A005025 -id:A005025 - cp's OEIS frontend

A122588 Expansion of x/(1 - 9x + 28x^2 - 35x^3 + 15x^4 - x^5).

1, 9, 53, 260, 1156, 4845, 19551, 76912, 297275, 1134705, 4292145, 16128061, 60304951, 224660626, 834641671, 3094322026, 11453607152, 42344301686, 156404021389, 577291806894, 2129654436910, 7853149169635, 28949515515376, 106692395098433, 393137817645838

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 19 2006

Essentially the same as A005025. - R. J. Mathar, Aug 02 2008

Colin Barker, Table of n, a(n) for n = 1..1000
MathPuzzle, Chebyshev Polynomials.
Index entries for linear recurrences with constant coefficients, signature (9,-28,35,-15,1).

Cf. A005021, A005025, A094256, A122589.

I:=[1,9,53,260,1156]; [n le 5 select I[n] else 9*Self(n-1) -28*Self(n-2) +35*Self(n-3) -15*Self(n-4) +Self(n-5): n in [1..30]]; // G. C. Greubel, Nov 29 2021

m = 10; p[x_]:= ExpandAll[x^m*ChebyshevU[m, 1/x]]; Table[SeriesCoefficient[ Series[2^(n+m-1)*x/p[x], {x,0,30}], n], {n,1,30,2}]

def A122588_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x/(1-9*x+28*x^2-35*x^3+15*x^4-x^5) ).list()
a=A122588_list(31); a[1:] # G. C. Greubel, Nov 29 2021

G.f.: x/(1 - 9*x + 28*x^2 - 35*x^3 + 15*x^4 - x^5).

Generating function corrected by R. J. Mathar, Jul 09 2009

New name (using g.f.) and editing by Joerg Arndt, Feb 12 2015

A188843 T(n,k) is the number of n X k binary arrays without the pattern 0 1 diagonally or vertically.

Original entry on oeis.org

2, 4, 3, 8, 8, 4, 16, 21, 13, 5, 32, 55, 40, 19, 6, 64, 144, 121, 66, 26, 7, 128, 377, 364, 221, 100, 34, 8, 256, 987, 1093, 728, 364, 143, 43, 9, 512, 2584, 3280, 2380, 1288, 560, 196, 53, 10, 1024, 6765, 9841, 7753, 4488, 2108, 820, 260, 64, 11, 2048, 17711, 29524

Offset: 1

R. H. Hardin, Apr 12 2011

Table starts
4   8   16   32    64    128    256     512     1024     2048      4096
8  21   55  144   377    987   2584    6765    17711    46368    121393
13  40  121  364  1093   3280   9841   29524    88573   265720    797161
19  66  221  728  2380   7753  25213   81927   266110   864201   2806272
26 100  364 1288  4488  15504  53296  182688   625184  2137408   7303360
34 143  560 2108  7752  28101 100947  360526  1282735  4552624  16131656
43 196  820 3264 12597  47652 177859  657800  2417416  8844448  32256553
53 260 1156 4845 19551  76912 297275 1134705  4292145 16128061  60304951
64 336 1581 6954 29260 119416 476905 1874730  7283640 28048800 107286661
76 425 2109 9709 42504 179630 740025 2991495 11920740 46981740 183579396

			Some solutions for 5 X 3:
  0 0 1    1 1 0    1 1 1    0 1 0    1 1 0    1 1 0    1 1 1
  0 0 0    1 0 0    1 1 0    0 0 0    1 1 0    1 1 0    1 1 1
  0 0 0    0 0 0    1 1 0    0 0 0    1 0 0    1 1 0    0 1 1
  0 0 0    0 0 0    1 1 0    0 0 0    1 0 0    1 0 0    0 0 0
  0 0 0    0 0 0    0 0 0    0 0 0    0 0 0    0 0 0    0 0 0

R. H. Hardin, Table of n, a(n) for n = 1..1741

Diagonal is A143388.
Column 2 is A034856(n+1).
Column 3 is A137742(n+1).
Row 2 is A001906(n+1).
Row 3 is A003462(n+1).
Row 4 is A005021.
Row 5 is A005022.
Row 6 is A005023.
Row 7 is A005024.
Row 8 is A005025.

Row recurrence
Empirical: T(n,k) = Sum_{i=1..floor((n+2)/2)} binomial(n+2-i,i)*T(n,k-i)*(-1)^(i-1).
E.g.,
empirical: T(1,k) = 2*T(1,k-1),
empirical: T(2,k) = 3*T(2,k-1) -    T(2,k-2),
empirical: T(3,k) = 4*T(3,k-1) -  3*T(3,k-2),
empirical: T(4,k) = 5*T(4,k-1) -  6*T(4,k-2) +    T(4,k-3),
empirical: T(5,k) = 6*T(5,k-1) - 10*T(5,k-2) +  4*T(5,k-3),
empirical: T(6,k) = 7*T(6,k-1) - 15*T(6,k-2) + 10*T(6,k-3) -    T(6,k-4),
empirical: T(7,k) = 8*T(7,k-1) - 21*T(7,k-2) + 20*T(7,k-3) -  5*T(7,k-4),
empirical: T(8,k) = 9*T(8,k-1) - 28*T(8,k-2) + 35*T(8,k-3) - 15*T(8,k-4) + T(8,k-5).
Columns are polynomials for n > k-3.
Empirical: T(n,1) = n + 1.
Empirical: T(n,2) = (1/2)*n^2 + (5/2)*n + 1.
Empirical: T(n,3) = (1/6)*n^3 + 2*n^2 + (35/6)*n.
Empirical: T(n,4) = (1/24)*n^4 + (11/12)*n^3 + (155/24)*n^2 + (163/12)*n - 6 for n > 1.
Empirical: T(n,5) = (1/120)*n^5 + (7/24)*n^4 + (89/24)*n^3 + (473/24)*n^2 + (1877/60)*n - 33 for n > 2.
Empirical: T(n,6) = (1/720)*n^6 + (17/240)*n^5 + (203/144)*n^4 + (647/48)*n^3 + (2659/45)*n^2 + (1379/20)*n - 143 for n > 3.
Empirical: T(n,7) = (1/5040)*n^7 + (1/72)*n^6 + (143/360)*n^5 + (53/9)*n^4 + (33667/720)*n^3 + (12679/72)*n^2 + (9439/70)*n - 572 for n > 4.
Empirical: T(n,8) = (1/40320)*n^8 + (23/10080)*n^7 + (17/192)*n^6 + (269/144)*n^5 + (43949/1920)*n^4 + (228401/1440)*n^3 + (1054411/2016)*n^2 + (9941/56)*n - 2210 for n > 5.

cp's OEIS Frontend

A122588 Expansion of x/(1 - 9x + 28x^2 - 35x^3 + 15x^4 - x^5).

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A188843 T(n,k) is the number of n X k binary arrays without the pattern 0 1 diagonally or vertically.

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cp's OEIS Frontend

A122588 Expansion of x/(1 - 9*x + 28*x^2 - 35*x^3 + 15*x^4 - x^5).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

Extensions

A188843 T(n,k) is the number of n X k binary arrays without the pattern 0 1 diagonally or vertically.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A122588 Expansion of x/(1 - 9x + 28x^2 - 35x^3 + 15x^4 - x^5).