A387152 -id:A387152 - cp's OEIS frontend

A387204 a(n) = n(n + 1)(n^2 + 17*n + 54)/24.

0, 6, 23, 57, 115, 205, 336, 518, 762, 1080, 1485, 1991, 2613, 3367, 4270, 5340, 6596, 8058, 9747, 11685, 13895, 16401, 19228, 22402, 25950, 29900, 34281, 39123, 44457, 50315, 56730, 63736, 71368, 79662, 88655, 98385, 108891, 120213, 132392, 145470, 159490, 174496

Offset: 0

Peter Luschny, Aug 27 2025

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A387152 (row 4).

a := n -> n*(n + 1)*(n^2 + 17*n + 54)/24: seq(a(n), n = 0..41);

A387204[n_] := n*(n + 1)*(n*(n + 17) + 54)/24;
Array[A387204, 50, 0] (* Paolo Xausa, Aug 29 2025 *)

a(n) = [x^n] (-2*x^3 + 7*x^2 - 6*x)/(x - 1)^5.

a(n) = [x^n] Pochhammer(x, 4)*(x + 1)^n.

A387205 a(n) = (n - 1)!*(2 + Harmonic(n - 1)) if n >= 1, and a(0) = 1.

Original entry on oeis.org

1, 2, 3, 7, 23, 98, 514, 3204, 23148, 190224, 1752336, 17886240, 200377440, 2444446080, 32256800640, 457822229760, 6954511737600, 112579862169600, 1934780446771200, 35181735469977600, 674855347635302400, 13618752053114880000, 288426695123589120000, 6396478234890670080000

Offset: 0

Peter Luschny, Aug 27 2025

Paolo Xausa, Table of n, a(n) for n = 0..400

Cf. A387152 (column 2), A001008, A130534.

a := n -> if n = 0 then 1 else (n-1)!*(2 + harmonic(n-1)) fi:
ser := series(LaguerreL(2, log(1 - x)), x, 24): a := n -> n! * coeff(ser, x, n):
seq(a(n), n = 0..23);

A387205[n_] := If[n == 0, 1, (n - 1)!*(2 + HarmonicNumber[n - 1])];
Array[A387205, 25, 0] (* Paolo Xausa, Aug 29 2025 *)

a(n) = if (n>0, (n-1)!*(2 + sum(i=1, n-1, 1/i)), 1); \\ Michel Marcus, Aug 27 2025

a(n) = 2*|Stirling1(n, 1)| + |Stirling1(n, 2)| for n >= 1.
a(n) = n! * [x^n] Laguerre(2, log(1 - x)).
a(n) = Gamma(n)*(PolyGamma(n) + EulerGamma + 2) for n >= 1.
Conjecture: Maple returns the exponential series expansion at x = 0:
a(n) = n! * [x^n] (1 + tau + (log(x - 1)^2 - (tau + 4)*log(x - 1) - Pi^2)/2) where tau = 2*Pi*I.

cp's OEIS Frontend

A387204 a(n) = n(n + 1)(n^2 + 17*n + 54)/24.

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Formula

A387205 a(n) = (n - 1)!*(2 + Harmonic(n - 1)) if n >= 1, and a(0) = 1.

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cp's OEIS Frontend

A387204 a(n) = n*(n + 1)*(n^2 + 17*n + 54)/24.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A387205 a(n) = (n - 1)!*(2 + Harmonic(n - 1)) if n >= 1, and a(0) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A387204 a(n) = n(n + 1)(n^2 + 17*n + 54)/24.