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A005582 n(n+1)(n+2)(n+7)/24.
(Formerly M1922)
8
0, 2, 9, 25, 55, 105, 182, 294, 450, 660, 935, 1287, 1729, 2275, 2940, 3740, 4692, 5814, 7125, 8645, 10395, 12397, 14674, 17250, 20150, 23400, 27027, 31059, 35525, 40455, 45880, 51832, 58344, 65450, 73185, 81585, 90687, 100529, 111150, 122590, 134890 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = number of Dyck (n+2)-paths with exactly 2 rows of peaks. A row of peaks is a maximal sequence of peaks all at the same height and 2 units apart. For example, UDUDUD ( = /\/\/\ ) contains exactly one row of peaks, as does UUUDDD, but UDUUDDUD has three and a(1)=2 counts UDUUDD, UUDDUD. - David Callan, Mar 02 2005

If X is an n-set and Y a fixed 2-subset of X then a(n-4) is equal to the number of (n-4)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007

Let I=I_n be the nXn identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n>=7, a(n-7) is the number of (0,1) nXn matrices A<=P^(-1)+I+P having exactly two 1's in every row and column with perA=16. [Vladimir Shevelev, Apr 12 2010]

Row 2 of the convolution array A213550.  [Clark Kimberling, Jun 20 2012]

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 797.

F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, http://www.math-cs.ucmo.edu/~curtisc/articles/howardcooper/genfib4.pdf.

P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, Vol. II, pp. 354-382] [See p. 301]

V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3(1992),15-19. [From Vladimir Shevelev, Apr 12 2010]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #51 (the case k=4) (First published: San Francisco: Holden-Day, Inc., 1964)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Milan Janjic, Two Enumerative Functions

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.

FORMULA

a(n) = binomial(n+3, n-1)+binomial(n+2, n-1).

a(n) = binomial(n,4) + 2*binomial(n,3), n>=2. - Zerinvary Lajos, Jul 26 2006

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). G.f.: x*(2-x)/(1-x)^5. [Colin Barker, Jan 28 2012]

a(n) = sum_{k=1..n} ( sum_{i=1..k} i(n-k+2) ). - Wesley Ivan Hurt, Sep 26 2013

MAPLE

[seq(binomial(n, 4)+2*binomial(n, 3), n=2..43)]; - Zerinvary Lajos, Jul 26 2006

seq((n+4)*binomial(n, 4)/n, n=3..43); - Zerinvary Lajos, Feb 28 2007

A005582:=(-2+z)/(z-1)**5; [Conjectured by Simon Plouffe in his 1992 dissertation.]

MATHEMATICA

Table[n(n+1)(n+2)(n+7)/24, {n, 0, 40}] (* Harvey P. Dale, Jun 01 2012 *)

CROSSREFS

Cf. A005581, A176222, A000211, A052928, A128209.

Sequence in context: A097346 A226388 A053194 * A173965 A116454 A124633

Adjacent sequences:  A005579 A005580 A005581 * A005583 A005584 A005585

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000

STATUS

approved

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Last modified July 29 04:40 EDT 2014. Contains 245018 sequences.