A223659 -id:A223659 - cp's OEIS frontend

A224050 T(n,k)=Number of nXk 0..3 arrays with rows and columns unimodal and antidiagonals nondecreasing.

4, 16, 16, 50, 160, 50, 130, 984, 984, 130, 296, 4580, 9731, 4580, 296, 610, 17723, 67585, 67585, 17723, 610, 1163, 59792, 376734, 638996, 376734, 59792, 1163, 2083, 180821, 1801402, 4646480, 4646480, 1801402, 180821, 2083, 3544, 499357, 7655477

Offset: 1

R. H. Hardin Apr 01 2013

Table starts
....4......16........50.........130..........296...........610...........1163
...16.....160.......984........4580........17723.........59792.........180821
...50.....984......9731.......67585.......376734.......1801402........7655477
..130....4580.....67585......638996......4646480......28403642......153482649
..296...17723....376734.....4646480.....41991129.....310514900.....1999959111
..610...59792...1801402....28403642....310514900....2701569493....20096076442
.1163..180821...7655477...153482649...1999959111...20096076442...169750613182
.2083..499357..29502561...753824187..11666303576..133729094355..1265415997425
.3544.1276595.104437965..3415377142..63064614327..821109158656..8609328529478
.5776.3053471.342818189.14392067725.319676018383.4743773544158.54810160645347

			Some solutions for n=3 k=4
..0..2..0..0....1..1..2..0....1..1..1..1....0..3..1..1....0..0..3..1
..3..2..2..1....3..3..2..2....2..2..3..1....3..2..2..2....2..3..3..1
..3..3..3..1....3..3..3..1....2..3..2..2....2..2..2..1....3..3..3..1

R. H. Hardin, Table of n, a(n) for n = 1..161

Column 1 is A223659

Column 2 is A224058

Empirical: columns k=1..5 are polynomials of degree 6*k for n>0,0,1,3,5

A224064 T(n,k)=Number of nXk 0..3 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

Original entry on oeis.org

4, 16, 16, 50, 160, 50, 130, 984, 984, 130, 296, 4580, 8854, 4580, 296, 610, 17723, 58814, 58814, 17723, 610, 1163, 59792, 324702, 506513, 324702, 59792, 1163, 2083, 180821, 1557606, 3509115, 3509115, 1557606, 180821, 2083, 3544, 499357, 6643979

Offset: 1

R. H. Hardin Mar 30 2013

Table starts
....4......16........50.........130..........296..........610.........1163
...16.....160.......984........4580........17723........59792.......180821
...50.....984......8854.......58814.......324702......1557606......6643979
..130....4580.....58814......506513......3509115.....21167501....114643788
..296...17723....324702.....3509115.....28682690....200974242...1274747540
..610...59792...1557606....21167501....200974242...1573171210..11060805360
.1163..180821...6643979...114643788...1274747540..11060805360..83942450048
.2083..499357..25596389...564290412...7460451193..72498474377.591725806925
.3544.1276595..90177585..2542634801..40485099654.448019196499
.5776.3053471.293585050.10557558941.203984697906

			Some solutions for n=3 k=4
..1..2..2..1....0..0..1..1....0..0..2..2....0..1..0..0....0..1..1..0
..0..2..2..2....0..1..1..1....0..0..2..2....0..2..1..0....0..1..2..2
..0..3..3..2....0..1..1..1....0..0..1..3....1..2..3..1....0..1..2..2

R. H. Hardin, Table of n, a(n) for n = 1..112

Column 1 is A223659

Empirical: columns k=1..5 are polynomials of degree 6*k for n>0,0,3,7,11

A224123 T(n,k)=Number of nXk 0..3 arrays with row sums nondecreasing and column sums unimodal.

Original entry on oeis.org

4, 16, 10, 50, 150, 20, 130, 1747, 1080, 35, 296, 16782, 47059, 6627, 56, 610, 140210, 1703178, 1070499, 36552, 84, 1163, 1050460, 53355889, 145901795, 21632718, 187000, 120, 2083, 7227405, 1491827492, 17292618579, 11066001160, 400993828

Offset: 1

R. H. Hardin Mar 31 2013

Table starts
...4......16.........50..........130...........296...........610
..10.....150.......1747........16782........140210.......1050460
..20....1080......47059......1703178......53355889....1491827492
..35....6627....1070499....145901795...17292618579.1830213565657
..56...36552...21632718..11066001160.4964366292229
..84..187000..400993828.766386005577
.120..905440.6962188526
.165.4206453
.220

