A154322 a(n) = 1 + n + binomial(n+3,5).

1, 2, 4, 10, 26, 62, 133, 260, 471, 802, 1298, 2014, 3016, 4382, 6203, 8584, 11645, 15522, 20368, 26354, 33670, 42526, 53153, 65804, 80755, 98306, 118782, 142534, 169940, 201406, 237367, 278288, 324665, 377026, 435932, 501978, 575794, 658046, 749437, 850708

Offset: 0

Author

Paul Barry, Jan 07 2009

Keywords

• easy
• nonn

Comments

Row sums of number triangle A113582.

It appears that the sequence is the pairwise sum of terms in A101338 and A000389 with offsets as follows:

1, 2, 4, 10, 26, 62, 133, 260, 471, 802, 1298, ... =

1, 2, 4, 9, 20, 41, 77, 134, 219, 340, 506, ... +

0, 0, 0, 1, 6, 21, 56, 126, 252, 462, 792, ...

- Gary W. Adamson, Oct 08 2015

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A000389, A101338, A113582.

Programs

Magma

[1+n+Binomial(n+3,5) : n in [0..50]]; // Wesley Ivan Hurt, Oct 08 2015

Magma

I:=[1,2,4,10,26,62]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Oct 09 2015

Maple

A154322:=n->1+n+binomial(n+3,5): seq(A154322(n), n=0..50); # Wesley Ivan Hurt, Oct 08 2015

Mathematica

CoefficientList[Series[(1 - 4*x + 7*x^2 - 4*x^3 + x^4)/(1 - x)^6, {x, 0, 40}], x] (* Wesley Ivan Hurt, Oct 08 2015 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 2, 4, 10, 26, 62}, 50] (* Vincenzo Librandi, Oct 09 2015 *)
 Table[ (n + 1)*(n^4 + 4*n^3 + n^2 - 6*n + 120)/120 , {n, 0, 25}] (* G. C. Greubel, Sep 10 2016 *)
Table[1+n+Binomial[n+3,5],{n,0,40}] (* Harvey P. Dale, Jan 19 2023 *)

PARI

Vec((1-4*x+7*x^2-4*x^3+x^4)/(1-x)^6 + O(x^100)) \\ Altug Alkan, Oct 18 2015

Formula

G.f.: (1 - 4*x + 7*x^2 - 4*x^3 + x^4)/(1-x)^6;

a(n) = n + 1 + Sum_{k=0..n} binomial(k+1,2) * binomial(n-k+1,2).

a(n) = (n+1)*(n^4 +4*n^3 +n^2 -6*n +120)/120.

a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6) for n>5. - Wesley Ivan Hurt, Oct 08 2015

E.g.f.: (1/120)*(120 + 120*x + 60*x^2 + 60*x^3 + 15*x^4 + x^5)*exp(x). - G. C. Greubel, Sep 10 2016