A027306 -id:A027306 - cp's OEIS frontend

A260419 Square array T(n,m) read by antidiagonals, T(n,m) is the number of (m,n)-parking functions.

1, 1, 1, 1, 3, 1, 1, 3, 4, 1, 1, 5, 16, 11, 1, 1, 5, 16, 27, 16, 1, 1, 7, 25, 125, 81, 42, 1, 1, 7, 49, 125, 256, 378, 64, 1, 1, 9, 49, 243, 1296, 1184, 729, 163, 1, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 1, 11, 100, 729, 2401, 16807, 15625, 27213, 9529, 638, 1

Offset: 1

Michel Marcus, Jul 25 2015

T(n,2) appears to be A027306(n).

			Table starts (see Table 1 in Aval & Bergeron link):
n/m  1   2    3    4     5
------------------------------
1   |1,  1,   1,   1,    1, ...
2   |1,  3,   3,   5,    5, ...
3   |1,  4,  16,  16,   25, ...
4   |1, 11,  27, 125,  125, ...
5   |1, 16,  81, 256, 1296, ...
6   |...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Jean-Christophe Aval, François Bergeron, Interlaced rectangular parking functions, arXiv:1503.03991 [math.CO], 2015.

Cf. A071201.

T(n,m) = m^(n-1), if m and n are coprime (see Lemma in Aval & Bergeron link).

More terms from Alois P. Heinz, Nov 30 2015

A307768 Number of n-step random walks on a line starting from the origin and returning to it at least once.

Original entry on oeis.org

0, 0, 2, 4, 10, 20, 44, 88, 186, 372, 772, 1544, 3172, 6344, 12952, 25904, 52666, 105332, 213524, 427048, 863820, 1727640, 3488872, 6977744, 14073060, 28146120, 56708264, 113416528, 228318856, 456637712, 918624304, 1837248608, 3693886906, 7387773812, 14846262964, 29692525928

Offset: 0

Robert FERREOL, Apr 27 2019

a(n)/2^n tends to 1 as n goes to infinity; this means that on the line any random walk returns sooner or later to its starting point with a probability 1.

a(n) is also the number of heads-or-tails games of length n during which at some point there are as many heads as tails.

			The a(3)=4 three-step walks returning to 0 are [0, -1, 0, -1], [0, -1, 0, 1], [0, 1, 0, -1], [0, 1, 0, 1].
The a(4)=10 three-step walks returning to 0 are [0, -1, -2, -1, 0], [0, -1, 0, -1, -2], [0, -1, 0, -1, 0], [0, -1, 0, 1, 0], [0, -1, 0, 1, 2], [0, 1, 0, -1, -2], [0, 1, 0, -1, 0], [0, 1, 0, 1, 0], [0, 1, 0, 1, 2], [0, 1, 2, 1, 0].

Robert Israel, Table of n, a(n) for n = 0..3318
Monique Laurent, Sven Polak, and Luis Felipe Vargas, Semidefinite approximations for bicliques and biindependent pairs, arXiv:2302.08886 [math.CO], 2023.

Cf. A027306, A063886, A045621, A128014, A284016.

b:=n->piecewise(n mod 2 = 0,binomial(n,n/2),2*binomial(n-1,(n-1)/2)):
seq(2^n-b(n),n=0..20);
# second program:
A307768 := series(exp(2*x) - int((1/x + 2)*BesselI(1,2*x),x) - BesselI(1,2*x), x = 0, 36): seq(n!*coeff(A307768, x, n), n = 0 .. 35); # Mélika Tebni, Jun 19 2024

a[n_] := If[n == 0, 0, 2^n - 2*Binomial[n-1, Floor[(n-1)/2]]];
Array[a, 36, 0] (* Jean-François Alcover, May 05 2019 *)

a(n) = 2^n - A063886(n).
a(n+1) = 2*A045621(n) = 2*(2^n - binomial(n,floor(n/2))).
a(2n) = 2^(2n) - binomial(2n,n); a(2n+1) = 2*a(2n).
G.f.: (1-sqrt(1-4*x^2))/(1-2*x). - Alois P. Heinz, May 05 2019
n*(a(n)-2*a(n-1)) - 4*(n-3)*(a(n-2)-2*a(n-3)) = 0. - Robert Israel, May 06 2019
a(2n+2) - 2*a(2n+1) = A284016(n) = 2*Catalan(n). - Robert FERREOL, Aug 26 2019
From Mélika Tebni, Jun 19 2024: (Start)
E.g.f.: exp(2*x) - Integral_{x=-oo..oo} (1/x + 2)*BesselI(1, 2*x) dx - BesselI(1, 2*x).
a(n) = 2*(A027306(n) - A128014(n)). (End)

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).

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

Alois P. Heinz, Jul 02 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..1000

Column k=8 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, 9)
    end:
a:= n-> coeff(b(n$2), x, 8):
seq(a(n), n=8..36);

a(n) = [x^n y^8] 1/Product_{j>=1} (1-y*x^j)^A027306(j).

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

Alois P. Heinz, Jul 02 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..1000

Column k=9 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, 10)
    end:
a:= n-> coeff(b(n$2), x, 9):
seq(a(n), n=9..37);

a(n) = [x^n y^9] 1/Product_{j>=1} (1-y*x^j)^A027306(j).

cp's OEIS Frontend

A260419 Square array T(n,m) read by antidiagonals, T(n,m) is the number of (m,n)-parking functions.

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A307768 Number of n-step random walks on a line starting from the origin and returning to it at least once.

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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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Maple

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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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Maple

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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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Maple

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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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Maple

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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.

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Keywords

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Crossrefs

Programs

Maple

Formula

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.

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Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

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.

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