A111373 -id:A111373 - cp's OEIS frontend

A111396 a(n) = n(n+7)(n+8)/6.

0, 12, 30, 55, 88, 130, 182, 245, 320, 408, 510, 627, 760, 910, 1078, 1265, 1472, 1700, 1950, 2223, 2520, 2842, 3190, 3565, 3968, 4400, 4862, 5355, 5880, 6438, 7030, 7657, 8320, 9020, 9758, 10535, 11352, 12210, 13110, 14053, 15040, 16072, 17150, 18275, 19448

Offset: 0

N. J. A. Sloane, Nov 11 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A111373.

I:=[0, 12, 30, 55]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 27 2012

Table[n(n+7)(n+8)/6, {n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)
CoefficientList[Series[x*(12-18*x+7*x^2)/(x-1)^4,{x,0,50}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{0,12,30,55},40] (* Vincenzo Librandi, Jun 27 2012 *)

a(n)=n*(n+7)*(n+8)/6 \\ Charles R Greathouse IV, Oct 16 2015

[n*(n+7)*(n+8)/6 for n in (0..50)] # G. C. Greubel, Jul 30 2022

a(n) = binomial(n+8,3) - 2*binomial(n+8,2). - Zerinvary Lajos, Nov 25 2006, corrected by R. J. Mathar, Mar 15 2011
G.f.: x*(12 - 18*x + 7*x^2) /(x-1)^4. - R. J. Mathar, Mar 15 2011
a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Vincenzo Librandi, Jun 27 2012
E.g.f.: (x/6)*(72 + 18*x + x^2)*exp(x). - G. C. Greubel, Jul 30 2022
From Amiram Eldar, Jul 30 2024: (Start)
Sum_{n>=1} 1/a(n) = 1443/7840.
Sum_{n>=1} (-1)^(n+1)/a(n) = 12*log(2)/7 - 1767/1568. (End)

A126042 Expansion of f(x^3)/(1-x*f(x^3)), where f(x) is the g.f. of A001764, whose n-th term is binomial(3n,n)/(2n+1).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 8, 13, 19, 38, 64, 98, 196, 337, 531, 1062, 1851, 2974, 5948, 10468, 17060, 34120, 60488, 99658, 199316, 355369, 590563, 1181126, 2115577, 3540464, 7080928, 12731141, 21430267, 42860534, 77306428, 130771376, 261542752, 473018396, 803538100

Offset: 0

Paul Barry, Dec 16 2006

Row sums of number triangle A111373.
Interleaves T(3n,2n), T(3n+1,2n+1) and T(3n+2,2n+2) for T(n,k) = A047089(n,k).
One step forward and two steps back: number of nonnegative walks of n steps where the steps are size 1 forwards and size 2 backwards. - David Scambler, Mar 15 2011
Brown's criterion ensures that the sequence is complete (see formulae). - Vladimir M. Zarubin, Aug 05 2019
Number of ordered trees with n+1 edges, having nonroot nodes of outdegree 0 or 3. - Emanuele Munarini, Jun 20 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, The Triple Riordan Group, arXiv:2412.05461 [math.CO], 2024. See pp. 7, 10.
Eric Weisstein's World of Mathematics, Brown's Criterion.

Cf. A047089, A111373.

[n lt 3 select 1 else Binomial(n, Floor(n/3)) - (&+[Binomial(n,j): j in [0..Floor(n/3)-1]]): n in [0..40]]; // G. C. Greubel, Jul 30 2022

a:= proc(n) option remember; `if`(n<4, [1$3, 2][n+1], (a(n-1)*
       2*(20*n^4-14*n^3-31*n^2-n+8)-6*(3*n-1)*(5*n-6)*a(n-2)
      +9*(n-2)*(15*n^3-48*n^2+15*n+14)*a(n-3)-54*(n-2)*(n-3)*
      (5*n^2-n-2)*a(n-4))/(2*(2*n+1)*(n+1)*(5*n^2-11*n+4)))
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Sep 07 2022

Table[Binomial[n, Floor[n/3]] -Sum[Binomial[n,i], {i,0,Floor[n/3] -1}], {n,0,40}] (* David Callan, Oct 26 2017 *)
a[n_] := Binomial[n, Floor[n/3]] (1 + Hypergeometric2F1[1, -n + Floor[n/3], 1 + Floor[n/3], -1]) - 2^n; Table[a[n], {n, 0, 38}] (* Peter Luschny, Jun 20 2024 *)

{a(n)=polcoeff((1/x)*serreverse(x*(1+x)^2/((1+x)^3+x^3+x*O(x^n))),n)}

n=30;
{a0=1;a1=1;a2=1;for(k=1, n/3,print1(a0,", ",a1,", ",a2,", ");
a0=2*a2;a1=2*a0-binomial(3*k,k)/(2*k+1);a2=2*a1-binomial(3*k+1,k)/(k+1))
} \\ Vladimir M. Zarubin, Aug 05 2019

