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

Author

Wolfdieter Lang, Sep 15 2008

Keywords

• nonn
• easy

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

Crossrefs

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

Cf. A049451 (scaled k=2 case).

Programs

Mathematica

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

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