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A002419 4-dimensional figurate numbers: (6*n-2)*binomial(n+2,3)/4.
(Formerly M4699 N2008)
17

%I M4699 N2008

%S 1,10,40,110,245,476,840,1380,2145,3190,4576,6370,8645,11480,14960,

%T 19176,24225,30210,37240,45430,54901,65780,78200,92300,108225,126126,

%U 146160,168490,193285,220720,250976,284240,320705,360570,404040,451326

%N 4-dimensional figurate numbers: (6*n-2)*binomial(n+2,3)/4.

%C a(n) is the n-th antidiagonal sum of the convolution array A213761. - _Clark Kimberling_, Jul 04 2012

%C Convolution of A000027 with A000567 (excluding 0). - _Bruno Berselli_, Dec 07 2012

%C a(n) = the sum of all the ways of adding the k-tuples of A016777(0) to A016777(n-1). For n=4, the terms are 1,4,7,10 giving (1)+(4)+(7)+(10)=22; (1+4)+(4+7)+(7+10)=33; (1+4+7)+(4+7+10)=33; (1+4+7+10)=22; adding 22+33+33+22=110. - _J. M. Bergot_, Jun 26 2017

%C Also the number of chordless cycles in the (n+3)-crown graph. - _Eric W. Weisstein_, Jan 02 2018

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%H Vincenzo Librandi, <a href="/A002419/b002419.txt">Table of n, a(n) for n = 1..1000</a>

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChordlessCycle.html">Chordless Cycle</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CrownGraph.html">Crown Graph</a>

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = (3*n-1)*binomial(n+2, 3)/2.

%F G.f.: x*(1+5*x)/(1-x)^5. - _Simon Plouffe_ in his 1992 dissertation.

%F Sum_{n>=1} 1/a(n) = (-24+81*log(3) -9*Pi*sqrt(3))/14 = 1.143929... - _R. J. Mathar_, Mar 29 2011

%F a(n) = (3*n^4 + 8*n^3 + 3*n^2 - 2*n)/12. - _Chai Wah Wu_, Jan 24 2016

%F a(n) = A080852(6,n-1). - _R. J. Mathar_, Jul 28 2016

%F E.g.f.: x*(12 + 48*x + 26*x^2 + 3*x^3)*exp(x)/12. - _G. C. Greubel_, Jul 03 2019

%t CoefficientList[Series[(1+5*x)/(1-x)^5, {x,0,40}], x] (* _Vincenzo Librandi_, Jun 20 2013 *)

%t LinearRecurrence[{5, -10, 10, -5, 1}, {1, 10, 40, 110, 245}, 40] (* _Harvey P. Dale_, Nov 30 2014 *)

%t Table[n(n+1)(n+2)(3n-1)/12, {n, 40}] (* _Eric W. Weisstein_, Jan 02 2018 *)

%o (MAGMA) /* A000027 convolved with A000567 (excluding 0): */ A000567:=func<n | n*(3*n-2)>; [&+[(n-i+1)*A000567(i): i in [1..n]]: n in [1..40]]; // _Bruno Berselli_, Dec 07 2012

%o (PARI) a(n)=(3*n-1)*binomial(n+2,3)/2 \\ _Charles R Greathouse IV_, Sep 24 2015

%o (Python)

%o A002419_list, m = [], [6, 1, 1, 1, 1]

%o for _ in range(10**2):

%o A002419_list.append(m[-1])

%o for i in range(4):

%o m[i+1] += m[i] # _Chai Wah Wu_, Jan 24 2016

%o (Sage) [n*(n+1)*(n+2)*(3*n-1)/12 for n in (1..40)] # _G. C. Greubel_, Jul 03 2019

%o (GAP) List([1..40], n-> n*(n+1)*(n+2)*(3*n-1)/12) # _G. C. Greubel_, Jul 03 2019

%Y Cf. A093563 ((6, 1) Pascal, column m=4).

%Y Cf. A000567, A002414 (first differences).

%Y Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

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Last modified January 29 08:12 EST 2020. Contains 331337 sequences. (Running on oeis4.)