A142962 Scaled convolution of (n^3)*A000984(n) with A000984(n).

4, 26, 81, 184, 350, 594, 931, 1376, 1944, 2650, 3509, 4536, 5746, 7154, 8775, 10624, 12716, 15066, 17689, 20600, 23814, 27346, 31211, 35424, 40000, 44954, 50301, 56056, 62234, 68850, 75919, 83456, 91476, 99994, 109025, 118584, 128686, 139346, 150579

Offset

1,1

Comments

S(3,n) := Sum_{j=0..n} j^3*binomial(2*j,j)*binomial(2*(n-j),n-j).

a(n) = 2^3*S(3,n)/4^n, n >= 1.

O.g.f. for S(3,n) is G(k=3,x). See triangle A142963 for the general G(k,x) formula.

The author was led to compute such sums by a question asked by M. Greiter, Jun 27 2008.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Formula

a(n) = n^2*(3+5*n)/2.

a(n) = (2^3)*S(3,n)/4^n with the convolution S(3,n) defined above.

G.f.: x*(4+10*x+x^2)/(1-x)^4. - Joerg Arndt, Jul 02 2023

Mathematica

Rest@ CoefficientList[Series[x (4 + 10 x + x^2)/(1 - x)^4, {x, 0, 39}], x] (* Michael De Vlieger, Jul 02 2023 *)

Crossrefs

Cf. A142961 triangle: row k=3: [3, 5], with the row polynomial 3+5*n.

Cf. A049451 (scaled k=2 case).

Sequence in context: A014450 A283573 A200058 * A247194 A102198 A100207

Adjacent sequences: A142959 A142960 A142961 * A142963 A142964 A142965

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 15 2008

Status

approved