436     58996    131328     266085 ...
0 1  72  1134   8320   39375   140616    412972   1050624    2394765 ...
0 1 136  3321  32896  195625   840456   2883601   8390656   21526641 ...
0 1 272  9963 131584  978125  5042736  20185207  67125248  193739769 ...
0 1 528 29646 524800 4884375 30236976 141246028 536887296 1743421725 ...
...

See A005418.

Robert A. Russell, Antidiagonals n=0..52, flattened (antidiagonals 1..50 from Andrew Howroyd)
C. G. Bower, Transforms (2)

Columns 0-6 are A000007, A000012, A005418(n+1), A032120, A032121, A032122, A056308.
Rows 0-20 are A000012, A001477, A000217 (triangular numbers), A002411 (pentagonal pyramidal numbers), A037270, A168178, A071232, A168194, A071231, A168372, A071236, A168627, A071235, A168663, A168664, A170779, A170780, A170790, A170791, A170801, A170802.
Main diagonal is A275549.
Transpose is A284979.
Cf. A284871, A284949.
Cf. A003992 (oriented), A293500 (chiral), A321391 (achiral).

[[n le 0 select 1 else ((n-k)^k + (n-k)^Ceiling(k/2))/2: k in [0..n]]: n in [0..15]]; // G. C. Greubel, Nov 15 2018

Table[If[n>0, ((n-k)^k + (n-k)^Ceiling[k/2])/2, 1], {n, 0, 15}, {k, 0, n}] // Flatten (* updated Jul 10 2018 *) (* Adapted to T(0,k)=1 by Robert A. Russell, Nov 13 2018 *)

for(n=0,15, for(k=0,n, print1(if(n==0,1, ((n-k)^k + (n-k)^ceil(k/2))/2), ", "))) \\ G. C. Greubel, Nov 15 2018

T(n,k) = {(k^n + k^ceil(n/2)) / 2} \\ Andrew Howroyd, Sep 13 2019

T(n,k) = [n==0] + [n>0] * (k^n + k^ceiling(n/2)) / 2. [Adapted to T(0,k)=1 by Robert A. Russell, Nov 13 2018]
G.f. for column k: (1 - binomial(k+1,2)*x^2) / ((1-k*x)*(1-k*x^2)). - Petros Hadjicostas, Jul 07 2018 [Adapted to T(0,k)=1 by Robert A. Russell, Nov 13 2018]
From Robert A. Russell, Nov 13 2018: (Start)
T(n,k) = (A003992(k,n) + A321391(n,k)) / 2.
T(n,k) = A003992(k,n) - A293500(n,k) = A293500(n,k) + A321391(n,k).
G.f. for row n: (Sum_{j=0..n} S2(n,j)*j!*x^j/(1-x)^(j+1) + Sum_{j=0..ceiling(n/2)} S2(ceiling(n/2),j)*j!*x^j/(1-x)^(j+1)) / 2, where S2 is the Stirling subset number A008277.
G.f. for row n>0: x*Sum_{k=0..n-1} A145882(n,k) * x^k / (1-x)^(n+1).
E.g.f. for row n: (Sum_{k=0..n} S2(n,k)*x^k + Sum_{k=0..ceiling(n/2)} S2(ceiling(n/2),k)*x^k) * exp(x) / 2, where S2 is the Stirling subset number A008277.
T(0,k) = 1; T(1,k) = k; T(2,k) = binomial(k+1,2); for n>2, T(n,k) = k*(T(n-3,k)+T(n-2,k)-k*T(n-1,k)).
For k>n, T(n,k) = Sum_{j=1..n+1} -binomial(j-n-2,j) * T(n,k-j). (End)

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

Origin changed to T(0,0) by Robert A. Russell, Nov 13 2018

A276544 Triangle read by rows: T(n,k) = number of primitive (aperiodic) reversible string structures with n beads using exactly k different colors.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 4, 1, 0, 9, 15, 6, 1, 0, 16, 49, 37, 9, 1, 0, 35, 160, 183, 76, 12, 1, 0, 66, 498, 876, 542, 142, 16, 1, 0, 133, 1544, 3930, 3523, 1346, 242, 20, 1, 0, 261, 4715, 17179, 21392, 11511, 2980, 390, 25, 1

Offset: 1

Andrew Howroyd, Apr 09 2017

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

			Triangle starts
1
0   1
0   2    1
0   4    4     1
0   9   15     6     1
0  16   49    37     9     1
0  35  160   183    76    12    1
0  66  498   876   542   142   16   1
0 133 1544  3930  3523  1346  242  20  1
0 261 4715 17179 21392 11511 2980 390 25 1
...
Primitive reversible word structures are:
n=1: a => 1
n=2: ab => 1
n=3: aab, aba; abc => 2 + 1
n=4: aaab, aaba, aabb, abba => 4 (k=2)
     aabc, abac, abbc, abca => 4 (k=3)

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Columns 2-6 are A056336, A056337, A056338, A056339, A056340.
Partial row sums include A056331, A056332, A056333, A056334, A056335.
Row sums are A276549.
Cf. A284871, A284949, A277504, A137651, A107424, A276543.

