A292506 -id:A292506 - cp's OEIS frontend

A292549 Number of multisets of exactly n nonempty binary words with a total of 2n letters such that no word has a majority of 0's.

1, 3, 10, 33, 98, 291, 826, 2320, 6342, 17188, 45750, 120733, 314690, 813854, 2085363, 5306878, 13406382, 33665476, 84031608, 208655086, 515469203, 1267600993, 3103490884, 7567559622, 18381579206, 44487740012, 107301636460, 257967350824, 618279370985

Offset: 0

Alois P. Heinz, Sep 18 2017

nonn

			a(0) = 1: {}.
a(1) = 3: {01}, {10}, {11}.
a(2) = 10: {1,011}, {1,101}, {1,110}, {1,111}, {01,01}, {01,10}, {01,11}, {10,10}, {10,11}, {11,11}.

Alois P. Heinz, Table of n, a(n) for n = 0..3163

Cf. A292506.

g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      g(d+1), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);

g[n_] := 2^(n-1) + If[OddQ[n], 0, Binomial[n, n/2]/2];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*g[d+1], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 11 2019, after Alois P. Heinz *)

a(n) = A292506(2n,n) = A292506(2n+j,n+j) for j >= 0.
G.f.: Product_{j>=1} 1/(1-x^j)^A027306(j+1).
Euler transform of j-> A027306(j+1).

A209406 Triangular array read by rows: T(n,k) is the number of multisets of exactly k nonempty binary words with a total of n letters.

Original entry on oeis.org

2, 4, 3, 8, 8, 4, 16, 26, 12, 5, 32, 64, 44, 16, 6, 64, 164, 132, 62, 20, 7, 128, 384, 376, 200, 80, 24, 8, 256, 904, 1008, 623, 268, 98, 28, 9, 512, 2048, 2632, 1792, 870, 336, 116, 32, 10, 1024, 4624, 6624, 5040, 2632, 1117, 404, 134, 36, 11

Offset: 1

Geoffrey Critzer, Mar 08 2012

Equivalently, T(n,k) is the number of partitions of the integer n with two types of 1's, four types of 2's, ..., 2^i types of i's...; having exactly k summands (of any type).

Row sums = A034899.

			Triangle T(n,k) begins:
    2;
    4,    3;
    8,    8,    4;
   16,   26,   12,    5;
   32,   64,   44,   16,   6;
   64,  164,  132,   62,  20,   7;
  128,  384,  376,  200,  80,  24,   8;
  256,  904, 1008,  623, 268,  98,  28,  9;
  512, 2048, 2632, 1792, 870, 336, 116, 32, 10;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Index entries for triangles generated by the Multiset Transformation

Cf. A034899, A055375, A208741, A290222, A292506.

T(2n,n) gives A359962.

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
       binomial(2^i+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Apr 13 2017

nn = 10; p[x_, y_] := Product[1/(1 - y x^i)^(2^i), {i, 1, nn}]; f[list_] := Select[lst, # > 0 &]; Map[f, Drop[CoefficientList[Series[p[x, y], {x, 0, nn}], {x, y}], 1]] // Flatten

O.g.f.: Product_{i>=1} 1/(1-y*x^i)^(2^i).

A290222 Multiset transform of A011782, powers of 2: 1, 2, 4, 8, 16, ...

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 2, 1, 0, 8, 7, 2, 1, 0, 16, 16, 7, 2, 1, 0, 32, 42, 20, 7, 2, 1, 0, 64, 96, 54, 20, 7, 2, 1, 0, 128, 228, 140, 59, 20, 7, 2, 1, 0, 256, 512, 360, 156, 59, 20, 7, 2, 1, 0, 512, 1160, 888, 422, 162, 59, 20, 7, 2, 1, 0, 1024, 2560, 2168, 1088, 442, 162, 59, 20, 7, 2, 1

Offset: 0

M. F. Hasler, Jul 24 2017

T(n,k) is the number of multisets of exactly k binary words with a total of n letters and each word beginning with 1. T(4,2) = 7: {1,100}, {1,101}, {1,110}, {1,111}, {10,10}, {10,11}, {11,11}. - Alois P. Heinz, Sep 18 2017

			The triangle starts:
1;
0    1;
0    2    1;
0    4    2    1;
0    8    7    2    1;
0   16   16    7    2   1;
0   32   42   20    7   2   1;
0   64   96   54   20   7   2  1;
0  128  228  140   59  20   7  2  1;
0  256  512  360  156  59  20  7  2  1;
0  512 1160  888  422 162  59 20  7  2  1;
0 1024 2560 2168 1088 442 162 59 20  7  2  1;
(...)

