A129995 a(n) = (n + 1)*(n^2 + 2)*(n^3 + 3)*(n^4 + 4)/4!.

1, 5, 165, 4675, 65325, 543456, 3155425, 14146210, 52259625, 166192975, 469090061, 1201490445, 2839166005, 6268589250, 13060542825, 25881747316, 49095506065, 89615392425, 158091087925, 270522770375, 450420100221, 731644012660, 1162094343345, 1808433948150

Offset: 0

Author

Alexander R. Povolotsky, Aug 19 2007, Aug 25 2007

Keywords

• nonn
• easy

Comments

Following my conjecture, computations by Peter J. C. Moses, mediation by Clark Kimberling and helpful comments from George E. Andrews, it is now known that a(n) = (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^k + k)/k! is an integer-valued sequence if and only if k belongs to {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21}; this is the case for k=4; see generalization in A131685.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Crossrefs

Cf. A000027 (k=1), A064808 (k=2), A131509 (k=3), this sequence (k=4), A131675 (k=5), ..., A131680 (k=10).

See A131685 for a generalization.

Programs

Magma

[(n^1 + 1)*(n^2 + 2)*(n^3 + 3)*(n^4 + 4)/24: 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,4),n=0..30)];
seq((n+1)*(n^2+2)*(n^3+3)*(n^4+4)/factorial(4), n = 0 .. 20) # Emeric Deutsch, Aug 26 2007

Mathematica

Table[x = 4; Product[(n^k) + k, {k, x}]/x!, {n, 0, 23}] (* Michael De Vlieger, Apr 24 2015 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,5,165,4675,65325,543456,3155425,14146210,52259625,166192975,469090061},30] (* Harvey P. Dale, Dec 07 2021 *)

PARI

vector(20,n,n--;(n+1)*(n^2+2)*(n^3+3)*(n^4+4)/4!) \\ Derek Orr, Apr 25 2015

PARI

A129995(n)=(n+1)*(n^2+2)*(n^3+3)*(n^4+4)/12 \\ M. F. Hasler, May 02 2015

Formula

G.f.: (1-6x+165x^2+2970x^3+22480x^4+55969x^5+51511x^6+16490x^7+1595x^8+25x^9)/(1-x)^11. - Emeric Deutsch, Aug 26 2007

G.f.: -(1 + x*(-6 + x*(165 + x*(2970 + x*(22480 + x*(55969 + x*(51511 + 5*x*(3298 + x*(319 + 5*x))))))))) / (x - 1)^11. - Peter J. C. Moses, Aug 29 2007