A293112
Number A(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 5, 2, 0, 1, 1, 2, 6, 10, 3, 0, 1, 1, 2, 6, 14, 23, 4, 0, 1, 1, 2, 6, 15, 39, 51, 5, 0, 1, 1, 2, 6, 15, 44, 104, 111, 6, 0, 1, 1, 2, 6, 15, 45, 129, 284, 243, 8, 0, 1, 1, 2, 6, 15, 45, 135, 386, 775, 530, 10, 0
Offset: 0
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 2, 2, 2, 2, 2, ...
0, 2, 5, 6, 6, 6, 6, 6, ...
0, 2, 10, 14, 15, 15, 15, 15, ...
0, 3, 23, 39, 44, 45, 45, 45, ...
0, 4, 51, 104, 129, 135, 136, 136, ...
0, 5, 111, 284, 386, 422, 429, 430, ...
Columns k=0-10 give:
A000007,
A000009,
A293741,
A293742,
A293743,
A293744,
A293745,
A293746,
A293747,
A293748,
A293749.
-
h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]
n, 0, g(n-i, i, [l[], i])))))
end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1, k)*binomial(g(i, k, []), j), j=0..n/i)))
end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);
-
h[l_] := Function[n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[ l[[k]] n, 0, g[n - i, i, Append[l, i]]]]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*Binomial[g[i, k, {}], j], {j, 0, n/i}]]];
A[n_, k_] := b[n, n, k];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)
A293883
Number of sets of nonempty words with a total of n letters over binary alphabet containing the second letter such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.
Original entry on oeis.org
1, 3, 8, 20, 47, 106, 237, 522, 1146, 2485, 5406, 11644, 25157, 53964, 116003, 247987, 530999, 1131889, 2415431, 5135838, 10927816, 23182209, 49199697, 104154808, 220543306, 465996956, 984704338, 2076988457, 4380764354, 9225209588, 19424813915, 40844509107
Offset: 2
A293884
Number of sets of nonempty words with a total of n letters over ternary alphabet containing the third letter such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.
Original entry on oeis.org
1, 4, 16, 53, 173, 532, 1615, 4785, 14066, 40908, 118438, 341253, 981200, 2815762, 8075265, 23149097, 66373778, 190376443, 546401592, 1569387414, 4511532695, 12980998062, 37385342522, 107771434819, 310967929569, 898108427259, 2596180252466, 7511411442182
Offset: 3
A305561
Expansion of 2*x*(1 - 2*x)/(1 + 2*x - 8*x^2 - sqrt(1 - 4*x^2)).
Original entry on oeis.org
1, 1, 3, 8, 23, 64, 182, 512, 1451, 4096, 11594, 32768, 92710, 262144, 741548, 2097152, 5931955, 16777216, 47454210, 134217728, 379628818, 1073741824, 3037013748, 8589934592, 24296051198, 68719476736, 194368201572, 549755813888, 1554944869676, 4398046511104
Offset: 0
-
m:=35; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 2*x*(1 - 2*x)/(1 + 2*x - 8*x^2 - Sqrt(1 - 4*x^2)))); // Vincenzo Librandi, Jan 27 2020
-
a:= proc(n) option remember; `if`(n=0, 1, add(
a(n-i)*binomial(i, floor(i/2)), i=1..n))
end:
seq(a(n), n=0..35); # Alois P. Heinz, Jun 21 2018
-
nmax = 29; CoefficientList[Series[2 x (1 - 2 x)/(1 + 2 x - 8 x^2 - Sqrt[1 - 4 x^2]), {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - Sum[Binomial[k, Floor[k/2]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[k, Floor[k/2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]
Showing 1-4 of 4 results.
Comments