A033487 a(n) = n*(n+1)*(n+2)*(n+3)/4.

0, 6, 30, 90, 210, 420, 756, 1260, 1980, 2970, 4290, 6006, 8190, 10920, 14280, 18360, 23256, 29070, 35910, 43890, 53130, 63756, 75900, 89700, 105300, 122850, 142506, 164430, 188790, 215760, 245520, 278256, 314160, 353430, 396270, 442890, 493506

Offset

0,2

Comments

Non-vanishing diagonal of A132440^4/4. Third subdiagonal of unsigned A238363 without the zero. Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices of the complete graph K_4. - Tom Copeland, Apr 05 2014

Total number of pips on a set of trominoes (3-armed dominoes) with up to n pips on each arm. - Alan Shore and N. J. A. Sloane, Jan 06 2016

Also the number of minimum connected dominating sets in the (n+2)-crown graph. - Eric W. Weisstein, Jun 29 2017

Crossing number of the (n+3)-cocktail party graph (conjectured). - Eric W. Weisstein, Apr 29 2019

Sum of all numbers in ordered triples (x,y,z) where 0 <= x <= y <= z <= n. - Edward Krogius, Jul 31 2022

References

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..690

Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq., Vol. 13 (2010), Article 10.4.4.

Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.

Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.

Aleksandar Petojević and Nenad Đapić, The vAm(a,b,c;z) function, Preprint 2013.

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

Hasan Unal, Proof Without Words: Sums of Products of Three Consecutive Integers, Mathematics Magazine, Vol. 88, No. 1 (February 2015), pp. 37-38.

Eric Weisstein's World of Mathematics, Cocktail Party Graph.

Eric Weisstein's World of Mathematics, Connected Dominating Set.

Eric Weisstein's World of Mathematics, Crown Party Graph.

Eric Weisstein's World of Mathematics, Graph Crossing Number.

Index entries for sequences related to Bessel functions or polynomials

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

From Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 10 2001: (Start)

G.f.: -6*x/(x-1)^5.

a(n) = 6*binomial(n+3, 4) = a(n-1) + A007531(n+1) = 6*A000332(n) = Sum_{i=0..n} i*(i+1)*(i+2). (End)

Constant term in Bessel polynomial {y_n(x)}''.

a(n) = binomial(n+1,2)*binomial(n+3,2) = A000217(n)*A000217(n+2). - Zerinvary Lajos, May 25 2005

a(n) = binomial(n+2,2)^2 - binomial(n+2,2). - Zerinvary Lajos, May 17 2006

a(n-1) = Sum_{j=1..n} Sum_{i=2..n} i*j = Sum_{j=1..n} j*(n+2)*(n-1)/2. - Zerinvary Lajos, May 11 2007

Sum_{n>0} 1/a(n) = 2/9. - Enrique Pérez Herrero, Nov 10 2013

a(-3-n) = a(n) = 2 * binomial(binomial(n+2, 2), 2). - Michael Somos, Apr 06 2014

a(n) = A002378(binomial(n+2,2)-1). - Salvador Cerdá, Nov 04 2016

a(n) = Sum_{k=0..n} A007531(k+2). See Proof Without Words link. - Michel Marcus, Oct 29 2021

Sum_{n>=1} (-1)^(n+1)/a(n) = 16*log(2)/3 - 32/9. - Amiram Eldar, Nov 02 2021

Example

G.f. = 6*x + 30*x^2 + 90*x^3 + 210*x^4 + 420*x^5 + 756*x^6 + 1260*x^7 + ...

Maple

[seq(binomial(n+3, 4)*6, n=0..40)]; # Zerinvary Lajos, Jul 18 2006

Mathematica

Table[Times @@ (n + Range[0, 3])/4, {n, 0, 40}] (* Harvey P. Dale, Nov 27 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 6, 30, 90, 210}, 40] (* Harvey P. Dale, Nov 27 2013 *)
Table[6 Binomial[n + 3, 4], {n, 0, 20}] (* Eric W. Weisstein, Jun 29 2017 *)
Times @@@ Table[n + k, {n, 0, 20}, {k, 0, 3}]/4 (* Eric W. Weisstein, Apr 29 2019 *)

Prog

(Magma) [n*(n+1)*(n+2)*(n+3)/4: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011
(PARI) a(n)=6*binomial(n+3, 4) \\ Charles R Greathouse IV, Apr 17 2012
(PARI) concat(0, Vec(-6*x/(x-1)^5 + O(x^100))) \\ Altug Alkan, Nov 29 2015

Crossrefs

Partial sums of A007531.

Cf. A050534, A034827.

Cf. A033486, A033488. - Zerinvary Lajos, Aug 26 2008

A row of the array in A129533.

A column of the triangle in A331430.

Sequence in context: A297570 A119536 A107394 * A061138 A073948 A101855

Adjacent sequences: A033484 A033485 A033486 * A033488 A033489 A033490

Keyword

nonn,easy,changed

Author

N. J. A. Sloane

Status

approved