cp's OEIS Frontend

A361528 a(n) = (2+n)*(2*a(n-1) - (n-2)*a(n-2)) with a(0)=a(1)=1.

%I A361528 #61 Mar 26 2025 08:31:42
%S A361528 1,1,8,75,804,9681,129168,1889379,30037500,515342817,9484627608,
%T A361528 186305208219,3888697965012,85920579594225,2002828537732896,
%U A361528 49107722192594739,1263165207424720812,34004577057249890241,955970215914084949800,28011115058953357075563,853924857091970071203972
%N A361528 a(n) = (2+n)*(2*a(n-1) - (n-2)*a(n-2)) with a(0)=a(1)=1.
%H A361528 Andrew Howroyd, <a href="/A361528/b361528.txt">Table of n, a(n) for n = 0..200</a>
%H A361528 I. V. Statsenko, <a href="https://www.mathnet.ru/eng/pdm757">Application of multiharmonic numbers for the synthesis of closed forms of parametrically modified factorial generating sequences</a>, Applied Discrete Mathematics No. 55, Tomsk State University Publishing House, 2022, pp. 5-13.
%F A361528 a(n) = (m+n-1)*(2*a(n-1) - (n-2)*a(n-2)) where m=3, a(0)=a(1)=1.
%F A361528 a(n) = Sum_{i=0..n-1} binomial(n-1,i) * binomial(n+m-1,n-i)*(n-i)!*m^(i-1) where m = 3 for n >= 1.
%F A361528 a(n) = (n + 2)!*hypergeom([1 - n], [3], -3) / 6 for n >= 1. - _Peter Luschny_, Mar 23 2023
%F A361528 From _Vaclav Kotesovec_, Mar 23 2023: (Start)
%F A361528 E.g.f.: 23/27 + (4 + 3*x + 2*x^3) * exp(3*x/(1-x)) / (27*(1-x)^3).
%F A361528 a(n) ~ exp(2*sqrt(3*n) - n - 3/2) * n^(n + 5/4) / (sqrt(2) * 3^(9/4)). (End)
%p A361528 # For recursion:
%p A361528 N:=10;a[0]:=1;a[1]:=1;for n from 1 to N do
%p A361528 a[n+1]:=(n+3)*(2*a[n]-(n-1)*a[n-1]);od;
%p A361528 # For closed form:
%p A361528 C := binomial:
%p A361528 a := n -> `if`(n=0, 1, add(C(n-1, i)*C(n+2, n-i)*(n-i)!*3^(i-1), i = 0..n-1)):
%p A361528 seq(a(n), n = 0..20);
%p A361528 # Alternative:
%p A361528 a := n -> `if`(n=0, 1, (n + 2)!*hypergeom([1 - n], [3], -3) / 6):
%p A361528 seq(simplify(a(n)), n = 0..20); # _Peter Luschny_, Mar 23 2023
%t A361528 nmax = 20; CoefficientList[Series[23/27 + (4 + 3*x + 2*x^3)*E^(3*x/(1 - x))/(27*(1 - x)^3), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Mar 23 2023 *)
%o A361528 (PARI) a(n) = if(n==0, 1, my(m=3); sum(i=0, n-1, binomial(n-1, i)*binomial(n+m-1, n-i)*(n-i)!*m^(i-1))) \\ _Andrew Howroyd_, Mar 23 2023
%Y A361528 For m=1 the formula gives the sequence A052852.
%Y A361528 Cf. A288268. For m=2 the formula gives the sequence A361649.
%K A361528 nonn
%O A361528 0,3
%A A361528 _Igor Victorovich Statsenko_, Mar 23 2023
%E A361528 Terms a(12) and beyond from _Andrew Howroyd_, Mar 23 2023