A223620 -id:A223620 - cp's OEIS frontend

A086601 Triangular numbers + 1 squared.

1, 4, 16, 49, 121, 256, 484, 841, 1369, 2116, 3136, 4489, 6241, 8464, 11236, 14641, 18769, 23716, 29584, 36481, 44521, 53824, 64516, 76729, 90601, 106276, 123904, 143641, 165649, 190096, 217156, 247009, 279841, 315844, 355216, 398161

Offset: 0

Jon Perry, Jul 23 2003

Also number of n X 2 0..1 arrays with rows and columns unimodal (cf. A223620, column 2). - Georg Fischer, Nov 03 2021

			a(5) = (t(5)+1)^2 = 16^2 = 256.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
R. J. Mathar, The number of binary nxm matrices with at most k 1's in each row or column, (2014) Table 2 column 2.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000124, A000217, A061316, A223620.

A086601 := proc(n)
    (n+2+n^2)^2 /4 ;
end proc:
seq(A086601(n),n=0..20) ; # R. J. Mathar, May 14 2014

(Accumulate[Range[0,40]]+1)^2 (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,4,16,49,121},40] (* Harvey P. Dale, Jan 14 2020 *)

PARI
```
w=vector(40,i,(t(i)+1)^2)
```

a(n) = (A000217(n) + 1)^2.
a(n) = (binomial(2+n,2) - binomial(n,1))^2. - Zerinvary Lajos, May 30 2006, corrected by R. J. Mathar, May 14 2014
a(n) = A000124(n)^2. - Omar E. Pol, Oct 30 2007
a(n) = 1 + A061316(n). Zerinvary Lajos, Apr 25 2008
G.f.: ( -1+x-6*x^2+x^3-x^4 ) / (x-1)^5. - R. J. Mathar, May 14 2014

A223615 Number of n X 3 0..1 arrays with rows and columns unimodal.

Original entry on oeis.org

7, 49, 240, 876, 2582, 6504, 14547, 29659, 56161, 100123, 169786, 276030, 432888, 658106, 973749, 1406853, 1990123, 2762677, 3770836, 5068960, 6720330, 8798076, 11386151, 14580351, 18489381, 23235967, 28958014, 35809810, 43963276, 53609262

Offset: 1

R. H. Hardin, Mar 24 2013

nonn

Column 3 of A223620.

			Some solutions for n=4:
..0..0..1....0..0..1....1..0..0....0..0..1....0..1..0....0..0..0....0..1..1
..0..1..1....0..1..1....1..1..1....0..0..1....0..1..1....1..0..0....0..1..0
..1..1..1....0..0..1....1..1..0....0..1..0....0..0..1....1..0..0....0..0..0
..1..1..1....0..0..1....0..1..0....0..1..0....0..0..0....1..1..0....1..0..0

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

Cf. A223620.

Empirical: a(n) = (23/360)*n^6 + (31/120)*n^5 + (8/9)*n^4 + (31/24)*n^3 + (737/360)*n^2 + (29/20)*n + 1.
Conjectures from Colin Barker, Aug 21 2018: (Start)
G.f.: x*(7 + 44*x^2 - 20*x^3 + 20*x^4 - 6*x^5 + x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

A223616 Number of n X 4 0..1 arrays with rows and columns unimodal.

Original entry on oeis.org

11, 121, 876, 4466, 17594, 57238, 160883, 403159, 921181, 1951247, 3879910, 7312800, 13164932, 22776596, 38059285, 61676477, 97264447, 149698645, 225411536, 332768158, 482506014, 688246274, 967083623, 1340262451, 1833947441

Offset: 1

R. H. Hardin, Mar 24 2013

nonn

Column 4 of A223620.

			Some solutions for n=4:
..0..0..1..0....0..0..1..1....1..0..0..0....1..0..0..0....0..0..1..0
..1..1..1..1....0..1..1..1....0..0..0..0....1..1..1..0....0..0..1..1
..1..1..1..0....0..0..1..0....0..0..1..0....0..0..0..0....0..1..1..1
..0..0..1..0....0..0..1..0....0..0..0..1....0..0..0..0....0..1..1..1

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

Cf. A223620.

Empirical: a(n) = (1/112)*n^8 + (79/1260)*n^7 + (59/180)*n^6 + (299/360)*n^5 + (259/144)*n^4 + (821/360)*n^3 + (7219/2520)*n^2 + (767/420)*n + 1.
Conjectures from Colin Barker, Aug 21 2018: (Start)
G.f.: x*(11 + 22*x + 183*x^2 + 14*x^3 + 158*x^4 - 56*x^5 + 35*x^6 - 8*x^7 + x^8) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>9.
(End)

A223617 Number of n X 5 0..1 arrays with rows and columns unimodal.

Original entry on oeis.org

16, 256, 2582, 17594, 89630, 367216, 1271969, 3857481, 10504174, 26169386, 60506898, 131289614, 269716032, 528358366, 992720569, 1797620937, 3149899316, 5359276060, 8879561670, 14362835354, 22729680458, 35259087700, 53702216265

Offset: 1

R. H. Hardin Mar 24 2013

nonn

Column 5 of A223620.

