A054340 10-fold convolution of A000302 (powers of 4).

1, 40, 880, 14080, 183040, 2050048, 20500480, 187432960, 1593180160, 12745441280, 96865353728, 704475299840, 4931327098880, 33381291130880, 219362770288640, 1403921729847296, 8774510811545600, 53679360258867200, 322076161553203200, 1898554215471513600

Offset: 0

Author

Wolfdieter Lang, Mar 13 2000

Keywords

• easy
• nonn

Comments

With a different offset, number of n-permutations (n>=9) of 5 objects: u, v, z, x, y with repetition allowed, containing exactly nine (9) u's. - Zerinvary Lajos, Jul 02 2008

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (40,-720,7680,-53760,258048,-860160,1966080,-2949120,2621440,-1048576).

Crossrefs

Cf. A000302, A054335.

Programs

GAP

List([0..20], n-> 4^n*Binomial(n+9, 9)); # G. C. Greubel, Jul 21 2019

Magma

[4^n*Binomial(n+9, 9): n in [0..20]]; // Vincenzo Librandi, Oct 15 2011

Maple

seq(binomial(n+9,9)*4^n,n=0..20); # Zerinvary Lajos, Jul 02 2008

Mathematica

Table[4^n*Binomial[n+9,9], {n,0,20}] (* G. C. Greubel, Jul 21 2019 *)

PARI

vector(20, n, n--; 4^n*binomial(n+9, 9)) \\ G. C. Greubel, Jul 21 2019

Formula

a(n) = binomial(n+9, 9)*4^n.

G.f.: 1/(1 - 4*x)^10.

a(n) = A054335(n+19, 19).

E.g.f.: (2^7/9!)*(2835 + 102060*x + 816480*x^2 + 2540160*x^3 + 3810240*x^4 + 3048192*x^5 + 1354752*x^6 + 331776*x^7 + 41472*x^8 + 2048*x^9)*exp(4*x).

From Amiram Eldar, Mar 27 2022: (Start)

Sum_{n>=0} 1/a(n) = 236196*log(4/3) - 4756383/70.

Sum_{n>=0} (-1)^n/a(n) = 14062500*log(5/4) - 43931373/14. (End)