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

%I A033487 #139 Jul 05 2025 14:29:42
%S A033487 0,6,30,90,210,420,756,1260,1980,2970,4290,6006,8190,10920,14280,
%T A033487 18360,23256,29070,35910,43890,53130,63756,75900,89700,105300,122850,
%U A033487 142506,164430,188790,215760,245520,278256,314160,353430,396270,442890,493506,548340,607620
%N A033487 a(n) = n*(n+1)*(n+2)*(n+3)/4.
%C A033487 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 A033487 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 A033487 Also the number of minimum connected dominating sets in the (n+2)-crown graph. - _Eric W. Weisstein_, Jun 29 2017
%C A033487 Crossing number of the (n+3)-cocktail party graph (conjectured). - _Eric W. Weisstein_, Apr 29 2019
%C A033487 Sum of all numbers in ordered triples (x,y,z) where 0 <= x <= y <= z <= n. - _Edward Krogius_, Jul 31 2022
%D A033487 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
%H A033487 Vincenzo Librandi, <a href="/A033487/b033487.txt">Table of n, a(n) for n = 0..690</a>
%H A033487 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 A033487 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 A033487 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 A033487 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 A033487 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 A033487 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 A033487 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CocktailPartyGraph.html">Cocktail Party Graph</a>.
%H A033487 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ConnectedDominatingSet.html">Connected Dominating Set</a>.
%H A033487 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CrownGraph.html">Crown Party Graph</a>.
%H A033487 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCrossingNumber.html">Graph Crossing Number</a>.
%H A033487 <a href="/index/Be#Bessel">Index entries for sequences related to Bessel functions or polynomials</a>
%H A033487 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A033487 From Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 10 2001: (Start)
%F A033487 G.f.: 6*x/(1-x)^5.
%F A033487 a(n) = 6*binomial(n+3, 4) = 6*A000332(n+3).
%F A033487 a(n) = a(n-1) + A007531(n+1).
%F A033487 a(n) = Sum_{i=0..n} i*(i+1)*(i+2). (End)
%F A033487 Constant term in Bessel polynomial {y_n(x)}''.
%F A033487 a(n) = binomial(n+1,2)*binomial(n+3,2) = A000217(n)*A000217(n+2). - _Zerinvary Lajos_, May 25 2005
%F A033487 a(n) = binomial(n+2,2)^2 - binomial(n+2,2). - _Zerinvary Lajos_, May 17 2006
%F A033487 From _Zerinvary Lajos_, May 11 2007: (Start)
%F A033487 a(n-1) = Sum_{j=1..n} Sum_{i=2..n} i*j.
%F A033487 a(n) = Sum_{j=1..n} j*(n+2)*(n-1)/2. (End)
%F A033487 Sum_{n>0} 1/a(n) = 2/9. - _Enrique Pérez Herrero_, Nov 10 2013
%F A033487 a(-3-n) = a(n) = 2 * binomial(binomial(n+2, 2), 2). - _Michael Somos_, Apr 06 2014
%F A033487 a(n) = A002378(binomial(n+2,2)-1). - _Salvador Cerdá_, Nov 04 2016
%F A033487 a(n) = Sum_{k=0..n} A007531(k+2). See Proof Without Words link. - _Michel Marcus_, Oct 29 2021
%F A033487 Sum_{n>=1} (-1)^(n+1)/a(n) = 16*log(2)/3 - 32/9. - _Amiram Eldar_, Nov 02 2021
%F A033487 E.g.f.: exp(x)*x*(24 + 36*x + 12*x^2 + x^3)/4. - _Stefano Spezia_, Jul 03 2025
%e A033487 G.f. = 6*x + 30*x^2 + 90*x^3 + 210*x^4 + 420*x^5 + 756*x^6 + 1260*x^7 + ...
%p A033487 [seq(binomial(n+3,4)*6, n=0..40)]; # _Zerinvary Lajos_, Jul 18 2006
%t A033487 Table[Times @@ (n + Range[0, 3])/4, {n, 0, 40}] (* _Harvey P. Dale_, Nov 27 2013 *)
%t A033487 LinearRecurrence[{5, -10, 10, -5, 1}, {0, 6, 30, 90, 210}, 40] (* _Harvey P. Dale_, Nov 27 2013 *)
%t A033487 Table[6 Binomial[n+3, 4], {n,0,40}] (* _Eric W. Weisstein_, Jun 29 2017 *)
%t A033487 Times @@@ Table[n+k, {n, 0, 40}, {k, 0, 3}]/4 (* _Eric W. Weisstein_, Apr 29 2019 *)
%o A033487 (Magma) [n*(n+1)*(n+2)*(n+3)/4: n in [0..40]]; // _Vincenzo Librandi_, Apr 28 2011
%o A033487 (PARI) a(n)=6*binomial(n+3,4) \\ _Charles R Greathouse IV_, Apr 17 2012
%o A033487 (PARI) concat(0, Vec(6*x/(1-x)^5 + O(x^100))) \\ _Altug Alkan_, Nov 29 2015
%o A033487 (SageMath)
%o A033487 def A033487(n): return 6*binomial(n+3,4)
%o A033487 print([A033487(n) for n in range(41)]) # _G. C. Greubel_, Feb 08 2025
%Y A033487 Cf. A000217, A000332, A002378, A033486, A033488, A034827, A050534, A130534, A132440, A238363.
%Y A033487 Partial sums of A007531.
%Y A033487 A row of the array in A129533.
%Y A033487 A column of the triangle in A331430.
%Y A033487 Sequences of the form binomial(n+k,k)*binomial(n+k+2,k): A000012 (k=0), A005563 (k=1), this sequence (k=2), A027790 (k=3), A107395 (k=4), A107396 (k=5), A107397 (k=6), A107398 (k=7), A107399 (k=8).
%K A033487 nonn,easy
%O A033487 0,2
%A A033487 _N. J. A. Sloane_