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a(n) = n*(n+1)*(n+2)*(n+3)/4.
24

%I #125 Jul 08 2024 10:37:38

%S 0,6,30,90,210,420,756,1260,1980,2970,4290,6006,8190,10920,14280,

%T 18360,23256,29070,35910,43890,53130,63756,75900,89700,105300,122850,

%U 142506,164430,188790,215760,245520,278256,314160,353430,396270,442890,493506

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

%C 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

%C 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

%C Also the number of minimum connected dominating sets in the (n+2)-crown graph. - _Eric W. Weisstein_, Jun 29 2017

%C Crossing number of the (n+3)-cocktail party graph (conjectured). - _Eric W. Weisstein_, Apr 29 2019

%C Sum of all numbers in ordered triples (x,y,z) where 0 <= x <= y <= z <= n. - _Edward Krogius_, Jul 31 2022

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

%H Vincenzo Librandi, <a href="/A033487/b033487.txt">Table of n, a(n) for n = 0..690</a>

%H Steve Butler and Pavel Karasik, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Butler/butler7.html">A note on nested sums</a>, J. Int. Seq., Vol. 13 (2010), Article 10.4.4.

%H Ömür Deveci and Anthony G. Shannon, <a href="https://doi.org/10.20948/mathmontis-2021-50-4">Some aspects of Neyman triangles and Delannoy arrays</a>, Mathematica Montisnigri (2021) Vol. L, 36-43.

%H Sela Fried, <a href="https://arxiv.org/abs/2406.18923">Counting r X s rectangles in nondecreasing and Smirnov words</a>, arXiv:2406.18923 [math.CO], 2024. See p. 9.

%H Aleksandar Petojević and Nenad Đapić, <a href="http://www.mi.sanu.ac.rs/~gvm/radovi/AP-Budva.pdf">The vAm(a,b,c;z) function</a>, Preprint 2013.

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

%H Hasan Unal, <a href="https://www.jstor.org/stable/10.4169/math.mag.88.1.37">Proof Without Words: Sums of Products of Three Consecutive Integers</a>, Mathematics Magazine, Vol. 88, No. 1 (February 2015), pp. 37-38.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConnectedDominatingSet.html">Connected Dominating Set</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CrownGraph.html">Crown Party Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCrossingNumber.html">Graph Crossing Number</a>.

%H <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

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

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

%F 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)

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

%F a(n) = binomial(n+1,2)*binomial(n+3,2) = A000217(n)*A000217(n+2). - _Zerinvary Lajos_, May 25 2005

%F a(n) = binomial(n+2,2)^2 - binomial(n+2,2). - _Zerinvary Lajos_, May 17 2006

%F 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

%F Sum_{n>0} 1/a(n) = 2/9. - _Enrique Pérez Herrero_, Nov 10 2013

%F a(-3-n) = a(n) = 2 * binomial(binomial(n+2, 2), 2). - _Michael Somos_, Apr 06 2014

%F a(n) = A002378(binomial(n+2,2)-1). - _Salvador Cerdá_, Nov 04 2016

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

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

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

%p [seq(binomial(n+3,4)*6, n=0..40)]; # _Zerinvary Lajos_, Jul 18 2006

%t Table[Times @@ (n + Range[0, 3])/4, {n, 0, 40}] (* _Harvey P. Dale_, Nov 27 2013 *)

%t LinearRecurrence[{5, -10, 10, -5, 1}, {0, 6, 30, 90, 210}, 40] (* _Harvey P. Dale_, Nov 27 2013 *)

%t Table[6 Binomial[n + 3, 4], {n, 0, 20}] (* _Eric W. Weisstein_, Jun 29 2017 *)

%t Times @@@ Table[n + k, {n, 0, 20}, {k, 0, 3}]/4 (* _Eric W. Weisstein_, Apr 29 2019 *)

%o (Magma) [n*(n+1)*(n+2)*(n+3)/4: n in [0..40]]; // _Vincenzo Librandi_, Apr 28 2011

%o (PARI) a(n)=6*binomial(n+3,4) \\ _Charles R Greathouse IV_, Apr 17 2012

%o (PARI) concat(0, Vec(-6*x/(x-1)^5 + O(x^100))) \\ _Altug Alkan_, Nov 29 2015

%Y Partial sums of A007531.

%Y Cf. A050534, A034827.

%Y Cf. A033486, A033488. - _Zerinvary Lajos_, Aug 26 2008

%Y A row of the array in A129533.

%Y A column of the triangle in A331430.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_