			Some solutions for n=4
..1..1..1..1..1....1..1..1..0..0....1..1..0..0..0....0..1..1..1..1
..0..1..1..1..1....0..1..1..1..0....0..0..0..1..0....0..1..1..1..0
..0..0..0..0..0....0..0..1..1..0....0..0..0..1..1....1..1..1..0..0
..0..0..0..0..0....0..0..0..0..0....0..0..1..1..1....1..1..1..0..0

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

Cf. A223620.

Empirical: a(n) = (359/453600)*n^10 + (251/30240)*n^9 + (3817/60480)*n^8 + (73/280)*n^7 + (1127/1350)*n^6 + (485/288)*n^5 + (532219/181440)*n^4 + (25699/7560)*n^3 + (30811/8400)*n^2 + (451/210)*n + 1.

A223618 Number of nX6 0..1 arrays with rows and columns unimodal.

Original entry on oeis.org

22, 484, 6504, 57238, 367216, 1856100, 7795951, 28248007, 90732638, 263650284, 703980614, 1748547620, 4079672416, 9012596316, 18975151965, 38281575869, 74344179766, 139521245076, 253869738212, 449160402594, 774624001120

Offset: 1

R. H. Hardin Mar 24 2013

nonn

Column 6 of A223620

			Some solutions for n=4
..1..1..0..0..0..0....0..0..1..0..0..0....0..0..1..1..1..1....1..1..1..1..1..1
..0..1..0..0..0..0....0..0..0..1..0..0....0..0..1..1..1..1....1..1..1..1..1..1
..0..0..1..0..0..0....0..0..0..1..1..0....0..0..0..1..1..1....0..0..0..1..1..1
..0..0..1..1..0..0....1..1..0..0..0..0....0..0..0..0..1..1....0..0..0..1..1..1

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

Empirical: a(n) = (271/5443200)*n^12 + (2327/3326400)*n^11 + (39107/5443200)*n^10 + (1579/36288)*n^9 + (52553/259200)*n^8 + (192371/302400)*n^7 + (8636441/5443200)*n^6 + (56113/20160)*n^5 + (2901677/680400)*n^4 + (523927/113400)*n^3 + (23969/5400)*n^2 + (33487/13860)*n + 1

A223619 Number of nX7 0..1 arrays with rows and columns unimodal.

Original entry on oeis.org

29, 841, 14547, 160883, 1271969, 7795951, 39162020, 167679972, 629671260, 2118764468, 6495046664, 18377663488, 48503155274, 120433134332, 283334050243, 635338765283, 1364719186775, 2820129263963, 5627010952649, 10875524219289

Offset: 1

R. H. Hardin Mar 24 2013

nonn

Column 7 of A223620

			Some solutions for n=4
..0..0..0..0..0..0..0....0..0..0..0..0..1..1....0..0..0..0..0..0..0
..0..0..0..1..1..1..0....0..0..1..1..1..1..1....0..0..1..1..1..1..1
..1..0..0..0..0..0..0....0..1..1..1..1..0..0....1..1..1..1..1..1..1
..0..1..0..0..0..0..0....0..1..1..1..1..0..0....0..0..1..1..1..1..0

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

Empirical: a(n) = (503/209563200)*n^14 + (6527/155675520)*n^13 + (17/31185)*n^12 + (37901/8553600)*n^11 + (76799/2721600)*n^10 + (46471/362880)*n^9 + (3497567/7620480)*n^8 + (936493/777600)*n^7 + (467419/181440)*n^6 + (320807/77760)*n^5 + (69073271/11975040)*n^4 + (29326111/4989600)*n^3 + (60167903/11642400)*n^2 + (191717/72072)*n + 1

A223614 Number of n X n 0..1 arrays with rows and columns unimodal.

Original entry on oeis.org

2, 16, 240, 4466, 89630, 1856100, 39162020, 840122330, 18350758538, 409124396540, 9333564282142, 218396645933584, 5252293916506694

Offset: 1

R. H. Hardin Mar 24 2013

nonn

Diagonal of A223620

			Some solutions for n=4
..1..1..0..0....0..1..0..0....0..0..0..0....1..0..0..0....0..1..1..0
..1..1..0..0....1..1..0..0....0..1..0..0....1..1..0..0....0..1..1..0
..1..1..0..0....0..1..0..0....0..1..1..0....1..1..1..1....0..1..1..0
..0..0..0..1....0..0..1..1....0..1..1..0....1..1..1..0....0..1..1..1

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A086601 Triangular numbers + 1 squared.

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A223615 Number of n X 3 0..1 arrays with rows and columns unimodal.

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A223616 Number of n X 4 0..1 arrays with rows and columns unimodal.

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A223617 Number of n X 5 0..1 arrays with rows and columns unimodal.

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A223618 Number of nX6 0..1 arrays with rows and columns unimodal.

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A223619 Number of nX7 0..1 arrays with rows and columns unimodal.

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A223614 Number of n X n 0..1 arrays with rows and columns unimodal.

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