A300254 a(n) = 25*(n + 1)*(4*n + 3)*(5*n + 4)/3.

100, 1050, 3850, 9500, 19000, 33350, 53550, 80600, 115500, 159250, 212850, 277300, 353600, 442750, 545750, 663600, 797300, 947850, 1116250, 1303500, 1510600, 1738550, 1988350, 2261000, 2557500, 2878850, 3226050, 3600100, 4002000, 4432750, 4893350, 5384800, 5908100, 6464250

Offset

0,1

Comments

Hirschhorn has discovered that p(20*n+11,4) + p(20*n+12,4) + p(20*n+13,4) = 25*(n + 1)*(4*n + 3)*(5*n + 4)/3, where p(m,k) denote the number of partitions of m into at most k parts. Therefore, p(20*n+11,4) + p(20*n+12,4) + p(20*n+13,4) == 0 (mod 50) [see Hirschhorn's paper in References section].

a(n) == 0 (mod 3) if n is of the form 2*h + 3*floor(h/3 + 2/3) + 1.

a(n) == 0 (mod 7) if n is a member of A047278.

References

Michael D. Hirschhorn, Congruences modulo 5 for partitions into at most four parts, The Fibonacci Quarterly, Vol. 56, Number 1, 2018, pages 32-37 [the equation 1.7 contains a typo].

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

O.g.f.: 50*(2 + 13*x + 5*x^2)/(1 - x)^4 [formula 4.3 in Hirschhorn's paper].

E.g.f.: 25*(12 + 114*x + 111*x^2 + 20*x^3)*exp(x)/3.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4)

a(n) = A014112(10*n+8) = A212964(10*n+9) = A228958(10*n+8) = A268684(5*n+4).

Mathematica

Table[25 (n + 1) (4 n + 3) (5 n + 4)/3, {n, 0, 40}]

Prog

(PARI) vector(40, n, n--; 25*(n+1)*(4*n+3)*(5*n+4)/3)
(Sage) [25*(n+1)*(4*n+3)*(5*n+4)/3 for n in (0..40)]
(Maxima) makelist(25*(n+1)*(4*n+3)*(5*n+4)/3, n, 0, 40);
(GAP) List([0..40], n -> 25*(n+1)*(4*n+3)*(5*n+4)/3);
(Magma) [25*(n+1)*(4*n+3)*(5*n+4)/3: n in [0..40]];
(Python) [25*(n+1)*(4*n+3)*(5*n+4)/3 for n in range(40)]
(Julia) [div(25*(n+1)*(4*n+3)*(5*n+4), 3) for n in 0:40] |> println
(PARI) Vec(50*(2 + 13*x + 5*x^2) / (1 - x)^4 + O(x^60)) \\ Colin Barker, Mar 13 2018

Crossrefs

Subsequence of A014112, A212964, A228958, A268684.

Cf. A300522, A300523.

Sequence in context: A267153 A343294 A025421 * A124166 A103175 A163450

Adjacent sequences: A300251 A300252 A300253 * A300255 A300256 A300257

Keyword

nonn,easy

Author

Bruno Berselli, Mar 12 2018

Status

approved