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URL: https://oeis.org/A034009

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A034009
Convolution of A000295(n+2) (n>=0) with itself.
9
1, 8, 38, 140, 443, 1268, 3384, 8584, 20965, 49744, 115402, 262996, 590831, 1311900, 2884956, 6293040, 13633305, 29362200, 62916910, 134220380, 285215651, 603983108, 1275072128, 2684358680, 5637149133, 11811165088
OFFSET
0,2
LINKS
Florent Hivert, Vincent Pilaud, and Ludovic Schwob, Heaps of rhombic dodecahedra, catalan congruences on alternating sign matrices, and bases of the Temperley-Lieb algebra, arXiv:2511.06968 [math.CO], 2025. See Table 3 p. 32.
FORMULA
(2^(n+2)-n-3) '*' (2^(n+2)-n-3) where '*' denotes the convolution product.
G.f.: 1/((1-2*x)*(1-x)^2)^2.
Partial sums of A045889.
a(n) = (n-3)*2^(n+4)+binomial(n+3,3)+4*(binomial(n+1,2)+4*n+12)
= 2^(n+4)*(n-3)+(n+7)*(n*(n+11)+42)/6.
a(n) = binomial(n+3,3)*hypergeom([2,-n],[-n-3],2). - Peter Luschny, Sep 19 2014
a(n) = Sum_{k=0..n+4} Sum_{i=0..n+4} (i-k) * C(n-k+4,i+2). - Wesley Ivan Hurt, Sep 19 2017
MAPLE
seq(16*(n-3)*2^n+(n+7)*(n^2+11*n+42)/6, n=0..100); # Robert Israel, Sep 19 2014
MATHEMATICA
Table[Sum[ k Binomial[n + 5, k + 4], {k, 0, n+1}], {n, 0, 26}] (* Zerinvary Lajos, Jul 08 2009 *)
Table[(16 (n-3) 2^n + (n + 7) (n^2 + 11 n + 42) / 6), {n, 0, 40}] (* Vincenzo Librandi, Sep 20 2014 *)
PROG
(Magma) [(16*(n-3)*2^n+(n+7)*(n^2+11*n+42) div 6): n in [0..30]]; // Vincenzo Librandi, Sep 20 2014
CROSSREFS
Sequence in context: A359931 A211063 A065762 * A038732 A038799 A156934
KEYWORD
easy,nonn
EXTENSIONS
Edited by Peter Luschny, Sep 20 2014
STATUS
approved