cp's OEIS Frontend

A005718 Quadrinomial coefficients: C(2+n,n) + C(3+n,n) + C(4+n,n).

%I A005718 M2918 #69 May 09 2024 13:27:01
%S A005718 3,12,31,65,120,203,322,486,705,990,1353,1807,2366,3045,3860,4828,
%T A005718 5967,7296,8835,10605,12628,14927,17526,20450,23725,27378,31437,35931,
%U A005718 40890,46345,52328,58872,66011,73780,82215,91353,101232,111891,123370,135710,148953,163142,178321,194535
%N A005718 Quadrinomial coefficients: C(2+n,n) + C(3+n,n) + C(4+n,n).
%C A005718 If Y is an (n-3)-subset of an n-set X then, for n>=5, a(n-5) is the number of 4-subsets of X having at least two elements in common with Y. - _Milan Janjic_, Dec 16 2007
%C A005718 This equation represents the number of numbers with <=n digits such that the sum of the digits is between 1 and 4 inclusive and no digit is larger than 3. - _David Consiglio, Jr._, Oct 27 2008
%C A005718 Row 2 of the convolution array A213548. - _Clark Kimberling_, Jun 20 2012
%D A005718 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
%D A005718 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005718 Vincenzo Librandi, <a href="/A005718/b005718.txt">Table of n, a(n) for n = 0..1000</a>
%H A005718 Niall Byrnes, Gary R. W. Greaves, and Matthew R. Foreman, <a href="https://arxiv.org/abs/2405.02541">Bootstrapping cascaded random matrix models: correlations in permutations of matrix products</a>, arXiv:2405.02541 [math-ph], 2024. See p. 7.
%H A005718 R. K. Guy, <a href="/A005712/a005712.pdf">Letter to N. J. A. Sloane, 1987</a>.
%H A005718 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A005718 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A005718 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A005718 a(n) = binomial(n, 2)*(n^2+7*n+18)/12, n >= 2.
%F A005718 G.f.: (3-3*x+x^2)/(1-x)^5. (numerator polynomial is N4(4, x) from A063421).
%F A005718 a(n) = A008287(n, 4), n >= 2 (fifth column of quadrinomial coefficients).
%F A005718 a(n) = A062745(n, 4), n >= 2 (fifth column).
%F A005718 a(n) = 3*C(n+2,2) + 3*C(n+2,3) + C(n+2,4) (see comment in A071675). - _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 22 2012
%F A005718 E.g.f.: exp(x)*(72 + 216*x + 120*x^2 + 20*x^3 + x^4)/24. - _Stefano Spezia_, May 09 2024
%p A005718 A005718:=-(3-3*z+z**2)/(z-1)**5; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t A005718 Table[Plus@@Table[Binomial[i + n, n], {i, 2, 4}], {n, 0, 43}] (* From _Alonso del Arte_, Jun 14 2011 *)
%o A005718 (PARI) a(n)=(((n+14)*n+71)*n+130)*n/24+3 \\ _Charles R Greathouse IV_, Jun 14 2011
%o A005718 (Magma) [(((n+14)*n+71)*n+130)*n/24+3: n in [0..45]]; // _Vincenzo Librandi_, Jun 15 2011
%Y A005718 Cf. A008287, A062745, A063421, A071675, A213548.
%K A005718 nonn,easy
%O A005718 0,1
%A A005718 _N. J. A. Sloane_
%E A005718 Better description from _Zerinvary Lajos_, Dec 02 2005