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A002419 4-dimensional figurate numbers: a(n) = (6*n-2)*binomial(n+2,3)/4.

%I A002419 M4699 N2008 #92 Jun 20 2025 10:44:46
%S A002419 1,10,40,110,245,476,840,1380,2145,3190,4576,6370,8645,11480,14960,
%T A002419 19176,24225,30210,37240,45430,54901,65780,78200,92300,108225,126126,
%U A002419 146160,168490,193285,220720,250976,284240,320705,360570,404040,451326,502645,558220
%N A002419 4-dimensional figurate numbers: a(n) = (6*n-2)*binomial(n+2,3)/4.
%C A002419 a(n) is the n-th antidiagonal sum of the convolution array A213761. - _Clark Kimberling_, Jul 04 2012
%C A002419 Convolution of A000027 with A000567 (excluding 0). - _Bruno Berselli_, Dec 07 2012
%C A002419 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 A002419 Also the number of chordless cycles in the (n+2)-crown graph. - _Eric W. Weisstein_, Jan 02 2018
%D A002419 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.
%D A002419 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002419 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002419 Vincenzo Librandi, <a href="/A002419/b002419.txt">Table of n, a(n) for n = 1..1000</a>
%H A002419 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 A002419 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A002419 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChordlessCycle.html">Chordless Cycle</a>.
%H A002419 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CrownGraph.html">Crown Graph</a>.
%H A002419 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%H A002419 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>.
%F A002419 a(n) = (3*n-1)*binomial(n+2, 3)/2.
%F A002419 G.f.: x*(1+5*x)/(1-x)^5. - _Simon Plouffe_ in his 1992 dissertation.
%F A002419 Sum_{n>=1} 1/a(n) = (-24+81*log(3) -9*Pi*sqrt(3))/14 = 1.143929... - _R. J. Mathar_, Mar 29 2011
%F A002419 a(n) = (3*n^4 + 8*n^3 + 3*n^2 - 2*n)/12. - _Chai Wah Wu_, Jan 24 2016
%F A002419 a(n) = A080852(6,n-1). - _R. J. Mathar_, Jul 28 2016
%F A002419 E.g.f.: x*(12 + 48*x + 26*x^2 + 3*x^3)*exp(x)/12. - _G. C. Greubel_, Jul 03 2019
%F A002419 Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(3*sqrt(3)*Pi - 32*log(2) + 8)/7. - _Amiram Eldar_, Feb 11 2022
%t A002419 CoefficientList[Series[(1+5*x)/(1-x)^5, {x,0,40}], x] (* _Vincenzo Librandi_, Jun 20 2013 *)
%t A002419 LinearRecurrence[{5, -10, 10, -5, 1}, {1, 10, 40, 110, 245}, 40] (* _Harvey P. Dale_, Nov 30 2014 *)
%t A002419 Table[n(n+1)(n+2)(3n-1)/12, {n, 40}] (* _Eric W. Weisstein_, Jan 02 2018 *)
%t A002419 Table[Sum[2 x + 3 x^2 - 2 y, {x, 0, g}, {y, x, g}], {g, 1, 20}] (* _Horst H. Manninger_, Jun 20 2025 *)
%o A002419 (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 A002419 (PARI) a(n)=(3*n-1)*binomial(n+2,3)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%o A002419 (Python)
%o A002419 A002419_list, m = [], [6, 1, 1, 1, 1]
%o A002419 for _ in range(10**2):
%o A002419     A002419_list.append(m[-1])
%o A002419     for i in range(4):
%o A002419         m[i+1] += m[i] # _Chai Wah Wu_, Jan 24 2016
%o A002419 (Sage) [n*(n+1)*(n+2)*(3*n-1)/12 for n in (1..40)] # _G. C. Greubel_, Jul 03 2019
%o A002419 (GAP) List([1..40], n-> n*(n+1)*(n+2)*(3*n-1)/12); # _G. C. Greubel_, Jul 03 2019
%Y A002419 Cf. A093563 ((6, 1) Pascal, column m=4).
%Y A002419 Cf. A000027, A000567, A002414 (first differences), A016777, A080852, A213761.
%Y A002419 Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.
%K A002419 nonn,easy
%O A002419 1,2
%A A002419 _N. J. A. Sloane_