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A001296
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4-dimensional pyramidal numbers: (3n+1)*C(n+2,3)/4. Also Stirling2(n+2,n).
(Formerly M4385 N1845)
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25
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0, 1, 7, 25, 65, 140, 266, 462, 750, 1155, 1705, 2431, 3367, 4550, 6020, 7820, 9996, 12597, 15675, 19285, 23485, 28336, 33902, 40250, 47450, 55575, 64701, 74907, 86275, 98890, 112840, 128216, 145112, 163625, 183855, 205905, 229881, 255892, 284050, 314470
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Permutations avoiding 12-3 that contain the pattern 31-2 exactly once.
Kekule numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
If Y is a 3-subset of an n-set X then, for n>=6, a(n-5) is the number of 6-subsets of X having at least two elements in common with Y. - Milan R. Janjic (agnus(AT)blic.net), Nov 23 2007
Rephrasing the Perry formula of 2003: a(n) is the sum of all products of all two numbers less than or equal to n, including the squares. Example: for n=3 the sum of these products is 1*1+1*2+1*3+2*2+2*3+3*3=25.- J. M. Bergot, Jul 16 2011
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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. 835.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.
S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 166, Table 10.4/I/3).
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
M. Griffiths, Remodified Bessel Functions via Coincidences and Near Coincidences, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.
L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 36.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
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].
T. Mansour, Restricted permutations by patterns of type 2-1.
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Stirling numbers of the 2nd kind.
Index entries for sequences related to linear recurrences with constant coefficients, signature (5,-10,10,-5,1)
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FORMULA
| a(n) = n*(1+n)*(2+n)*(1+3*n)/24. - T. D. Noe, Jan 21 2008
G.f.: x*(1+2*x)/(1-x)^5 - Paul Barry, Jul 23 2003
a(n) = sum(j=1..n, j*triangle(j) ) - Jon Perry, Jul 28 2003
E.g.f. with offset -1: exp(x)*(1*(x^2)/2! + 4*(x^3)/3! + 3*(x^4)/4!). For the coefficients [1, 4, 3] see triangle A112493.
E.g.f. x*exp(x)*(24 + 60*x + 28*x^2 + 3*x^3)/24 (above e.g.f. differentiated).
Partial sums of A002411. - Jonathan Vos Post, Mar 16 2006
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4)+3. [From Kieren MacMillan, Sep 29 2008]
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) [From Jaume Oliver Lafont, Nov 23 2008]
Half of the partial sums of A011379. [Jolley, Summation of Series, Dover (1961), page 12 eq (66)]
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MAPLE
| A001296:=-(1+2*z)/(z-1)**5; [S. Plouffe in his 1992 dissertation for sequence without the leading zero.]
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MATHEMATICA
| Table[n*(1+n)*(2+n)*(1+3*n)/24, {n, 0, 100}]
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PROG
| (PARI) t(n)=n*(n+1)/2 for(i=1, 30, print1(", "sum(j=1, i, j*t(j))))
(Sage) [stirling_number2(n+2, n) for n in xrange(0, 38)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 14 2009]
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CROSSREFS
| a(n)=f(n, 2) where f is given in A034261.
Cf. A008277, A094262, A001297, A001298.
a(n)= A093560(n+3, 4), (3, 1)-Pascal column.
Cf. A002411.
Sequence in context: A155305 A155290 A056685 * A000970 A155245 A155291
Adjacent sequences: A001293 A001294 A001295 * A001297 A001298 A001299
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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