A034687 Related to quintic factorial numbers A008548.

1, 15, 275, 5500, 115500, 2502500, 55412500, 1246781250, 28398906250, 653174843750, 15141780468750, 353308210937500, 8289154179687500, 195387205664062500, 4624163867382812500, 109823891850341796875

Offset: 1

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Convolution of A034688(n-1) with A025750(n), n >= 1.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..717 (first 500 terms from G. C. Greubel).
• Wolfdieter Lang, On generalizations of the Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
• Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Crossrefs

Cf. A008548, A025750, A034255, A034688, A175380.

Programs

GAP

List([1..20], n-> 5^(n-1)*Product([0..n-1], k-> 5*k+1)/Factorial(n)); # G. C. Greubel, Aug 17 2019

Magma

[5^(n-1)*(&*[5*k+1: k in [0..n-1]])/Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 17 2019

Maple

seq(5^(n-1)*(product(5*k+1, k = 0..n-1))/factorial(n), n = 1..20); # G. C. Greubel, Aug 17 2019

Mathematica

Table[5^(2*n-1)*Pochhammer[1/5, n]/n!, {n, 20}] (* G. C. Greubel, Aug 17 2019 *)

PARI

vector(20, n, 5^(n-1)*prod(k=0, n-1, 5*k+1)/n!) \\ G. C. Greubel, Aug 17 2019

Sage

[5^(n-1)*product(5*k+1 for k in (0..n-1))/factorial(n) for n in (1..20)] # G. C. Greubel, Aug 17 2019

Formula

a(n) = 5^(n-1)*A008548(n)/n!, where A008548(n) = (5*n-4)(!^5) = Product_{j=1..n} (5*j-4).

G.f.: (-1 + (1-25*x)^(-1/5))/5.

E.g.f.: (1/5)*L_{-1/5}(25*x) - 1, where L_{k}(x) is the Laguerre polynomial. - Stefano Spezia, Aug 17 2019

a(n) ~ 5^(2*n-1) * n^(-4/5) / Gamma(1/5). - Amiram Eldar, Aug 17 2025