			Some solutions for n=3 k=4
..0..2..2..0....2..0..0..1....2..2..0..0....0..2..3..0....0..0..2..1
..2..2..2..1....2..2..2..0....2..3..1..0....2..0..3..0....0..0..0..3
..2..3..1..3....0..2..2..3....0..2..2..3....0..3..2..0....0..1..3..3

R. H. Hardin, Table of n, a(n) for n = 1..49

Column 1 is A000292(n+1)
Column 2 is A223069
Row 1 is A223659

A223756 Number of n X 2 0..3 arrays with rows, antidiagonals and columns unimodal.

Original entry on oeis.org

16, 256, 2500, 16900, 87616, 372100, 1352569, 4338889, 12559936, 33362176, 82373776, 190992400, 419266576, 877225924, 1759047481, 3396208729, 6338070544, 11471266816, 20192978404, 34657779556, 58123423744, 95427859396

Offset: 1

R. H. Hardin, Mar 27 2013

nonn

Column 2 of A223762.

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

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

Empirical: a(n) = (1/518400)*n^12 + (1/17280)*n^11 + (91/103680)*n^10 + (71/8640)*n^9 + (9101/172800)*n^8 + (457/1920)*n^7 + (81397/103680)*n^6 + (497/270)*n^5 + (203687/64800)*n^4 + (1933/540)*n^3 + (2533/720)*n^2 + (11/6)*n + 1.
a(n) = A223659(n)^2. - Mark van Hoeij, May 14 2013
Conjectures from Colin Barker, Feb 19 2018: (Start)
G.f.: x*(16 + 48*x + 420*x^2 - 208*x^3 + 1140*x^4 - 1260*x^5 + 1401*x^6 - 1044*x^7 + 597*x^8 - 244*x^9 + 69*x^10 - 12*x^11 + x^12) / (1 - x)^13.
a(n) = 13*a(n-1) - 78*a(n-2) + 286*a(n-3) - 715*a(n-4) + 1287*a(n-5) - 1716*a(n-6) + 1716*a(n-7) - 1287*a(n-8) + 715*a(n-9) - 286*a(n-10) + 78*a(n-11) - 13*a(n-12) + a(n-13) for n>13.
(End)

A223811 T(n,k)=Number of nXk 0..3 arrays with rows, columns, diagonals and antidiagonals unimodal.

Original entry on oeis.org

4, 16, 16, 50, 256, 50, 130, 2500, 2500, 130, 296, 16900, 58806, 16900, 296, 610, 87616, 825896, 825896, 87616, 610, 1163, 372100, 8165133, 20847008, 8165133, 372100, 1163, 2083, 1352569, 62305953, 342521725, 342521725, 62305953, 1352569, 2083

Offset: 1

R. H. Hardin Mar 27 2013

Table starts
....4.......16..........50............130.............296.............610
...16......256........2500..........16900...........87616..........372100
...50.....2500.......58806.........825896.........8165133........62305953
..130....16900......825896.......20847008.......342521725......4146732319
..296....87616.....8165133......342521725......8597979566....151474085262
..610...372100....62305953.....4146732319....151474085262...3678996027680
.1163..1352569...388531932....39816673636...2058985931297..66795003874023
.2083..4338889..2057610878...317796753758..22901512677629.975436194200049
.3544.12559936..9513089522..2176384736806.216485354275124
.5776.33362176.39201336756.13081738670880

			Some solutions for n=3 k=4
..0..0..3..0....1..1..1..1....0..1..0..0....0..0..2..1....0..0..3..1
..0..0..2..2....0..2..3..1....1..2..3..2....1..3..2..1....0..1..3..1
..0..2..2..0....0..2..2..1....1..2..2..3....0..2..3..3....1..3..0..0

R. H. Hardin, Table of n, a(n) for n = 1..97

Column 1 is A223659

Column 2 is A223756

Empirical: columns k=1..5 are polynomials of degree 6*k for n>0,0,0,8,15

A279006 Alternating Jacobsthal triangle read by rows (second version).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, 1, 1, 1, -2, 2, 0, 1, 1, -3, 4, -2, 1, 1, 1, -4, 7, -6, 3, 0, 1, 1, -5, 11, -13, 9, -3, 1, 1, 1, -6, 16, -24, 22, -12, 4, 0, 1, 1, -7, 22, -40, 46, -34, 16, -4, 1, 1, 1, -8, 29, -62, 86, -80, 50, -20, 5, 0, 1, 1, -9, 37, -91, 148, -166, 130, -70, 25, -5, 1, 1

