A223663 -id:A223663 - cp's OEIS frontend

A223659 Number of unimodal maps [1..n]->[0..3].

1, 4, 16, 50, 130, 296, 610, 1163, 2083, 3544, 5776, 9076, 13820, 20476, 29618, 41941, 58277, 79612, 107104, 142102, 186166, 241088, 308914, 391967, 492871, 614576, 760384, 933976, 1139440, 1381300, 1664546, 1994665, 2377673, 2820148, 3329264

Offset: 0

R. H. Hardin, Mar 25 2013

nonn

Column 1 of A223663.

Apparently also column 4 of A071920. - R. J. Mathar, May 17 2014

			Some solutions for n=3:
  2  2  0  1  1  3  1  0  3  1  2  1  2  1  0  2
  2  2  1  3  3  3  3  2  2  2  2  3  0  1  1  1
  2  0  2  2  0  1  3  3  1  0  3  1  0  1  1  0

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..210 from R. H. Hardin)
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

Cf. A071920, A223663, A223718.

Empirical: a(n) = (1/720)*n^6 + (1/48)*n^5 + (23/144)*n^4 + (9/16)*n^3 + (241/180)*n^2 + (11/12)*n + 1 = 1 + n*(n+1)*(n^4 + 14*n^3 + 101*n^2 + 304*n + 660)/720.

Empirical g.f.: 1-x*(x^2-2*x+2)*(x^4-4*x^3+6*x^2-4*x+2) / (x-1)^7. - R. J. Mathar, May 14 2014

a(0)=1 prepended by Alois P. Heinz, Feb 11 2024

A223660 Number of nX2 0..3 arrays with row sums unimodal and column sums inverted unimodal.

Original entry on oeis.org

16, 256, 3060, 29922, 252912, 1912914, 13254601, 85563043, 521069404, 3022541224, 16826714534, 90449485556, 471770734372, 2397374836954, 11909366979539, 57999389713133, 277578926336176, 1308191004875392, 6081976574677816, 27936365857925926, 126946765412455656

Offset: 1

R. H. Hardin, Mar 25 2013

nonn

			Some solutions for n=3:
..3..3....3..2....0..0....0..0....3..1....1..0....1..0....2..0....1..3....3..1
..1..3....2..2....1..3....0..1....0..2....2..1....3..1....0..2....2..3....0..3
..1..1....0..3....3..0....2..0....2..0....0..3....1..2....3..0....1..0....1..0

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

Column 2 of A223663.

Empirical: a(n) = 31*a(n-1) -437*a(n-2) +3707*a(n-3) -21099*a(n-4) +85029*a(n-5) -249431*a(n-6) +538841*a(n-7) -856504*a(n-8) +988504*a(n-9) -804432*a(n-10) +436752*a(n-11) -141696*a(n-12) +20736*a(n-13).

Empirical g.f.: -x*( 16 -240*x +2116*x^2 -12378*x^3 +51142*x^4 -153984*x^5 +342369*x^6 -562536*x^7 +675688*x^8 -578496*x^9 +336528*x^10 -120960*x^11 +20736*x^12) / ( (-1+4*x)^2 *(x-1)^3 *(3*x-1)^4 *(2*x-1)^4 ). - R. J. Mathar, May 17 2014

A223661 Number of nX3 0..3 arrays with row sums unimodal and column sums inverted unimodal.

Original entry on oeis.org

50, 3060, 141046, 5285887, 169756788, 4836834467, 125226945708, 2997363216275, 67192414310968, 1424909830742272, 28811928773038893, 559031955620162864, 10462533894185251783, 189688734342719058204, 3343610975591843927435

Offset: 1

R. H. Hardin Mar 25 2013

nonn

Column 3 of A223663

			Some solutions for n=3
..0..1..2....0..0..1....1..2..2....1..2..0....1..0..2....0..2..0....2..3..0
..2..1..3....0..0..3....1..2..3....2..1..0....3..1..2....3..0..3....3..2..0
..3..2..1....3..2..1....2..0..0....1..1..1....0..1..2....2..1..0....1..0..3

R. H. Hardin, Table of n, a(n) for n = 1..24

A223658 Number of n X n 0..3 arrays with row sums unimodal and column sums inverted unimodal.

Original entry on oeis.org

4, 256, 141046, 761845474, 44626165058452

Offset: 1

R. H. Hardin Mar 25 2013

nonn

Diagonal of A223663

			Some solutions for n=3
..1..0..2....0..0..1....0..0..1....0..1..0....3..0..2....2..1..3....2..0..2
..3..2..3....2..0..1....2..0..3....3..3..2....1..2..2....3..2..1....0..2..2
..0..1..2....2..1..2....0..2..2....3..1..1....2..2..0....0..2..2....3..1..3

A223662 Number of nX4 0..3 arrays with row sums unimodal and column sums inverted unimodal.

Original entry on oeis.org

130, 29922, 5285887, 761845474, 93974096640, 10253839694278, 1013084819998517, 92199924066276051, 7830543600441814997, 626952495503328450002

Offset: 1

R. H. Hardin Mar 25 2013

nonn

Column 4 of A223663

			Some solutions for n=3
..2..2..3..2....0..0..0..0....0..0..0..2....2..2..0..0....2..2..2..2
..2..2..0..2....1..0..0..2....3..0..0..2....2..2..1..0....2..2..1..2
..3..2..0..1....0..1..3..1....2..2..0..0....2..1..3..1....2..0..2..1

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A223659 Number of unimodal maps [1..n]->[0..3].

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A223660 Number of nX2 0..3 arrays with row sums unimodal and column sums inverted unimodal.

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A223661 Number of nX3 0..3 arrays with row sums unimodal and column sums inverted unimodal.

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A223658 Number of n X n 0..3 arrays with row sums unimodal and column sums inverted unimodal.

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A223662 Number of nX4 0..3 arrays with row sums unimodal and column sums inverted unimodal.

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