A327878
Irregular triangle read by rows: T(n,k) is the number of primitive (period n) periodic palindromes using exactly k different symbols, 1 <= k <= 1 + floor(n/2).
Original entry on oeis.org
1, 0, 1, 0, 2, 0, 3, 3, 0, 6, 6, 0, 7, 21, 12, 0, 14, 36, 24, 0, 18, 90, 132, 60, 0, 28, 150, 240, 120, 0, 39, 339, 900, 960, 360, 0, 62, 540, 1560, 1800, 720, 0, 81, 1149, 4968, 9300, 7920, 2520, 0, 126, 1806, 8400, 16800, 15120, 5040, 0, 175, 3765, 24588, 71400, 103320, 73080, 20160
Offset: 1
Triangle begins:
1;
0, 1;
0, 2;
0, 3, 3;
0, 6, 6;
0, 7, 21, 12;
0, 14, 36, 24;
0, 18, 90, 132, 60;
0, 28, 150, 240, 120;
0, 39, 339, 900, 960, 360;
0, 62, 540, 1560, 1800, 720;
0, 81, 1149, 4968, 9300, 7920, 2520;
0, 126, 1806, 8400, 16800, 15120, 5040;
0, 175, 3765, 24588, 71400, 103320, 73080, 20160;
...
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T(n,k) = {sumdiv(n, d, moebius(n/d) * k! * (stirling((d+1)\2,k,2) + stirling(d\2+1,k,2)))/2}
A056463
Number of primitive (aperiodic) palindromes using exactly two different symbols.
Original entry on oeis.org
0, 0, 2, 2, 6, 4, 14, 12, 28, 24, 62, 54, 126, 112, 246, 240, 510, 476, 1022, 990, 2030, 1984, 4094, 4020, 8184, 8064, 16352, 16254, 32766, 32484, 65534, 65280, 131006, 130560, 262122, 261576, 524286, 523264, 1048446, 1047540, 2097150, 2094988, 4194302, 4192254
Offset: 1
- 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]
-
seq(n)={Vec(sum(k=1, n\3, moebius(k)*2*x^(3*k)/((1 - 2*x^(2*k))*(1 - x^k)) + O(x*x^n)), -n)} \\ Andrew Howroyd, Sep 29 2019
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from sympy import mobius, divisors
def A056463(n): return sum(mobius(n//d)*((1<<(d+1>>1))-2) for d in divisors(n, generator=True)) # Chai Wah Wu, Feb 18 2024
A056464
Number of primitive (aperiodic) palindromes using exactly three different symbols.
Original entry on oeis.org
0, 0, 0, 0, 6, 6, 36, 36, 150, 144, 540, 534, 1806, 1770, 5790, 5760, 18150, 17994, 55980, 55830, 170970, 170466, 519156, 518580, 1569744, 1567944, 4733670, 4732014, 14250606, 14244660, 42850116, 42844320, 128746410, 128728800
Offset: 1
- 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]
-
with(numtheory):with(combinat,stirling2):A056454:=n->3!*stirling2(floor((n+1)/2),3);A056464:=n->add(mobius(d)*A056454(n/d),d=divisors(n)); # C. Ronaldo
A056465
Number of primitive (aperiodic) palindromes using exactly four different symbols.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 24, 24, 240, 240, 1560, 1560, 8400, 8376, 40824, 40800, 186480, 186240, 818520, 818280, 3497976, 3496440, 14676024, 14674440, 60780720, 60772320, 249401640, 249393480, 1016542800, 1016501736, 4123173624, 4123132800, 16664093400, 16663908480
Offset: 1
- 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]
A056466
Number of primitive (aperiodic) palindromes using exactly five different symbols.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 120, 120, 1800, 1800, 16800, 16800, 126000, 126000, 834120, 834000, 5103000, 5102880, 29607600, 29605800, 165528000, 165526200, 901020120, 901003320, 4809004080, 4808987400, 25292030400, 25291904280, 131542866000, 131542740000
Offset: 1
- 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]
A056467
Number of primitive (aperiodic) palindromes using exactly six different symbols.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 720, 720, 15120, 15120, 191520, 191520, 1905120, 1905120, 16435440, 16435440, 129230640, 129229920, 953029440, 953028720, 6711344640, 6711329520, 45674188560, 45674173440, 302899156560, 302898965040, 1969147121760, 1969146930240
Offset: 1
- 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]
A327874
Number of length n primitive (aperiodic) palindromes with integer entries that cover an initial interval of positive integers.
Original entry on oeis.org
1, 1, 0, 2, 2, 12, 10, 74, 72, 538, 528, 4682, 4668, 47292, 47218, 545820, 545760, 7087260, 7086710, 102247562, 102247020, 1622632496, 1622627890, 28091567594, 28091562840, 526858348368, 526858301088, 10641342969902, 10641342923148, 230283190977852, 230283190431480
Offset: 0
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a(n) = {if(n<1, n==0, sumdiv(n, d, moebius(n/d)*sum(k=1, ceil(d/2), k!*stirling(ceil(d/2), k, 2))))}
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