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

0,2

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).

Formula

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

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 *)

Prog

(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
(Magma) [(n^1 + 1)*(n^2 + 2)*(n^3 + 3)/6: n in [0..30]]; // Vincenzo Librandi, Apr 25 2015
(PARI) A131509(n)=(n+1)*(n^2+2)*(n^3+3)/6 \\ M. F. Hasler, May 02 2015

Crossrefs

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

Sequence in context: A273700 A273708 A097705 * A221030 A081007 A213168

Adjacent sequences: A131506 A131507 A131508 * A131510 A131511 A131512

Keyword

nonn,easy

Author

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

Extensions

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

Status

approved