A125176 -id:A125176 - cp's OEIS frontend

A059738 Binomial transform of A054341 and inverse binomial transform of A049027.

1, 3, 10, 34, 117, 405, 1407, 4899, 17083, 59629, 208284, 727900, 2544751, 8898873, 31125138, 108881166, 380928795, 1332824049, 4663705782, 16319702046, 57109857519, 199859075307, 699435489795, 2447823832671, 8566818534141, 29982268505595, 104933418068332

Offset: 0

John W. Layman, Feb 09 2001

nonn

First column of the Riordan array ((1-2x)/(1+x+x^2),x/(1+x+x^2))^(-1). [Paul Barry, Nov 06 2008]
Apparently the Motzkin transform of A125176, supposed A125176 is interpreted with offset 0. [R. J. Mathar, Dec 11 2008]
a(n) is the number of Motzkin paths of length n in which the (1,0)-steps at level 0 come in 3 colors. Example: a(3)=34 because, denoting  U=(1,1), H=(1,0), and D=(1,-1), we have 3^3 = 27 paths of shape HHH, 3 paths of shape HUD, 3 paths of shape UDH, and 1 path of shape UHD. - Emeric Deutsch, May 02 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Taras Goy and Mark Shattuck, Determinants of Some Hessenberg-Toeplitz Matrices with Motzkin Number Entries, J. Int. Seq., Vol. 26 (2023), Article 23.3.4.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Table[SeriesCoefficient[2/(1-5*x+Sqrt[1-2*x-3*x^2]),{x,0,n}],{n,0,20}]

x='x+O('x^66); Vec(2/(1-5*x+sqrt(1-2*x-3*x^2))) \\ Joerg Arndt, May 06 2013

a(n) = Sum[k=0..n, 2^(n-k)*A026300(n, k) ], where A026300 is the Motzkin triangle. - Ralf Stephan, Jan 25 2005 [Corrected by Philippe Deléham, Nov 29 2009]
a(n)= A126954(n,0). [Philippe Deléham, Nov 24 2009]
G.f.: 2/(1-5*x+sqrt(1-2*x-3*x^2)). - Emeric Deutsch, May 02 2011
Recurrence: 2*(n+1)*a(n) = (11*n+5)*a(n-1) - (8*n+5)*a(n-2) - 21*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 11 2012
a(n) ~ 3*7^n/2^(n+2). - Vaclav Kotesovec, Oct 11 2012
G.f.: 1/(1 - 3*x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Nov 19 2021

More terms from Vincenzo Librandi, May 06 2013

A047355 Numbers that are congruent to {0, 3} mod 7.

Original entry on oeis.org

0, 3, 7, 10, 14, 17, 21, 24, 28, 31, 35, 38, 42, 45, 49, 52, 56, 59, 63, 66, 70, 73, 77, 80, 84, 87, 91, 94, 98, 101, 105, 108, 112, 115, 119, 122, 126, 129, 133, 136, 140, 143, 147, 150, 154, 157, 161, 164, 168, 171, 175, 178, 182, 185, 189, 192, 196, 199, 203

Offset: 1

N. J. A. Sloane

Numbers k such that k^2/7 + k*(k + 1)/7 = k*(2*k + 1)/7 is a nonnegative integer. - Bruno Berselli, Feb 14 2017

David Lovler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A030123, A010702 (first differences).

Cf. A030308, A125176.

a047355 n = a047355_list !! (n-1)
a047355_list = scanl (+) 0 a010702_list -- Reinhard Zumkeller, Jul 05 2012

A047355:=n->(14*n - (-1)^n - 15)/4; seq(A047355(n), n=1..100); # Wesley Ivan Hurt, Jan 30 2014

Table[(14n - (-1)^n - 15)/4, {n, 100}] (* Wesley Ivan Hurt, Jan 30 2014 *)

a(n)=n\2*7 - 4 + n%2*4 \\ Charles R Greathouse IV, Aug 01 2016

a(n) = a(n-2) + 7 = a(n-1) + a(n-2) - a(n-3). - Henry Bottomley, Jan 19 2001
From Bruno Berselli, Sep 12 2011: (Start)
G.f.: x^2*(3 + 4*x)/((1 + x)*(1 - x)^2).
a(n) = (14*n - (-1)^n - 15)/4. (End)
a(n+1) = Sum_{k>=0} A030308(n,k)*A125176(k+2). - Philippe Deléham, Oct 17 2011
a(n) = 2*n - 2 + floor((3*n - 3)/2). - Wesley Ivan Hurt, Jan 30 2014
E.g.f.: 4 + ((14*x - 15)*exp(x) - exp(-x))/4. - David Lovler, Aug 31 2022

A135092 Binomial transform of [1, 6, 1, 6, 1, 6, ...].

