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%I #93 Sep 26 2023 15:54:00
%S 1,5,13,26,45,71,105,148,201,265,341,430,533,651,785,936,1105,1293,
%T 1501,1730,1981,2255,2553,2876,3225,3601,4005,4438,4901,5395,5921,
%U 6480,7073,7701,8365,9066,9805,10583,11401,12260,13161,14105,15093,16126,17205,18331
%N a(n) = (n+1)*(n^2 +8*n +6)/6. Number of n-dimensional partitions of 4. Number of terms in 4th derivative of a function composed with itself n times.
%C Let m(i,1)=i; m(1,j)=j; m(i,j)=m(i-1,j)-m(i-1,j-1); then a(n)=m(n+3,3) - _Benoit Cloitre_, May 08 2002
%C a(n) = number of (n+6)-bit binary sequences with exactly 6 1's none of which is isolated. - _David Callan_, Jul 15 2004
%C If a 2-set Y and 2-set Z, having one element in common, are subsets of an n-set X then a(n-4) is the number of 4-subsets of X intersecting both Y and Z. - _Milan Janjic_, Oct 03 2007
%C Sum of first n triangular numbers plus previous triangular number. - _Vladimir Joseph Stephan Orlovsky_, Oct 13 2009
%C a(n) = Sum of first (n+1) triangular numbers plus n-th triangular number (see penultimate formula by Henry Bottomley). - _Vladimir Joseph Stephan Orlovsky_, Oct 13 2009
%C For n > 0, a(n-1) is the number of compositions of n+6 into n parts avoiding the part 2. - _Milan Janjic_, Jan 07 2016
%C The binomial transform of [1,4,4,1,0,0,0,...], the 4th row in A116672. - _R. J. Mathar_, Jul 18 2017
%D G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 190 eq. (11.4.7).
%H Vincenzo Librandi, <a href="/A008778/b008778.txt">Table of n, a(n) for n = 0..10000</a>
%H P. Chinn and S. Heubach, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Heubach/heubach5.html">Integer Sequences Related to Compositions without 2's</a>, J. Integer Seqs., Vol. 6, 2003.
%H Francisco Javier de Vega, <a href="https://doi.org/10.3390/axioms12100905">Some Variants of Integer Multiplication</a>, Axioms (2023) Vol. 12, 905. See p. 15.
%H Milan Janjic, <a href="http://www.pmfbl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
%H László Németh, <a href="https://arxiv.org/abs/1905.13475">Tetrahedron trinomial coefficient transform</a>, arXiv:1905.13475 [math.CO], 2019.
%H W. C. Yang, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00412-4">Derivatives are essentially integer partitions</a>, Discrete Mathematics, 222(1-3), July 2000, 235-245.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = dot_product(n, n-1, ...2, 1)*(2, 3, ..., n, 1) for n = 2, 3, 4, ... [i.e., a(2) = (2, 1)*(2, 1), a(3) = (3, 2, 1)*(2, 3, 1)]. - _Clark Kimberling_
%F a(n) = a(n-1) + A034856(n+1) = A000297(n-1) + 1 = A000217(n) + A000292(n+1) = A000290(n-1) + A000292(n). - _Henry Bottomley_, Oct 25 2001
%F a(n) = Sum_{0<=k, l<=n; k+l|n} k*l. - _Ralf Stephan_, May 06 2005
%F G.f.: (1+x-x^2)/(1-x)^4. - _Colin Barker_, Jan 06 2012
%F a(n) = A000330(n+1) - A000292(n-1). - _Bruce J. Nicholson_, Jul 05 2018
%F E.g.f.: (6 +24*x +12*x^2 +x^3)*exp(x)/6. - _G. C. Greubel_, Sep 11 2019
%e G.f. = 1 + 5*x + 13*x^2 + 26*x^3 + 45*x^4 + 71*x^5 + 105*x^6 + 148*x^7 + 201*x^8 + ...
%p seq(1+4*k+4*binomial(k, 2)+binomial(k, 3), k=0..45);
%t Table[(n+1)*(n^2+8*n+6)/6, {n,0,50}] (* _Vladimir Joseph Stephan Orlovsky_, Oct 13 2009, modified by _G. C. Greubel_, Sep 11 2019 *)
%t LinearRecurrence[{4,-6,4,-1}, {1,5,13,26}, 51] (* _G. C. Greubel_, Sep 11 2019 *)
%o (Magma) [(n+1)*(n^2+8*n+6)/6: n in [0..50]]; // _Vincenzo Librandi_, May 21 2011
%o (PARI) Vec((1+x-x^2)/(1-x)^4 + O(x^50)) \\ _Altug Alkan_, Jan 07 2016
%o (Sage) [(n+1)*(n^2 +8*n +6)/6 for n in (0..50)] # _G. C. Greubel_, Sep 11 2019
%o (GAP) List([0..50], n-> (n+1)*(n^2 +8*n +6)/6); # _G. C. Greubel_, Sep 11 2019
%Y Cf. A022811-A022817, A024207-A024210.
%Y Column 1 of triangle A094415.
%Y Row n=4 of A022818.
%Y Cf. A002411, A008779, A005712 (partial sums), A034856 (first diffs).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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