A167772 -id:A167772 - cp's OEIS frontend

A166830 Number of n X 3 1..2 arrays containing at least one of each value, all equal values connected, rows considered as a single number in nondecreasing order, and columns considered as a single number in nonincreasing order.

2, 8, 18, 33, 54, 82, 118, 163, 218, 284, 362, 453, 558, 678, 814, 967, 1138, 1328, 1538, 1769, 2022, 2298, 2598, 2923, 3274, 3652, 4058, 4493, 4958, 5454, 5982, 6543, 7138, 7768, 8434, 9137, 9878, 10658, 11478, 12339, 13242

Offset: 1

R. H. Hardin, Oct 21 2009

nonn

			All solutions for n=3
...1.1.1...1.1.1...1.1.1...1.1.1...1.1.1...1.1.1...1.1.1...1.1.1...1.1.1
...1.1.1...1.1.1...1.1.1...2.1.1...2.1.1...2.1.1...2.2.1...2.2.1...2.2.2
...2.1.1...2.2.1...2.2.2...2.1.1...2.2.1...2.2.2...2.2.1...2.2.2...2.2.2
------
...2.1.1...2.1.1...2.1.1...2.1.1...2.1.1...2.1.1...2.2.1...2.2.1...2.2.1
...2.1.1...2.1.1...2.1.1...2.2.1...2.2.1...2.2.2...2.2.1...2.2.1...2.2.2
...2.1.1...2.2.1...2.2.2...2.2.1...2.2.2...2.2.2...2.2.1...2.2.2...2.2.2

lst={};Do[AppendTo[lst,n*(n+1)*(n+2)/6-2],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)

Empirical: a(n) = (n^3+6*n^2+11*n-6)/6.
a(n) = A167772(n+3,n). - Philippe Deléham, Nov 11 2009
a(n) = A227819(n+6,n+2). - Alois P. Heinz, Sep 22 2013
Empirical: a(n) = floor(A000292(n+1)^3/(A000292(n+1) + 1)^ 2). - Ivan N. Ianakiev, Nov 05 2013
From G. C. Greubel, May 25 2016: (Start)
Empirical G.f.: (-1 + 6*x - 6*x^2 + 2*x^3)/(1 - x)^4 + 1.
Empirical E.g.f.: (1/6)*(-6 + 18*x + 9*x^2 + x^3)*exp(x) + 1. (End)

A065602 Triangle T(n,k) giving number of hill-free Dyck paths of length 2n and having height of first peak equal to k.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 8, 6, 3, 1, 24, 18, 10, 4, 1, 75, 57, 33, 15, 5, 1, 243, 186, 111, 54, 21, 6, 1, 808, 622, 379, 193, 82, 28, 7, 1, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1, 9458, 7338, 4596, 2476, 1164, 474, 163, 45, 9, 1, 33062, 25724, 16266, 8928, 4332, 1856, 692, 218, 55, 10, 1

Offset: 2

N. J. A. Sloane, Dec 02 2001

A Riordan triangle.
Subtriangle of triangle in A167772. - Philippe Deléham, Nov 14 2009
Riordan array (f(x), x*g(x)) where f(x) is the g.f. of A000958 and g(x) is the g.f. of A000108. - Philippe Deléham, Jan 23 2010

			T(3,2)=1 reflecting the unique Dyck path (UUDUDD) of length 6, with no hills and height of first peak equal to 2.
Triangle begins:
     1;
     1,    1;
     3,    2,    1;
     8,    6,    3,   1;
    24,   18,   10,   4,   1;
    75,   57,   33,  15,   5,   1;
   243,  186,  111,  54,  21,   6,  1;
   808,  622,  379, 193,  82,  28,  7,  1;
  2742, 2120, 1312, 690, 311, 118, 36,  8,  1;

Reinhard Zumkeller, Rows n = 2..125 of triangle, flattened
Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Row sums give A000957 (the Fine sequence).
First column is A000958.
Cf. A000108, A007318, A167772.

a065602 n k = sum
   [(k-1+2*j) * a007318' (2*n-k-1-2*j) (n-1) `div` (2*n-k-1-2*j) |
    j <- [0 .. div (n-k) 2]]
a065602_row n = map (a065602 n) [2..n]
a065602_tabl = map a065602_row [2..]
-- Reinhard Zumkeller, May 15 2014

a := proc(n,k) if n=0 and k=0 then 1 elif k<2 or k>n then 0 else sum((k-1+2*j)*binomial(2*n-k-1-2*j,n-1)/(2*n-k-1-2*j),j=0..floor((n-k)/2)) fi end: seq(seq(a(n,k),k=2..n),n=1..14);

nmax = 12; t[n_, k_] := Sum[(k-1+2j)*Binomial[2n-k-1-2j, n-1] / (2n-k-1-2j), {j, 0, (n-k)/2}]; Flatten[ Table[t[n, k], {n, 2, nmax}, {k, 2, n}]] (* Jean-François Alcover, Nov 08 2011, after Maple *)

def T(n,k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) )
flatten([[T(n,k) for k in (2..n)] for n in (2..12)]) # G. C. Greubel, May 26 2022