Alois P. Heinz, Rows n = 0..140, flattened
Index entries for triangles generated by the Multiset Transformation

Cf. A034691 (row sums), A000007 (column k=0), A011782 (column k=1), A178945(n-1) (column k=2).
The reverse of the n-th row converges to A034899.
Cf. A000079, A209406, A292506.

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(binomial(2^(i-1)+j-1, j)*
         b(n-i*j, i-1, p-j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Sep 12 2017

b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[Binomial[2^(i - 1) + j - 1, j] b[n - i j, i - 1, p - j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

G.f.: 1 / Product_{j>=1} (1-y*x^j)^(2^(j-1)). - Alois P. Heinz, Sep 18 2017

A292548 Number of multisets of 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, 1, 4, 8, 25, 53, 148, 328, 858, 1938, 4862, 11066, 27042, 61662, 147774, 336854, 795678, 1810466, 4228330, 9597694, 22211897, 50279985, 115489274, 260686018, 594986149, 1339215285, 3040004744, 6823594396, 15416270130, 34510814918, 77644149076, 173368564396

Offset: 0

Alois P. Heinz, Sep 18 2017

nonn

			a(0) = 1: {}.
a(1) = 1: {1}.
a(2) = 4: {01}, {10}, {11}, {1,1}.
a(3) = 8: {011}, {101}, {110}, {111}, {1,01}, {1,10}, {1,11}, {1,1,1}.

Alois P. Heinz, Table of n, a(n) for n = 0..3213

Row sums of A292506.
Column k=2 of A292712.
Cf. A027306.

g:= n-> 2^(n-1)+`if`(n::odd, 0, binomial(n, n/2)/2):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      g(d), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);

g[n_] :=  2^(n-1) + If[OddQ[n], 0, Binomial[n, n/2]/2];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*
     g[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Apr 30 2022, after Alois P. Heinz *)

G.f.: Product_{j>=1} 1/(1-x^j)^A027306(j).

Euler transform of A027306.

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

Alois P. Heinz, Jul 02 2018

nonn

			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}.

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Column k=2 of A292506.

Cf. A027306, A292549.

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);

a(n) = [x^n y^2] 1/Product_{j>=1} (1-y*x^j)^A027306(j).

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

Alois P. Heinz, Jul 02 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A292506.

Cf. A027306, A292549.

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);

a(n) = [x^n y^3] 1/Product_{j>=1} (1-y*x^j)^A027306(j).

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

Alois P. Heinz, Jul 02 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Column k=4 of A292506.

Cf. A027306, A292549.

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);

a(n) = [x^n y^4] 1/Product_{j>=1} (1-y*x^j)^A027306(j).

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

Alois P. Heinz, Jul 02 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..1000

Column k=5 of A292506.

Cf. A027306, A292549.

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);

a(n) = [x^n y^5] 1/Product_{j>=1} (1-y*x^j)^A027306(j).

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

Alois P. Heinz, Jul 02 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=6 of A292506.

Cf. A027306, A292549.

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);

a(n) = [x^n y^6] 1/Product_{j>=1} (1-y*x^j)^A027306(j).

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

Alois P. Heinz, Jul 02 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..1000

Column k=7 of A292506.

Cf. A027306, A292549.

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);

a(n) = [x^n y^7] 1/Product_{j>=1} (1-y*x^j)^A027306(j).

cp's OEIS Frontend

A292549 Number of multisets of exactly n nonempty binary words with a total of 2n letters such that no word has a majority of 0's.

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A209406 Triangular array read by rows: T(n,k) is the number of multisets of exactly k nonempty binary words with a total of n letters.

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A290222 Multiset transform of A011782, powers of 2: 1, 2, 4, 8, 16, ...

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A292548 Number of multisets of nonempty binary words with a total of n letters such that no word has a majority of 0's.

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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.

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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.

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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.

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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.

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