Ach[n_, k_] := Ach[n, k] = Switch[k, 0, If[n == 0, 1, 0], 1, If[n > 0, 1, 0], _, If[OddQ[n], Sum[Binomial[(n - 1)/2, i] Ach[n - 1 - 2 i, k - 1], {i, 0, (n - 1)/2}], Sum[Binomial[n/2 - 1, i] (Ach[n - 2 - 2 i, k - 1] + 2^i Ach[n - 2 - 2 i, k - 2]), {i, 0, n/2 - 1}]]]
T[n_, k_] := DivisorSum[n, MoebiusMu[n/#] (StirlingS2[#, k] + Ach[#, k])/2& ];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 29 2018, after Robert A. Russell and Andrew Howroyd *)

\\ here Ach is A304972 as matrix.
Ach(n,m=n)={my(M=matrix(n, m, i, k, i>=k)); for(i=3, n, for(k=2, m, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
T(n,m=n)={my(M=matrix(n, m, i, k, stirling(i, k, 2)) + Ach(n,m)); matrix(n, m, i, k, sumdiv(i, d, moebius(i/d)*M[d,k]))/2}
{ my(A=T(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 09 2020

T(n, k) = Sum_{d|n} mu(n/d) * A284949(d, k).

A045625 Number of n-bead black-white reversible strings with fundamental period n.

Original entry on oeis.org

1, 2, 1, 4, 7, 18, 29, 70, 126, 266, 507, 1054, 2037, 4158, 8183, 16488, 32760, 65790, 131026, 262654, 524265, 1049524, 2097119, 4196350, 8388450, 16781292, 33554367, 67116784, 134217657, 268451838, 536870247, 1073774590, 2147483520

Offset: 0

David W. Wilson

nonn

Column 2 of A284871.

Moebius transform of A005418(n+1) (Christian Bower).

A056314 Number of primitive (aperiodic) reversible strings with n beads using a maximum of three different colors.

Original entry on oeis.org

3, 3, 15, 39, 132, 357, 1131, 3276, 9945, 29508, 88935, 265668, 798252, 2391441, 7177584, 21523320, 64579920, 193709763, 581160255, 1743392040, 5230264026, 15690529437, 47071855131, 141214764600

Offset: 1

Marks R. Nester

nonn

A string and its reverse are considered to be equivalent.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Column 3 of A284871.

Cf. A045625.

Sum mu(d)*A032120(n/d) where d|n.

A056315 Number of primitive (aperiodic) reversible strings with n beads using a maximum of four different colors.

Original entry on oeis.org

4, 6, 36, 126, 540, 2034, 8316, 32760, 131544, 524250, 2099196, 8388450, 33562620, 134217594, 536903100, 2147483520, 8590065660, 34359735816, 137439477756, 549755813250, 2199025344348, 8796093020154

Offset: 1

Marks R. Nester

nonn

A string and its reverse are considered to be equivalent.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Column 4 of A284871.

Cf. A045625, A056314.

Sum mu(d)*A032121(n/d) where d|n.

A056316 Number of primitive (aperiodic) reversible strings with n beads using a maximum of five different colors.

Original entry on oeis.org

5, 10, 70, 310, 1620, 7790, 39370, 195300, 978050, 4882740, 24421870, 122069940, 610390620, 3051757490, 15258982680, 76293945000, 381470703120, 1907348623450, 9536748046870, 47683715818440

Offset: 1

Marks R. Nester

nonn

A string and its reverse are considered to be equivalent.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Column 5 of A284871.

Cf. A045625.

Sum mu(d)*A032122(n/d) where d|n.

A056317 Number of primitive (aperiodic) reversible strings with n beads using a maximum of six different colors.

Original entry on oeis.org

6, 15, 120, 645, 3990, 23295, 140610, 839790, 5042610, 30232965, 181421850, 1088390415, 6530486970, 39182081385, 235093327980, 1410554953080, 8463334761210, 50779978307010, 304679900238330

Offset: 1

Marks R. Nester

nonn

A string and its reverse are considered to be equivalent.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Column 6 of A284871.

Cf. A045625.

sum mu(d)*A056308(n/d) where d|n.

cp's OEIS Frontend

A277504 Array read by descending antidiagonals: T(n,k) is the number of unoriented strings with n beads of k or fewer colors.

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A276544 Triangle read by rows: T(n,k) = number of primitive (aperiodic) reversible string structures with n beads using exactly k different colors.

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A045625 Number of n-bead black-white reversible strings with fundamental period n.

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A056314 Number of primitive (aperiodic) reversible strings with n beads using a maximum of three different colors.

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A056315 Number of primitive (aperiodic) reversible strings with n beads using a maximum of four different colors.

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A056316 Number of primitive (aperiodic) reversible strings with n beads using a maximum of five different colors.

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A056317 Number of primitive (aperiodic) reversible strings with n beads using a maximum of six different colors.

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