T(n, 2) = A000958(n-1).
Sum_{k=2..n} T(n, k) = A000957(n+1).
From Emeric Deutsch, Feb 23 2004: (Start)
T(n, k) = Sum_{j=0..floor((n-k)/2)} (k-1+2*j)*binomial(2*n-k-1-2*j, n-1)/(2*n-k-1-2*j).
G.f.: t^2*z^2*C/( (1-z^2*C^2)*(1-t*z*C) ), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function. (End)
T(n,k) = A167772(n-1,k-1), k=2..n. - Reinhard Zumkeller, May 15 2014
From G. C. Greubel, May 26 2022: (Start)
T(n, n-1) = A000027(n-2).
T(n, n-2) = A000217(n-2).
T(n, n-3) = A166830(n-3). (End)

More terms from Emeric Deutsch, Feb 23 2004

A237619 Riordan array (1/(1+xc(x)), xc(x)) where c(x) is the g.f. of Catalan numbers (A000108).

Original entry on oeis.org

1, -1, 1, 0, 0, 1, -1, 1, 1, 1, -2, 2, 3, 2, 1, -6, 6, 8, 6, 3, 1, -18, 18, 24, 18, 10, 4, 1, -57, 57, 75, 57, 33, 15, 5, 1, -186, 186, 243, 186, 111, 54, 21, 6, 1, -622, 622, 808, 622, 379, 193, 82, 28, 7, 1, -2120, 2120, 2742, 2120, 1312, 690, 311, 118, 36, 8, 1

Offset: 0

Philippe Deléham, Feb 10 2014

			Triangle begins:
    1;
   -1,  1;
    0,  0,  1;
   -1,  1,  1,  1;
   -2,  2,  3,  2,  1;
   -6,  6,  8,  6,  3,  1;
  -18, 18, 24, 18, 10,  4, 1;
  -57, 57, 75, 57, 33, 15, 5, 1;
Production matrix begins:
  -1, 1;
  -1, 1, 1;
  -1, 1, 1, 1;
  -1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1, 1;
  -1, 1, 1, 1, 1, 1, 1, 1, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Diagonals: A000012, A000217, A023443, A166830.
Cf. A000957, A000958, A001558, A001559, A104629, A126983.
Cf. A065602, A167772.

A065602[n_, k_]:= A065602[n, k]= Sum[(k-1+2*j)*Binomial[2*(n-j)-k-1, n-1]/(2*(n - j) -k-1), {j,0,(n-k)/2}];
T[n_, k_]:= If[k==0, A065602[n, 0], If[n==1 && k==1, 1, A065602[n, k]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 27 2022 *)

def A065602(n, k): return sum( (k+2*j-1)*binomial(2*n-2*j-k-1, n-1)/(2*n-2*j-k-1) for j in (0..(n-k)//2) )
def A237619(n, k):
    if (n<2): return (-1)^(n-k)
    elif (k==0): return A065602(n, 0)
    else: return A065602(n, k)
flatten([[A237619(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2022

Sum_{k=0..n} T(n,k)*x^k = A126983(n), A000957(n+1), A026641(n) for x = 0, 1, 2 respectively.
T(n, k) = A167772(n-1, k-1) for k > 0, with T(n, 0) = A167772(n, 0).
T(n, 0) = A126983(n).
T(n+1, 1) = A000957(n+1).
T(n+2, 2) = A000958(n+1).
T(n+3, 3) = A104629(n) = A000957(n+3).
T(n+4, 4) = A001558(n).
T(n+5, 5) = A001559(n).
T(n, k) = A065602(n, k) for k > 0, with T(n, k) = (-1)^(n-k), for n < 2, and T(n, 0) = A065602(n, 0). - G. C. Greubel, May 27 2022

cp's OEIS Frontend

A166830 Number of n X 3 1..2 arrays containing at least one of each value, all equal values connected, rows considered as a single number in nondecreasing order, and columns considered as a single number in nonincreasing order.

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A065602 Triangle T(n,k) giving number of hill-free Dyck paths of length 2n and having height of first peak equal to k.

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A237619 Riordan array (1/(1+xc(x)), xc(x)) where c(x) is the g.f. of Catalan numbers (A000108).

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cp's OEIS Frontend

A166830 Number of n X 3 1..2 arrays containing at least one of each value, all equal values connected, rows considered as a single number in nondecreasing order, and columns considered as a single number in nonincreasing order.

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Author

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Programs

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Formula

A065602 Triangle T(n,k) giving number of hill-free Dyck paths of length 2n and having height of first peak equal to k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

SageMath

Formula

Extensions

A237619 Riordan array (1/(1+x*c(x)), x*c(x)) where c(x) is the g.f. of Catalan numbers (A000108).

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Author

Keywords

Examples

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Crossrefs

Programs

Mathematica

SageMath

Formula

A237619 Riordan array (1/(1+xc(x)), xc(x)) where c(x) is the g.f. of Catalan numbers (A000108).