Offset: 0

N. J. A. Sloane, Dec 07 2016

			Triangle begins:
  1,
  1,   1,
  1,   0,   1,
  1,  -1,   1,   1,
  1,  -2,   2,   0,   1,
  1,  -3,   4,  -2,   1,   1,
  1,  -4,   7,  -6,   3,   0,   1,
  1,  -5,  11, -13,   9,  -3,   1,   1,
  1,  -6,  16, -24,  22, -12,   4,   0,   1,
  ...

Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

See A112468, A112555 and A108561 for other versions.

Columns give A000124, A003600, A223718, A257890, A223659.

T := (n, k) -> local j; add((-1)^j*binomial(n-k-1+j, j), j = 0..k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..9);  # Peter Luschny, Aug 30 2024

T[i_, i_] = T[, 0] = 1; T[i, j_] := T[i, j] = T[i-1, j] - T[i-1, j-1];
Table[T[i, j], {i, 0, 11}, {j, 0, i}] // Flatten (* Jean-François Alcover, Sep 06 2018 *)
T[n_, k_] := 2^k*Hypergeometric2F1[-n, -k, -k, 1/2]; Table[T[n, k], {n, 0, 11}, {k, 0, n}]//Flatten (* Detlef Meya, Aug 30 2024 *)

\\ using arxiv (3.1) and (3.7) formulas where A is A220074 and B is this sequence
A(i, j) = if ((i < 0), 0, if (j==0, 1, A(i - 1, j - 1) - A(i - 1, j))); \\ A220074
B(i, j) = A(i, i-j);
tabl(nn) = for (i=0, nn, for (j=0, i, print1(B(i,j), ", ")); print()); \\ Michel Marcus, Jun 17 2017

T(i, j) = A220074(i, i-j). See (3.7) in arxiv link. - Michel Marcus, Jun 17 2017

T(n, k) = 2^k*hypergeom([-n, -k], [-k], 1/2). - Detlef Meya, Aug 30 2024

More terms from Michel Marcus, Jun 17 2017

A224113 T(n,k)=Number of nXk 0..3 arrays with row sums and column sums unimodal.

Original entry on oeis.org

4, 16, 16, 50, 256, 50, 130, 3060, 3060, 130, 296, 29922, 141112, 29922, 296, 610, 252912, 5286827, 5286827, 252912, 610, 1163, 1912914, 169786856, 762031090, 169786856, 1912914, 1163, 2083, 13254601, 4837645361, 93991412090, 93991412090

Offset: 1

R. H. Hardin Mar 31 2013

Table starts
....4........16............50..............130...............296
...16.......256..........3060............29922............252912
...50......3060........141112..........5286827.........169786856
..130.....29922.......5286827........762031090.......93991412090
..296....252912.....169786856......93991412090....44633436796823
..610...1912914....4837645361...10255460738958.18747100782890395
.1163..13254601..125243425246.1013220280453733
.2083..85563043.2997702534361
.3544.521069404
.5776

			Some solutions for n=3 k=4
..2..0..0..2....0..2..3..0....2..2..0..0....2..0..3..2....2..0..3..0
..2..2..1..0....0..0..3..2....2..0..2..0....2..2..3..0....0..2..2..2
..3..1..2..0....3..3..3..0....0..3..0..2....0..3..1..2....3..3..2..3

R. H. Hardin, Table of n, a(n) for n = 1..59

Column 1 is A223659

Column 2 is A223660

cp's OEIS Frontend

A224050 T(n,k)=Number of nXk 0..3 arrays with rows and columns unimodal and antidiagonals nondecreasing.

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A224064 T(n,k)=Number of nXk 0..3 arrays with rows, columns and antidiagonals unimodal and diagonals nondecreasing.

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A224123 T(n,k)=Number of nXk 0..3 arrays with row sums nondecreasing and column sums unimodal.

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A223756 Number of n X 2 0..3 arrays with rows, antidiagonals and columns unimodal.

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A223811 T(n,k)=Number of nXk 0..3 arrays with rows, columns, diagonals and antidiagonals unimodal.

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A279006 Alternating Jacobsthal triangle read by rows (second version).

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A224113 T(n,k)=Number of nXk 0..3 arrays with row sums and column sums unimodal.

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