A361649 -id:A361649 - cp's OEIS frontend

A361528 a(n) = (2+n)(2a(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

Igor Victorovich Statsenko, Mar 23 2023

nonn

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.

For m=1 the formula gives the sequence A052852.

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

# 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

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 *)

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

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)

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

A377058 Triangle of generalized Stirling numbers of the lower level of the hierarchy (case m=2).

Original entry on oeis.org

1, 5, 1, 32, 11, 1, 248, 113, 18, 1, 2248, 1230, 263, 26, 1, 23272, 14534, 3765, 505, 35, 1, 270400, 186992, 55654, 9115, 865, 45, 1, 3479744, 2612000, 865186, 163779, 19110, 1372, 56, 1, 49079936, 39434448, 14235388, 3013164, 408569, 36288, 2058, 68, 1

Offset: 0

Igor Victorovich Statsenko, Oct 14 2024

nonn

These numbers are a subset of the generalized Stirling numbers introduced in A370518. Therefore, we assume them to be numbers of the lower level of hierarchy with respect to A370518.

			[0]           1;
[1]           5,          1;
[2]          32,         11,         1;
[3]         248,        113,        18,        1;
[4]        2248,       1230,       263,       26,       1;
[5]       23272,      14534,      3765,      505,      35,      1;
[6]      270400,     186992,     55654,     9115,     865,     45,     1;
[7]     3479744,    2612000,    865186,   163779,   19110,   1372,    56,    1;
[8]    49079936,   39434448,  14235388,  3013164,  408569,  36288,  2058,   68,    1;

Igor Victorovich Statsenko, Relationships of "P"-generalized Stirling numbers of the first kind with other generalized Stirling numbers, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2024, pp. 19-22. In Russian.

A361649 (row sums).
Triangle for m=0: A130534.
Triangle for m=1: A376863.

T := (m,n,k) -> add(add(Stirling1(n-j,k)*binomial(n+m,i)*binomial(n,j)*binomial(j,i)*i!*m^(j-i), j=i..n), i=0..n): m:=2: seq(seq(T(m,n,k), k=0..n), n=0..10);

T(m, n, k) = Sum_{i=0..n} Sum_{j=i..n} Stirling1(n-j, k)*binomial(n+m, i)*binomial(n, j)* binomial(j, i)*i!*m^(j-i), for m = 2.

cp's OEIS Frontend

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

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A377058 Triangle of generalized Stirling numbers of the lower level of the hierarchy (case m=2).

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cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A377058 Triangle of generalized Stirling numbers of the lower level of the hierarchy (case m=2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

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