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

3, 12, 31, 65, 120, 203, 322, 486, 705, 990, 1353, 1807, 2366, 3045, 3860, 4828, 5967, 7296, 8835, 10605, 12628, 14927, 17526, 20450, 23725, 27378, 31437, 35931, 40890, 46345, 52328, 58872, 66011, 73780, 82215, 91353, 101232, 111891, 123370, 135710, 148953, 163142, 178321, 194535

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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

Row 2 of the convolution array A213548. - Clark Kimberling, Jun 20 2012

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 78.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Niall Byrnes, Gary R. W. Greaves, and Matthew R. Foreman, Bootstrapping cascaded random matrix models: correlations in permutations of matrix products, arXiv:2405.02541 [math-ph], 2024. See p. 7.
• R. K. Guy, Letter to N. J. A. Sloane, 1987.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A008287, A062745, A063421, A071675, A213548.

Programs

Magma

[(((n+14)*n+71)*n+130)*n/24+3: n in [0..45]]; // Vincenzo Librandi, Jun 15 2011

Maple

A005718:=-(3-3*z+z**2)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Table[Plus@@Table[Binomial[i + n, n], {i, 2, 4}], {n, 0, 43}] (* From Alonso del Arte, Jun 14 2011 *)

PARI

a(n)=(((n+14)*n+71)*n+130)*n/24+3 \\ Charles R Greathouse IV, Jun 14 2011

Formula

a(n) = binomial(n, 2)*(n^2+7*n+18)/12, n >= 2.

G.f.: (3-3*x+x^2)/(1-x)^5. (numerator polynomial is N4(4, x) from A063421).

a(n) = A008287(n, 4), n >= 2 (fifth column of quadrinomial coefficients).

a(n) = A062745(n, 4), n >= 2 (fifth column).

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

E.g.f.: exp(x)*(72 + 216*x + 120*x^2 + 20*x^3 + x^4)/24. - Stefano Spezia, May 09 2024

Extensions

Better description from Zerinvary Lajos, Dec 02 2005