A131509 a(n) = (n + 1)*(n^2 + 2)*(n^3 + 3)/6.

1, 4, 33, 220, 1005, 3456, 9709, 23528, 50985, 101260, 187561, 328164, 547573, 877800, 1359765, 2044816, 2996369, 4291668, 6023665, 8303020, 11260221, 15047824, 19842813, 25849080, 33300025, 42461276, 53633529, 67155508, 83407045, 102812280, 125842981

Offset: 0

Author

Alexander R. Povolotsky, Aug 13 2007, Aug 25 2007

Keywords

• nonn
• easy

Comments

See also A131685(k) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^k+ k) / k! takes integral values for all i>=0. For k=3, A131685(k)=1, which implies that this is a well defined integer sequence. - Alexander R. Povolotsky, May 18 2015

Links

• M. F. Hasler, Table of n, a(n) for n = 0..100
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000027 (k=1), A064808 (k=2), this sequence (k=3), A129995 (k=4), A131675 (k=5), ..., A131680 (k=10).

Programs

Magma

[(n^1 + 1)*(n^2 + 2)*(n^3 + 3)/6: n in [0..30]]; // Vincenzo Librandi, Apr 25 2015

Maple

p:=proc(n,i) mul( n^j+j, j=1..i)/i!; end; [seq(p(n,3),n=0..30)];
seq((1/6)*(n+1)*(n^2+2)*(n^3+3),n=0..25); # Emeric Deutsch, Aug 23 2007

Mathematica

Table[x = 3; Product[(n^k) + k, {k, x}]/6, {n, 0, 27}] (* Michael De Vlieger, Apr 24 2015 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,4,33,220,1005,3456,9709},40] (* Harvey P. Dale, Oct 18 2016 *)

Maxima

A131509(n):=(n^1 + 1)*(n^2 + 2)*(n^3 + 3)/6$
makelist(A131509(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

PARI

vector(20,n,n--;(n+1)*(n^2+2)*(n^3+3)/3!) \\ Derek Orr, Apr 25 2015

PARI

A131509(n)=(n+1)*(n^2+2)*(n^3+3)/6 \\ M. F. Hasler, May 02 2015

Formula

G.f.: (1 -3x +26x^2 +38x^3 +53x^4 +5x^5)/(1-x)^7. - Emeric Deutsch, Aug 23 2007

Extensions

Corrected and extended by R. J. Mathar and Emeric Deutsch, Aug 21 2007