[binomial(n, (n//3)) - sum(binomial(n,j) for j in (0..(n//3)-1)) for n in (0..40)] # G. C. Greubel, Jul 30 2022

a(n) = Sum_{k=0..n} binomial(3*floor((n+2k)/3) - 2k, floor((n+2k)/3)-k)*(k+1)/(2*floor((n+2k)/3) - k + 1)(2*cos(2*Pi*(n-k)/3) + 1)/3.
G.f.: (1/x)*Series_Reversion( x*(1+x)^2/((1+x)^3+x^3) ). - Paul D. Hanna, Mar 15 2011
From Vladimir M. Zarubin, Aug 05 2019: (Start)
a(0) = 1, a(1) = 1, a(2) = 1 and for k>0
a(3*k) = 2*a(3*k-1),
a(3*k+1) = 2*a(3*k) - binomial(3*k,k)/(2*k+1),
a(3*k+2) = 2*a(3*k+1) - binomial(3*k+1,k)/(k+1),
where binomial(3*k,k)/(2*k+1) = A001764(k)
and binomial(3*k+1,k)/(k+1) = A006013(k). (End)
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (n-3*k+1) * binomial(n+1,k). - Seiichi Manyama, Jan 27 2024

A167543 a(n) = (n-5)(n-6)(n-7)*(n-16)/24.

Original entry on oeis.org

-2, -7, -15, -25, -35, -42, -42, -30, 0, 55, 143, 273, 455, 700, 1020, 1428, 1938, 2565, 3325, 4235, 5313, 6578, 8050, 9750, 11700, 13923, 16443, 19285, 22475, 26040, 30008, 34408, 39270, 44625, 50505, 56943, 63973, 71630, 79950, 88970, 98728, 109263, 120615

Offset: 8

Jamel Ghanouchi, Nov 06 2009

G. C. Greubel, Table of n, a(n) for n = 8..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A111373, A135929, A137276.

[Binomial(n-5,3)*(n-16)/4: n in [8..60]]; // G. C. Greubel, Jul 30 2022

Table[(n-5)*(n-6)*(n-7)*(n-16)/24, {n,8,60}] (* G. C. Greubel, Jun 15 2016 *)

[binomial(n-5,3)*(n-16)/4 for n in (8..60)] # G. C. Greubel, Jul 30 2022

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x^8*(-2+3*x)/(1-x)^5.
E.g.f.: (1/24)*( -3360 - 1800*x - 420*x^2 - 52*x^3 - 3*x^4 + (3360 - 1560*x + 300*x^2 - 28*x^3 + x^4)*exp(x) ). - G. C. Greubel, Jul 30 2022

Definition simplified, sequence extended by R. J. Mathar, Nov 12 2009

A126030 Riordan array (1/(1+x^3),x/(1+x^3)).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, -1, 0, 0, 1, 0, -2, 0, 0, 1, 0, 0, -3, 0, 0, 1, 1, 0, 0, -4, 0, 0, 1, 0, 3, 0, 0, -5, 0, 0, 1, 0, 0, 6, 0, 0, -6, 0, 0, 1, -1, 0, 0, 10, 0, 0, -7, 0, 0, 1, 0, -4, 0, 0, 15, 0, 0

Offset: 0

Paul Barry, Dec 15 2006

Inverse is A111373. Row sums are A050935(n+2). Diagonal sums are an alternating sign version of A000931(n+3) with g.f. 1/(1-x^2+x^3).

			Triangle begins
.1,
.0, 1,
.0, 0, 1,
.-1, 0, 0, 1,
.0, -2, 0, 0, 1,
.0, 0, -3, 0, 0, 1,
.1, 0, 0, -4, 0, 0, 1,
.0, 3, 0, 0, -5, 0, 0, 1,
.0, 0, 6, 0, 0, -6, 0, 0, 1,
.-1, 0, 0, 10, 0, 0, -7, 0, 0, 1,
.0, -4, 0, 0, 15, 0, 0, -8, 0, 0, 1

Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3

Number triangle T(n,k)=C(k+(n-k)/3,(n-k)/3)*(-1)^(n-k)*(2*cos(2*pi*(n-k)/3)+1)/3

cp's OEIS Frontend

A111396 a(n) = n*(n+7)*(n+8)/6.

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A126042 Expansion of f(x^3)/(1-x*f(x^3)), where f(x) is the g.f. of A001764, whose n-th term is binomial(3n,n)/(2n+1).

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A167543 a(n) = (n-5)*(n-6)*(n-7)*(n-16)/24.

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A126030 Riordan array (1/(1+x^3),x/(1+x^3)).

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A111396 a(n) = n(n+7)(n+8)/6.

A167543 a(n) = (n-5)(n-6)(n-7)*(n-16)/24.