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%I #32 Sep 08 2022 08:46:23
%S 1,1,15,337,10081,376461,16849351,878797165,52324954977,3501300491641,
%T 260062721279551,21228108532279881,1888618754806601665,
%U 181873163575529411077,18846187172580219099831,2090754000231731874682021,247221828043044971020645441
%N E.g.f.: exp(x/(1-7*x)).
%C For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k.
%H Ludovic Schwob, <a href="/A321850/b321850.txt">Table of n, a(n) for n = 0..199</a>
%F a(n) = Sum_{k=0..n} 7^(n-k)*(n!/k!)*binomial(n-1, k-1).
%F Recurrence: a(n) = (14*n-13)*a(n-1) - 49*(n-2)*(n-1)*a(n-2).
%F a(n) ~ n! * exp(2*sqrt(n/7) - 1/14) * 7^(n - 1/4) / (2 * sqrt(Pi) * n^(3/4)). - _Vaclav Kotesovec_, Nov 21 2018
%p seq(coeff(series(factorial(n)*exp(x/(1-7*x)),x,n+1), x, n), n = 0 .. 17); # _Muniru A Asiru_, Nov 24 2018
%t a[n_] := Sum[7^(n - k)*n!/k!*Binomial[n - 1, k - 1], {k, 0, n}]; Array[a, 20, 0] (* or *) a[0] = a[1] = 1; a[n_] := a[n] = (14n - 13)*a[n - 1] - 49(n - 2)(n - 1)*a[n - 2]; Array[a, 20, 0] (* _Amiram Eldar_, Nov 19 2018 *)
%o (PARI) my(x='x + O('x^20)); Vec(serlaplace(exp(x/(1-7*x)))) \\ _Michel Marcus_, Nov 25 2018
%o (Magma) [1] cat [&+[7^(n-k)*Factorial(n)/Factorial(k)*Binomial(n-1, k-1): k in [0..n]]: n in [1.. 18]]; // _Vincenzo Librandi_, Dec 08 2018
%Y Cf. A000262, A025168, A321837, A321847, A321848, A321849.
%K nonn
%O 0,3
%A _Ludovic Schwob_, Nov 19 2018
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