A292506
Number T(n,k) of multisets of exactly k nonempty binary words with a total of n letters such that no word has a majority of 0's; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 3, 1, 0, 4, 3, 1, 0, 11, 10, 3, 1, 0, 16, 23, 10, 3, 1, 0, 42, 59, 33, 10, 3, 1, 0, 64, 134, 83, 33, 10, 3, 1, 0, 163, 320, 230, 98, 33, 10, 3, 1, 0, 256, 699, 568, 270, 98, 33, 10, 3, 1, 0, 638, 1599, 1451, 738, 291, 98, 33, 10, 3, 1, 0, 1024, 3434, 3439, 1935, 798, 291, 98, 33, 10, 3, 1
Offset: 0
T(4,2) = 10: {1,011}, {1,101}, {1,110}, {1,111}, {01,01}, {01,10}, {01,11}, {10,10}, {10,11}, {11,11}.
Triangle T(n,k) begins:
1;
0, 1;
0, 3, 1;
0, 4, 3, 1;
0, 11, 10, 3, 1;
0, 16, 23, 10, 3, 1;
0, 42, 59, 33, 10, 3, 1;
0, 64, 134, 83, 33, 10, 3, 1;
0, 163, 320, 230, 98, 33, 10, 3, 1;
0, 256, 699, 568, 270, 98, 33, 10, 3, 1;
0, 638, 1599, 1451, 738, 291, 98, 33, 10, 3, 1;
...
Columns k=0-10 give:
A000007,
A027306 (for n>0),
A316403,
A316404,
A316405,
A316406,
A316407,
A316408,
A316409,
A316410,
A316411.
-
g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, x^n,
add(binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^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);
-
g[n_] := 2^(n-1) + If[OddQ[n], 0, Binomial[n, n/2]/2];
b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, x^n, Sum[Binomial[g[i] + j - 1, j]*b[n - i*j, i - 1]*x^j, {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 06 2018, from Maple *)
A316403
Number of multisets of exactly two nonempty binary words with a total of n letters such that no word has a majority of 0's.
Original entry on oeis.org
1, 3, 10, 23, 59, 134, 320, 699, 1599, 3434, 7682, 16246, 35762, 74892, 163032, 338771, 731051, 1510466, 3237206, 6658530, 14189790, 29083988, 61687496, 126076638, 266332390, 543061284, 1143207236, 2326521164, 4882706596, 9920514328, 20764519984, 42130081155
Offset: 2
a(4) = 10: {1,011}, {1,101}, {1,110}, {1,111}, {01,01}, {01,10}, {01,11}, {10,10}, {10,11}, {11,11}.
a(5) = 23: {1,0011}, {1,0101}, {1,0110}, {1,0111}, {1,1001}, {1,1010}, {1,1011}, {1,1100}, {1,1101}, {1,1110}, {1,1111}, {01,011}, {01,101}, {01,110}, {01,111}, {10,011}, {10,101}, {10,110}, {10,111}, {11,011}, {11,101}, {11,110}, {11,111}.
-
g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
b:= proc(n, i) option remember; series(`if`(n=0 or i=1, x^n, add(
binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 3)
end:
a:= n-> coeff(b(n$2), x, 2):
seq(a(n), n=2..33);
A316404
Number of multisets of exactly three nonempty binary words with a total of n letters such that no word has a majority of 0's.
Original entry on oeis.org
1, 3, 10, 33, 83, 230, 568, 1451, 3439, 8384, 19390, 45708, 103770, 238855, 534400, 1208485, 2672043, 5959769, 13051586, 28792488, 62551270, 136760659, 295115360, 640444498, 1374092646, 2963283862, 6326402780, 13569867602, 28846140436, 61586022487, 130422459008
Offset: 3
-
g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
b:= proc(n, i) option remember; series(`if`(n=0 or i=1, x^n, add(
binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 4)
end:
a:= n-> coeff(b(n$2), x, 3):
seq(a(n), n=3..33);
A316405
Number of multisets of exactly four nonempty binary words with a total of n letters such that no word has a majority of 0's.
Original entry on oeis.org
1, 3, 10, 33, 98, 270, 738, 1935, 5004, 12580, 31354, 76444, 185305, 441363, 1046837, 2447913, 5705753, 13143961, 30202325, 68719396, 156034994, 351348607, 789783351, 1762658134, 3928209272, 8700183502, 19244947618, 42340195770, 93049476310, 203518456343
Offset: 4
-
g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
b:= proc(n, i) option remember; series(`if`(n=0 or i=1, x^n, add(
binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 5)
end:
a:= n-> coeff(b(n$2), x, 4):
seq(a(n), n=4..33);
A316406
Number of multisets of exactly five nonempty binary words with a total of n letters such that no word has a majority of 0's.
