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

1, 1, 8, 75, 804, 9681, 129168, 1889379, 30037500, 515342817, 9484627608, 186305208219, 3888697965012, 85920579594225, 2002828537732896, 49107722192594739, 1263165207424720812, 34004577057249890241, 955970215914084949800, 28011115058953357075563, 853924857091970071203972

Offset: 0

Author

Igor Victorovich Statsenko, Mar 23 2023

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• I. V. Statsenko, Application of multiharmonic numbers for the synthesis of closed forms of parametrically modified factorial generating sequences, Applied Discrete Mathematics No. 55, Tomsk State University Publishing House, 2022, pp. 5-13.

Crossrefs

For m=1 the formula gives the sequence A052852.

Cf. A288268. For m=2 the formula gives the sequence A361649.

Programs

Maple

# For recursion:
N:=10;a[0]:=1;a[1]:=1;for n from 1 to N do
a[n+1]:=(n+3)*(2*a[n]-(n-1)*a[n-1]);od;
# For closed form:
C := binomial:
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)):
seq(a(n), n = 0..20);
# Alternative:
a := n -> `if`(n=0, 1, (n + 2)!*hypergeom([1 - n], [3], -3) / 6):
seq(simplify(a(n)), n = 0..20); # Peter Luschny, Mar 23 2023

Mathematica

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 *)

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

Formula

a(n) = (m+n-1)*(2*a(n-1) - (n-2)*a(n-2)) where m=3, a(0)=a(1)=1.

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.

a(n) = (n + 2)!*hypergeom([1 - n], [3], -3) / 6 for n >= 1. - Peter Luschny, Mar 23 2023

From Vaclav Kotesovec, Mar 23 2023: (Start)

E.g.f.: 23/27 + (4 + 3*x + 2*x^3) * exp(3*x/(1-x)) / (27*(1-x)^3).

a(n) ~ exp(2*sqrt(3*n) - n - 3/2) * n^(n + 5/4) / (sqrt(2) * 3^(9/4)). (End)

Extensions

Terms a(12) and beyond from Andrew Howroyd, Mar 23 2023