Original entry on oeis.org

1, 7, 14, 28, 56, 112, 224, 448, 896, 1792, 3584, 7168, 14336, 28672, 57344, 114688, 229376, 458752, 917504, 1835008, 3670016, 7340032, 14680064, 29360128, 58720256, 117440512, 234881024, 469762048, 939524096, 1879048192, 3758096384, 7516192768, 15032385536

Offset: 0

Gary W. Adamson, Nov 18 2007

			a(3) = (1, 3, 3, 1) dot (1, 6, 1, 6) = (1 + 18 + 3 + 6) = 28 = 7*2^2.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (2).

Essentially identical to A005009 and to A125176.

Join[{1},NestList[2#&,7,50]] (* Harvey P. Dale, Aug 30 2015 *)

a(n) = 7*2^(n-1) for n>0, a(0)=1.
a(n) = Sum_{k=0..n} A097805(n,k)*7^k*(-5)^(n-k). - Philippe Deléham, Nov 19 2007
G.f.: (1+5*x)/(1-2*x). - Bruno Berselli, Sep 20 2011
E.g.f.: (1/2)*(7*exp(2*x) - 5). - G. C. Greubel, Sep 22 2016
a(n) = A125176(n+2) for n >= 1. - Georg Fischer, Nov 02 2018

Corrected and extended by Philippe Deléham and N. J. A. Sloane, Dec 15 2007

A125175 Triangle T(n,k) = |A053123(n/2+k/2,k)| for even n+k, T(n,k)= A082985((n+k-1)/2,k) for odd n+k; read by rows, 0<=k<=n.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 10, 7, 5, 1, 6, 14, 20, 9, 6, 1, 7, 21, 30, 35, 11, 7, 1, 8, 27, 56, 55, 56, 13, 8, 1, 9, 36, 77, 126, 91, 84, 15, 9, 1, 10, 44, 120, 182, 252, 140, 120, 17, 10, 1, 11, 55, 156, 330, 378, 462, 204, 165, 19, 11

Offset: 0

Gary W. Adamson, Nov 22 2006

			First few rows of the triangle are:
  1;
  1, 2;
  1, 3,  3;
  1, 4,  5,  4;
  1, 5, 10,  7,   5;
  1, 6, 14, 20,   9,  6;
  1, 7, 21, 30,  35, 11,  7;
  1, 8, 27, 56,  55, 56, 13,  8;
  1, 9, 36, 77, 126, 91, 84, 15, 9; ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A053123, A082985, A125176 (row sums).

[[ k eq n select n+1 else (n+k mod 2) eq 0 select Binomial(n+1,k) else Binomial(n-1, k)*(n+k)/(n-k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jun 05 2019

A125175 := proc(n,k)
        if type(n+k,'even') then
                binomial(n+1,k) ;
        else
                binomial(n-1,k)*(n+k)/(n-k) ;
        end if;
end proc: # R. J. Mathar, Sep 08 2013

Table[If[EvenQ[n+k], Binomial[n+1, k], Binomial[n-1, k]*(n+k)/(n-k)], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 05 2019 *)

{T(n,k) = if((n+k)%2==0, binomial(n+1,k), binomial(n-1, k)* (n+k)/(n-k))}; \\ G. C. Greubel, Jun 05 2019

def T(n, k):
    if (mod(n+k,2)==0): return binomial(n+1,k)
    else: return binomial(n-1, k)* (n+k)/(n-k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 05 2019

T(n,k) = binomial(n+1,k) if n+k even. T(n,k) = binomial(n-1,k)*(n+k)/(n-k) if n+k odd. - R. J. Mathar, Sep 08 2013

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A059738 Binomial transform of A054341 and inverse binomial transform of A049027.

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A047355 Numbers that are congruent to {0, 3} mod 7.

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A135092 Binomial transform of [1, 6, 1, 6, 1, 6, ...].

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A125175 Triangle T(n,k) = |A053123(n/2+k/2,k)| for even n+k, T(n,k)= A082985((n+k-1)/2,k) for odd n+k; read by rows, 0<=k<=n.

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