A075915 Seventh column of triangle A075500.

1, 140, 11550, 735000, 39991875, 1960612500, 89303500000, 3853850000000, 159664583203125, 6409926960937500, 251055710800781250, 9641722822265625000, 364483553427490234375, 13602971247133789062500, 502386213470141601562500, 18394848021467285156250000

Offset: 0

Author

Wolfdieter Lang, Oct 02 2002

Keywords

• nonn
• easy

Comments

The e.g.f. given below is Sum_{m=0..6}(A075513(7,m)exp(5*(m+1)*x))/6!.

Links

• Colin Barker, Table of n, a(n) for n = 0..646
• Index entries for linear recurrences with constant coefficients, signature (140,-8050,245000,-4230625,41037500,-204187500,393750000).

Crossrefs

Cf. A000351, A016164, A075911, A075912, A075913, A075914.

Programs

Mathematica

Table[5^(n-1) * (1 - 3*2^(7 + n) - 5*2^(14 + 2*n) + 5*3^(7 + n) + 3*5^(7 + n) - 6^(7 + n) + 7^(6 + n))/144, {n, 0, 20}] (* Vaclav Kotesovec, Dec 12 2015 *)

PARI

Vec(1/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)*(1-35*x)) + O(x^30)) \\ Colin Barker, Dec 12 2015

Formula

a(n) = A075500(n+7, 7) = (5^n)S2(n+7, 7) with S2(n, m) = A008277(n, m) (Stirling2).

a(n) = Sum_{m=0..6}(A075513(7, m)*(5*(m+1))^n)/6!.

G.f.: 1/Product_{k=1..7}(1-5k*x).

E.g.f.: (d^7/dx^7)((((exp(5x)-1)/5)^7)/7!) = (exp(5*x) - 384*exp(10*x) + 10935*exp(15*x) - 81920*exp(20*x) + 234375*exp(25*x) - 279936*exp(30*x) + 117649*exp(35*x))/6!.

G.f.: 1 / ((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)*(1-35*x)). - Colin Barker, Dec 12 2015