A104722 -id:A104722 - cp's OEIS frontend

A059348 Third diagonal of array in A059347.

2, 2, 3, 4, 7, 10, 19, 28, 56, 84, 174, 264, 561, 858, 1859, 2860, 6292, 9724, 21658, 33592, 75582, 117572, 266798, 416024, 950912, 1485800, 3417340, 5348880, 12369285, 19389690, 45052515, 70715340, 165002460, 259289580, 607283490

Offset: 2

N. J. A. Sloane, Jan 27 2001

Same as A104722 except for leading term (both have rational g.f.s). - R. J. Mathar and Franklin T. Adams-Watters, May 19 2008

G. C. Greubel, Table of n, a(n) for n = 2..1000
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.

Cf. A059347, A104722.

CoefficientList[Series[1 + ( (1 + x)*(1 - Sqrt[1 - 4*x^2])/(2*x^2))^2, {x, 0, 50}], x] (* G. C. Greubel, Jan 08 2017 *)

Vec( 1 + ( (1 + x)*(1 - sqrt(1 - 4*x^2))/(2*x^2))^2 + O(x^25)) \\ G. C. Greubel, Jan 08 2017

G.f.: 1 + ( (1 + x)*(1 - sqrt(1 - 4*x^2))/(2*x^2))^2. - G. C. Greubel, Jan 08 2017

a(n) ~ (9 + (-1)^n) * 2^(n - 3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 31 2025

A256644 Numbers of alternating permutations where numbers at odd positions and even positions are monotone respectively.

Original entry on oeis.org

1, 1, 1, 2, 5, 6, 9, 12, 21, 30, 58, 86, 176, 266, 563, 860, 1861, 2862, 6294, 9726, 21660, 33594, 75584, 117574, 266800, 416026, 950914, 1485802, 3417342, 5348882, 12369287, 19389692, 45052517, 70715342, 165002462, 259289582, 607283492, 955277402, 2244901892

Offset: 0

Ran Pan, Apr 07 2015

nonn

			a(5) = 6: (1,3,2,5,4), (1,4,2,5,3), (1,5,2,4,3), (3,4,2,5,1), (3,5,2,4,1), (4,5,2,3,1).
a(6) = 9: (1,3,2,5,4,6), (1,4,2,5,3,6), (1,6,2,5,3,4), (3,4,2,5,1,6), (3,6,2,5,1,4), (4,6,2,5,1,3), (4,6,3,5,1,2), (5,6,2,4,1,3), (5,6,3,4,1,2).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Ran Pan, Exercise Q, Project P

Cf. A104722, A000111, A000108.

[1,1,1,2] cat [Catalan(Floor(n/2))+ Catalan(Floor((n-1)/2))+2: n in [4..40]]; // Vincenzo Librandi, Apr 08 2015

C:= n-> binomial(2*n, n)/(n+1):
a:= n-> `if`(n<4, [1$3, 2][n+1], C(iquo(n, 2))+C(iquo(n-1, 2))+2):
seq(a(n), n=0..40);  # Alois P. Heinz, Apr 08 2015

Table[Which[n < 3, 1, n == 3, 2, True, CatalanNumber[Floor[n/2]] + CatalanNumber[Floor[(n - 1)/2]] + 2], {n, 0, 38}] (* Michael De Vlieger, Apr 07 2015 *)

C(n) = binomial(2*n, n)/(n+1);
a(n) = if (n<3, 1, if (n==3, 2, C(n\2)+ C((n-1)\2)+2)); \\ Michel Marcus, Apr 07 2015

a(n) = if (n<4, return(max(1,n-1))); binomial(n\2*2, n\2)/(n\2+1)*if(n%2, 2, (5*n-2)/(4*n-4)) + 2 \\ Charles R Greathouse IV, Apr 07 2015

For n>3, a(n) = C(floor(n/2))+ C(floor((n-1)/2))+2, where C(n) is the n-th Catalan number, with a(0)=a(1)=a(2)=1 and a(3)=2.

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A059348 Third diagonal of array in A059347.

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A256644 Numbers of alternating permutations where numbers at odd positions and even positions are monotone respectively.

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