A071920 -id:A071920 - cp's OEIS frontend

A227406 Number of unimodal functions f:[n]->[2^n].

1, 2, 16, 372, 24616, 5014592, 3349471840, 7649590386464, 61356625102897216, 1758844330913892684288, 182379122144778004351027200, 69026760045145802122822210022400, 96048744530120196897251255933762037760, 494360393380904255996973467025921794482614272

Offset: 0

Alois P. Heinz, Sep 21 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A071920, A227402.

a:= n-> sum(binomial(n+2*j-1, 2*j), j=0..2^n-1):
seq(a(n), n=0..20);

Table[Sum[Binomial[n+2*j-1,2*j],{j,0,2^n-1}],{n,0,15}] (* Vaclav Kotesovec, Sep 22 2013 *)

a(n) = Sum_{j=0..2^n-1} C(n+2*j-1,2*j).
a(n) = A071921(n,2^n).
a(n) ~ 2^(n^2+n-1)/n!. - Vaclav Kotesovec, Sep 22 2013

A225011 Number of 4 X n 0..1 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

5, 25, 95, 295, 791, 1897, 4166, 8518, 16414, 30086, 52834, 89402, 146446, 233108, 361711, 548591, 815083, 1188679, 1704377, 2406241, 3349193, 4601059, 6244892, 8381596, 11132876, 14644540, 19090180, 24675260, 31641640, 40272566, 50898157

Offset: 1

R. H. Hardin, Apr 23 2013

nonn

Row 4 of A225010.

Apparently also column 5 of A071920. - R. J. Mathar, May 17 2014

			Some solutions for n=3
..0..1..0....0..0..0....0..0..1....0..0..0....1..0..0....1..0..0....0..0..1
..0..1..1....0..1..0....0..1..1....0..0..0....1..0..0....1..0..0....0..0..1
..1..1..1....0..1..0....1..1..1....0..0..0....1..1..0....1..0..0....0..0..1
..1..1..1....0..1..0....1..1..1....0..0..1....1..1..0....1..1..1....0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A071920, A225010.

Empirical: a(n) = (1/40320)*n^8 + (1/1440)*n^7 + (3/320)*n^6 + (5/72)*n^5 + (629/1920)*n^4 + (1279/1440)*n^3 + (16763/10080)*n^2 + (25/24)*n + 1 = 1 + n* (n+1) *(n^6 + 27*n^5 + 351*n^4 + 2449*n^3 + 10760*n^2 + 25052*n + 42000)/40320.

Empirical: G.f.: -x*(x^4 - 5*x^3 + 10*x^2 - 10*x + 5) *(x^4 - 3*x^3 + 4*x^2 - 2*x + 1) / (x-1)^9. - R. J. Mathar, May 17 2014

A225012 Number of 5 X n 0..1 arrays with rows unimodal and columns nondecreasing.

Original entry on oeis.org

6, 36, 161, 581, 1792, 4900, 12174, 27966, 60172, 122464, 237590, 442118, 793092, 1377174, 2322967, 3817351, 6126818, 9624964, 14827487, 22436251, 33394208, 48953224, 70757132, 100942636, 142261016, 198223936, 273277036, 373005396

Offset: 1

R. H. Hardin, Apr 23 2013

nonn

Row 5 of A225010.

Apparently column 6 of A071920. - R. J. Mathar, May 17 2014

			Some solutions for n=3
..0..0..0....0..1..0....0..1..0....1..0..0....0..0..0....0..0..0....0..0..0
..1..0..0....0..1..0....1..1..0....1..1..0....0..0..1....0..0..0....0..1..1
..1..0..0....0..1..0....1..1..1....1..1..0....0..1..1....0..1..0....0..1..1
..1..1..0....0..1..1....1..1..1....1..1..1....0..1..1....0..1..1....1..1..1
..1..1..0....0..1..1....1..1..1....1..1..1....0..1..1....1..1..1....1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = (1/3628800)*n^10 + (1/80640)*n^9 + (1/3780)*n^8 + (19/5760)*n^7 + (4633/172800)*n^6 + (331/2304)*n^5 + (191249/362880)*n^4 + (24421/20160)*n^3 + (10897/5600)*n^2 + (137/120)*n + 1 = 1 + n*(n+1)* (n^8 + 44*n^7 + 916*n^6 + 11054*n^5 + 86239*n^4 + 435086*n^3 + 1477404*n^2 + 2918376*n + 4142880)/ 3628800.

Empirical: G.f.: -x*(x^2 - 3*x + 3) *(x^2 - 2*x + 2) *(x^2 - x + 1) *(x^4 - 4*x^3 + 5*x^2 - 2*x + 1) / (x-1)^11. - R. J. Mathar, May 17 2014

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A227406 Number of unimodal functions f:[n]->[2^n].

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A225011 Number of 4 X n 0..1 arrays with rows unimodal and columns nondecreasing.

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Crossrefs

Formula

A225012 Number of 5 X n 0..1 arrays with rows unimodal and columns nondecreasing.

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Author

Keywords

Comments

Examples

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Formula