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A284877 Irregular triangle T(n,k) for 1 <= k <= n/2 + 1: T(n,k) = number of double palindrome structures of length n using exactly k different symbols. 7
1, 1, 1, 1, 3, 1, 7, 2, 1, 15, 5, 1, 25, 21, 3, 1, 49, 42, 7, 1, 79, 122, 44, 4, 1, 129, 225, 90, 9, 1, 211, 570, 375, 80, 5, 1, 341, 990, 715, 165, 11, 1, 517, 2321, 2487, 930, 132, 6, 1, 819, 3913, 4550, 1820, 273, 13, 1, 1275, 8827, 14350, 8330, 2009, 203, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

A double palindrome is the concatenation of two palindromes. Permuting the symbols will not change the structure. For the purposed of this sequence, valid palindromes include both the empty word and a singleton symbol.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..960

FORMULA

T(n, k) = (Sum_{j=0..k} (-1)^j * binomial(k, j) * A284873(n, k-j)) / k!.

T(n, k) = r(n, k) - Sum_{d|n, d<n} phi(n/d) * T(d,k) where r(2d, k) = d *(stirling2(d,k)+stirling2(d,k+1)), r(2d+1, k) = (2d+1)*stirling2(d+1,k).

EXAMPLE

Triangle starts:

1

1     1

1     3

1     7      2

1    15      5

1    25     21       3

1    49     42       7

1    79    122      44       4

1   129    225      90       9

1   211    570     375      80       5

1   341    990     715     165      11

1   517   2321    2487     930     132      6

1   819   3913    4550    1820     273     13

1  1275   8827   14350    8330    2009    203      7

1  1863  14480   25515   15750    3990    420     15

1  2959  31802   75724   64004   23296   3920    296     8

1  4335  51425  132090  118167   44982   7854    612    17

1  6703 110928  376779  445275  229257  57078   7074   414   9

1  9709 177270  647995  807975  433713 111720  14250   855  19

1 15067 377722 1798175 2892470 2023135 698670 126300 12000 560 10

....

The first few structures are:

n = 1: a => 1

n = 2: aa; ab => 1 + 1

n = 3: aaa; aab, aba, abb => 1 + 3

n = 4: aaaa; aaab, aaba, aabb, abaa, abab, abba, abbb; abac, abcb => 1 + 7 + 2

MATHEMATICA

r[d_, k_]:=If[OddQ[d], d*k^((d + 1)/2), (d/2)*(k + 1)*k^(d/2)]; a[n_, k_]:= r[n, k] - Sum[If[d<n, EulerPhi[n/d] a[d, k], 0], {d, Divisors[n]}]; T[n_, k_]:=(Sum[(-1)^j * Binomial[k, j] * a[n, k - j], {j, 0, k}]) / k!; Table[T[n, k], {n, 25}, {k, n/2 + 1}] // Flatten (* Indranil Ghosh, Apr 07 2017 *)

PROG

(PARI)

r(d, k)=if (d % 2 == 0, (d/2)*(stirling(d/2, k, 2)+stirling(d/2+1, k, 2)), d*stirling((d+1)/2, k, 2));

a(n, k) = r(n, k) - sumdiv(n, d, if (d<n, eulerphi(n/d)*a(d, k), 0));

for(n=1, 15, for(k=1, n/2+1, print1( a(n, k), ", "); ); print(); );

(Python)

from sympy import totient, divisors, binomial, factorial

def r(d, k): return (d//2)*(k + 1)*k**(d//2) if d%2 == 0 else d*k**((d + 1)//2)

def a(n, k): return r(n, k) - sum([totient(n//d)*a(d, k) for d in divisors(n) if d<n])

def T(n, k): return (sum([(-1)**j * binomial(k, j) * a(n, k - j) for j in range(k + 1)]))//factorial(k)

for n in range(1, 21): print([T(n, k) for k in range(1, n//2 + 2)]) # Indranil Ghosh, Apr 07 2017

CROSSREFS

Columns k=2..4 are A328688, A328689, A328690.

Row sums are A165137.

Partial row sums include A180249, A328692, A328693.

Cf. A284873, A284826.

Sequence in context: A146007 A061782 A074625 * A169659 A091724 A016576

Adjacent sequences:  A284874 A284875 A284876 * A284878 A284879 A284880

KEYWORD

nonn,tabf

AUTHOR

Andrew Howroyd, Apr 04 2017

STATUS

approved

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Last modified June 13 00:57 EDT 2021. Contains 344980 sequences. (Running on oeis4.)