A226874
Number T(n,k) of n-length words w over a k-ary alphabet {a1, a2, ..., ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 1, where #(w,x) counts the letters x in word w; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 2, 0, 1, 3, 6, 0, 1, 10, 12, 24, 0, 1, 15, 50, 60, 120, 0, 1, 41, 180, 300, 360, 720, 0, 1, 63, 497, 1260, 2100, 2520, 5040, 0, 1, 162, 1484, 6496, 10080, 16800, 20160, 40320, 0, 1, 255, 5154, 20916, 58464, 90720, 151200, 181440, 362880
Offset: 0
T(4,2) = 10: aaab, aaba, aabb, abaa, abab, abba, baaa, baab, baba, bbaa.
T(4,3) = 12: aabc, aacb, abac, abca, acab, acba, baac, baca, bcaa, caab, caba, cbaa.
T(5,2) = 15: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, abbaa, baaaa, baaab, baaba, babaa, bbaaa.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 2;
0, 1, 3, 6;
0, 1, 10, 12, 24;
0, 1, 15, 50, 60, 120;
0, 1, 41, 180, 300, 360, 720;
0, 1, 63, 497, 1260, 2100, 2520, 5040;
0, 1, 162, 1484, 6496, 10080, 16800, 20160, 40320;
...
Columns k=0-10 give:
A000007,
A057427,
A226881,
A226882,
A226883,
A226884,
A226885,
A226886,
A226887,
A226888,
A226889.
-
b:= proc(n, i, t) option remember;
`if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
end:
T:= (n, k)-> `if`(n*k=0, `if`(n=k, 1, 0), n!*b(n, 1, k)):
seq(seq(T(n, k), k=0..n), n=0..12);
# second Maple program:
b:= proc(n, i) option remember; expand(
`if`(n=0, 1, `if`(i<1, 0, add(x^j*b(n-i*j, i-1)*
combinat[multinomial](n, n-i*j, i$j), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);
-
b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]]; t[n_, k_] := If[n*k == 0, If[n == k, 1, 0], n!*b[n, 1, k]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from first Maple *)
-
T(n)={Vec(serlaplace(prod(k=1, n, 1/(1-y*x^k/k!) + O(x*x^n))))}
{my(t=T(10)); for(n=1, #t, for(k=0, n-1, print1(polcoeff(t[n], k), ", ")); print)} \\ Andrew Howroyd, Dec 20 2017
A213942
a(n) is the number of representative two-color bracelets (necklaces with turnover allowed) with n beads for n >= 2.
Original entry on oeis.org
1, 1, 3, 3, 7, 8, 18, 22, 46, 62, 136, 189, 409, 611, 1344, 2055, 4535, 7154, 15881, 25481, 56533, 92204, 204759, 337593, 748665, 1246862, 2762111, 4636389, 10253938, 17334800, 38278784, 65108061, 143534770, 245492243, 540353057, 928772649, 2041154125
Offset: 2
a(5) = A213939(5,2) + A213939(5,3) = 1 + 2 = 3 from the representative bracelets (with colors j for c[j], j=1,2) cyclic(11112), cyclic(11122) and cyclic(11212). The first one has color signature (exponents) [4,1] and the two others have signature [3,2]. For the number of all two-color 5-bracelets with beads of five colors available see A214308(5) = 60.
a(8) = 18 = 1 + 4 + 5 + 8 for the partitions of 8 with 2 parts (7,1), (6, 2), (5,3), (4,4), respectively. see A213939(5, k), k = 2..5). The 8 representative bracelets for the exponents (signature) from partition (4,4) are B1 = (11112222), B2 = (11121222), B3 = (11212122), B4 = (11212212), B5 = (11221122), B6 = (12121212), B7 = (11122122) and B8 = (11211222). B1 to B6 are color exchange (1 <-> 2) invariant (modulo D_8 symmetry, i.e., cyclic or anti-cyclic operations). B7 is equivalent to B8 under color exchange.
This explains why A056357(8) = 17. The difference between the present sequence and A056357 is that there, besides D_n symmetry, also color exchange is allowed. Here only color exchange compatible with D_n symmetry is allowed. - _Wolfdieter Lang_, Sep 28 2017
-
a29[n_] := (1/4)*(Mod[n, 2] + 3)*2^Quotient[n, 2] + DivisorSum[n, EulerPhi[#]*2^(n/#)&]/(2*n);
a5648[n_] := 1/2*(Binomial[2*Quotient[n, 2], Quotient[n, 2]] + DivisorSum[n, EulerPhi[#]*Binomial[2*n/#, n/#]&]/(2*n));
a[n_] := a29[n]/2 - 1 + If[EvenQ[n], a5648[n/2]/2, 0];
Array[a, 37, 2] (* Jean-François Alcover, Nov 05 2017, after Andrew Howroyd *)
A191313
Sum of the abscissae of the first returns to the horizontal axis (assumed to be 0 if there are no such returns) in all dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights).
Original entry on oeis.org
0, 0, 2, 5, 15, 30, 71, 134, 296, 551, 1188, 2211, 4720, 8815, 18722, 35105, 74307, 139842, 295223, 557366, 1174031, 2222606, 4672473, 8866776, 18607461, 35384676, 74139407, 141248270, 295524297, 563959752, 1178389423, 2252131246, 4700155088, 8995122383, 18751860084
Offset: 0
a(4)=15 because the sum of the abscissae of the first returns in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD is 0+4+3+2+2+4=15; here H=(1,0), U=(1,1), and D=(1,-1).
-
g := z*(4*z-1+sqrt(1-4*z^2))/((1-z)^2*sqrt(1-4*z^2)*(1-2*z+sqrt(1-4*z^2))): gser := series(g, z = 0, 37): seq(coeff(gser, z, n), n = 0 .. 34);
-
CoefficientList[Series[x*(4*x-1+Sqrt[1-4*x^2])/((1-x)^2*Sqrt[1-4*x^2]*(1-2*x+Sqrt[1-4*x^2])), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
Showing 1-3 of 3 results.
Comments