Original entry on oeis.org
1, 3, 10, 33, 98, 291, 798, 2200, 5804, 15275, 39014, 99214, 247065, 612090, 1492837, 3622213, 8682565, 20711303, 48923317, 115048586, 268374750, 623503251, 1438753371, 3307821910, 7560955644, 17225642730, 39047321794, 88249150462, 198572820286, 445610719629
Offset: 5
-
g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
b:= proc(n, i) option remember; series(`if`(n=0 or i=1, x^n, add(
binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 6)
end:
a:= n-> coeff(b(n$2), x, 5):
seq(a(n), n=5..34);
A316407
Number of multisets of exactly six nonempty binary words with a total of n letters such that no word has a majority of 0's.
Original entry on oeis.org
1, 3, 10, 33, 98, 291, 826, 2284, 6185, 16471, 43156, 111446, 284517, 717486, 1793081, 4434929, 10887761, 26495243, 64069055, 153761086, 366992020, 870215947, 2053484109, 4818104922, 11256015936, 26164409278, 60583174348, 139655557194, 320805463602
Offset: 6
-
g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
b:= proc(n, i) option remember; series(`if`(n=0 or i=1, x^n, add(
binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 7)
end:
a:= n-> coeff(b(n$2), x, 6):
seq(a(n), n=6..34);
A316408
Number of multisets of exactly seven nonempty binary words with a total of n letters such that no word has a majority of 0's.
Original entry on oeis.org
1, 3, 10, 33, 98, 291, 826, 2320, 6297, 16989, 44828, 117352, 302429, 773496, 1954845, 4905939, 12195457, 30123762, 73825711, 179891662, 435427632, 1048510795, 2510267189, 5981859208, 14182293004, 33482368279, 78690956088, 184229429914, 429570180998
Offset: 7
-
g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
b:= proc(n, i) option remember; series(`if`(n=0 or i=1, x^n, add(
binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 8)
end:
a:= n-> coeff(b(n$2), x, 7):
seq(a(n), n=7..35);
A316409
Number of multisets of exactly eight nonempty binary words with a total of n letters such that no word has a majority of 0's.
Original entry on oeis.org
1, 3, 10, 33, 98, 291, 826, 2320, 6342, 17133, 45504, 119580, 310416, 798196, 2033289, 5136803, 12878647, 32056022, 79277444, 194822462, 476101571, 1156995495, 2797803485, 6731961588, 16126628466, 38459836055, 91355046531, 216126089962, 509445131238
Offset: 8
-
g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
b:= proc(n, i) option remember; series(`if`(n=0 or i=1, x^n, add(
binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 9)
end:
a:= n-> coeff(b(n$2), x, 8):
seq(a(n), n=8..36);
A316410
Number of multisets of exactly nine nonempty binary words with a total of n letters such that no word has a majority of 0's.
Original entry on oeis.org
1, 3, 10, 33, 98, 291, 826, 2320, 6342, 17188, 45684, 120435, 313280, 808581, 2065885, 5241557, 13191343, 32992806, 81964072, 202499115, 497418503, 1215823396, 2956890329, 7159215090, 17256728038, 41428552721, 99060756883, 235997525351, 560191343126
Offset: 9
-
g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
b:= proc(n, i) option remember; series(`if`(n=0 or i=1, x^n, add(
binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 10)
end:
a:= n-> coeff(b(n$2), x, 9):
seq(a(n), n=9..37);
A316411
Number of multisets of exactly ten nonempty binary words with a total of n letters such that no word has a majority of 0's.
Original entry on oeis.org
1, 3, 10, 33, 98, 291, 826, 2320, 6342, 17188, 45750, 120655, 314335, 812161, 2078985, 5283157, 13326283, 33400066, 83195864, 206069915, 507722068, 1244740868, 3037497201, 7379529734, 17854498058, 43026654989, 103302756909, 247127149283, 589196413579
Offset: 10
-
g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
b:= proc(n, i) option remember; series(`if`(n=0 or i=1, x^n, add(
binomial(g(i)+j-1, j)*b(n-i*j, i-1)*x^j, j=0..n/i)), x, 11)
end:
a:= n-> coeff(b(n$2), x, 10):
seq(a(n), n=10..38);
Showing 1-10 of 10 results.