A050168 -id:A050168 - cp's OEIS frontend

A026010 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n and s(0) = 2. Also a(n) = sum of numbers in row n+1 of array T defined in A026009.

1, 2, 4, 7, 14, 25, 50, 91, 182, 336, 672, 1254, 2508, 4719, 9438, 17875, 35750, 68068, 136136, 260338, 520676, 999362, 1998724, 3848222, 7696444, 14858000, 29716000, 57500460, 115000920, 222981435, 445962870, 866262915, 1732525830, 3370764540

Offset: 0

Clark Kimberling

nonn

Conjecture: a(n) is the number of integer compositions of n + 2 in which the even parts appear as often at even positions as at odd positions (confirmed up to n = 19). - Gus Wiseman, Mar 17 2018

			The a(3) = 7 compositions of 5 in which the even parts appear as often at even positions as at odd positions are (5), (311), (131), (113), (221), (122), (11111). Missing are (41), (14), (32), (23), (212), (2111), (1211), (1121), (1112). - _Gus Wiseman_, Mar 17 2018

Michael De Vlieger, Table of n, a(n) for n = 0..3326
Christian Krattenthaler, Daniel Yaqubi, Some determinants of path generating functions, II, arXiv:1802.05990 [math.CO], 2018; Adv. Appl. Math. 101 (2018), 232-265.

First differences of A050168. Pairwise sums of A037952.

Cf. A000712, A001405, A005774, A045931, A063886, A097613, A130780, A171966, A239241, A299926, A300061, A300787, A300788, A300789.

[(&+[Binomial(Floor((n+k)/2), Floor(k/2)): k in [0..n]]): n in [0..40]]; // G. C. Greubel, Nov 08 2018

Array[Sum[Binomial[Floor[(# + k)/2], Floor[k/2]], {k, 0, #}] &, 34, 0] (* Michael De Vlieger, May 16 2018 *)
Table[2^(-1 + n)*(((2 + 3*#)*Gamma[(1 + #)/2])/(Sqrt[Pi]*Gamma[2 + #/2]) &[n + Mod[n, 2]]), {n,0,40}] (* Peter Pein, Nov 08 2018 *)
Table[(1/2)^((5 - (-1)^n)/2)*(6*n + 7 - 3*(-1)^n)*CatalanNumber[(2*n + 1 - (-1)^n)/4], {n, 0, 40}] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, n--; sum(k=0,n, binomial(floor((n+k)/2), floor(k/2)))) \\ G. C. Greubel, Nov 08 2018

a(2*n) = ((3*n + 1)/(2*n + 1))*C(2*n + 1, n)= A051924(1+n), n>=0, a(2*n-1) = a(2*n)/2 = A097613(1+n), n >= 1. - Herbert Kociemba, May 08 2004
a(n) = Sum_{k=0..n} binomial(floor((n+k)/2), floor(k/2)). - Paul Barry, Jul 15 2004
Inverse binomial transform of A005774: (1, 3, 9, 26, 75, 216, ...). - Gary W. Adamson, Oct 22 2007
Conjecture: (n+3)*a(n) - 2*a(n-1) + (-5*n-3)*a(n-2) + 2*a(n-3) + 4*(n-3)*a(n-4) = 0. - R. J. Mathar, Jun 20 2013
a(n) = (1/2)^((5 - (-1)^n)/2)*(6*n + 7 - 3*(-1)^n)*Catalan((2*n + 1 - (-1)^n)/4), where Catalan is the Catalan number = A000108. - G. C. Greubel, Nov 08 2018

A208976 Row sums of the triangle in A208101.

Original entry on oeis.org

1, 2, 4, 8, 15, 29, 54, 104, 195, 377, 713, 1385, 2639, 5147, 9866, 19304, 37179, 72929, 140997, 277133, 537471, 1058147, 2057509, 4056233, 7904455, 15600899, 30458899, 60174899, 117675359, 232676279, 455657714, 901620584, 1767883499, 3500409329, 6871173869

Offset: 0

Reinhard Zumkeller, Mar 04 2012

nonn

a(n) = A050168(n+1) - 1.

Number of Dyck (n+1)-paths that are symmetric after deleting all leading hills. - David Scambler, Aug 23 2012

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

a208976 n = a208976_list !! n
a208976_list = map (subtract 1) $ tail a050168_list

T[, 0] = 1; T[n, 1] := n; T[n_, n_] := T[n - 1, n - 2]; T[n_, k_] /; 1 < k < n := T[n, k] = T[n - 1, k] + T[n - 1, k - 2];
a[n_] := Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 03 2018, from A208101 *)

a(n) = A001405(n) + A001405(n+1) - 1.

A122908 A central binomial scaling of the Riordan array (1+x,x) (A097806).

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 0, 3, 6, 0, 0, 0, 0, 6, 10, 0, 0, 0, 0, 0, 10, 20, 0, 0, 0, 0, 0, 0, 20, 35, 0, 0, 0, 0, 0, 0, 0, 35, 70, 0, 0, 0, 0, 0, 0, 0, 0, 70, 126, 0, 0, 0, 0, 0, 0, 0, 0, 0, 126, 252

Offset: 0

Paul Barry, Sep 18 2006

Row sums are A050168. Diagonal sums are A001045 doubled. Row sums of inverse are 1/C(2n,n) aerated. Applications to other sequences are obvious.

			Triangle begins
.1,
.1, 1,
.0, 1, 2,
.0, 0, 2, 3,
.0, 0, 0, 3, 6,
.0, 0, 0, 0, 6, 10,
.0, 0, 0, 0, 0, 10, 20,
.0, 0, 0, 0, 0, 0, 20, 35,
.0, 0, 0, 0, 0, 0, 0, 35, 70,
.0, 0, 0, 0, 0, 0, 0, 0, 70, 126,
.0, 0, 0, 0, 0, 0, 0, 0, 0, 126, 252

Number triangle T(n,k)=C(k,floor(k/2))*sum{j=0..n, (-1)^(n-j)C(n,j)C(j+1,k+1)}

cp's OEIS Frontend

A026010 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n and s(0) = 2. Also a(n) = sum of numbers in row n+1 of array T defined in A026009.

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A208976 Row sums of the triangle in A208101.

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Haskell

Mathematica

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A122908 A central binomial scaling of the Riordan array (1+x,x) (A097806).

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Examples

Formula