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%I M4617 N1970 #89 Feb 11 2022 04:48:12
%S 0,1,9,35,95,210,406,714,1170,1815,2695,3861,5369,7280,9660,12580,
%T 16116,20349,25365,31255,38115,46046,55154,65550,77350,90675,105651,
%U 122409,141085,161820,184760,210056,237864,268345,301665,337995
%N 4-dimensional figurate numbers: a(n) = (5*n-1)*binomial(n+2,3)/4.
%C Partial sums of A002413.
%C Principal diagonal of the convolution array A213550, for n>0. - _Clark Kimberling_, Jun 17 2012
%C Convolution of A000027 with A000566. - _Bruno Berselli_, Dec 06 2012
%C Coefficients in the hypergeometric series identity 1 - 9*(x - 1)/(4*x + 1) + 35*(x - 1)*(x - 2)/((4*x + 1)*(4*x + 2)) - 95*(x - 1)*(x - 2)*(x - 3)/((4*x + 1)*(4*x + 2)*(4*x + 3)) + ... = 0, valid for Re(x) > 1. Cf. A000326 and A002412. Column 4 of A103450. - _Peter Bala_, Mar 14 2019
%D Albert 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="/A002418/b002418.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Luis Verde-Star <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Verde/verde4.html">A Matrix Approach to Generalized Delannoy and Schröder Arrays</a>, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5, 1).
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>.
%F G.f.: x*(1+4*x)/(1-x)^5. - _Simon Plouffe_ in his 1992 dissertation.
%F Starting (1, 9, 35, 95, ...), = A128064 * A000332, (A000332 starting 1, 5, 15, 35, 70, ...), such that a(n) = n*C((n+3),4)) - (n-1)*C((n+2),4)). E.g., a(5) = 210 = 5*C(8,4) - 4*C(7,4) = 5*70 - 4*35. - _Gary W. Adamson_, Dec 28 2007
%F Unit digit, A010879(a(n)), is one of {0,1,9,5,6,4} [Eric Desbiaux] because a(n) mod 5 = 0,1,4,0,0, periodic with period 5. [Proof: A002413(n) mod 5 = 1,3,1,0,0 with period 5 and a(n) are the partial sums of A002413.] - _R. J. Mathar_, Mar 19 2008
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Harvey P. Dale_, Oct 16 2012
%F a(n) = A080852(5,n-1). - _R. J. Mathar_, Jul 28 2016
%F a(n) = Sum_{i=0..n} (n-i) * Sum_{j=i..n} j. - _J. M. Bergot_, May 30 2017
%F E.g.f.: x*(24 + 84*x + 44*x^2 + 5*x^3)*exp(x)/4!. - _G. C. Greubel_, Jul 03 2019
%F Sum_{n>=1} 1/a(n) = (50*sqrt(5)*log(phi) + 125*log(5) - 50*sqrt(1+2/sqrt(5))*Pi - 26)/11, where phi is the golden ratio (A001622). - _Amiram Eldar_, Feb 11 2022
%t Table[(5n-1) Binomial[n+2,3]/4,{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{0,1,9,35,95},40] (* _Harvey P. Dale_, Oct 16 2012 *)
%t CoefficientList[Series[x*(1 + 4*x)/(1 - x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 17 2012 *)
%o (Magma) [(5*n - 1)*Binomial(n + 2, 3)/4: n in [0..40]]; // _Vincenzo Librandi_, Oct 17 2012
%o (Magma) /* A000027 convolved with A000566: */ A000566:=func<n | n*(5*n-3)/2>; [&+[(n-i+1)*A000566(i): i in [0..n]]: n in [0..35]]; // _Bruno Berselli_, Dec 06 2012
%o (PARI) a(n)=(5*n-1)*binomial(n+2,3)/4 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (GAP) List([0..40],n->(5*n-1)*Binomial(n+2,3)/4); # _Muniru A Asiru_, Mar 18 2019
%o (Sage) [(5*n-1)*binomial(n+2,3)/4 for n in (0..40)] # _G. C. Greubel_, Jul 03 2019
%Y Cf. A093562 ((5, 1) Pascal, column m=4).
%Y Cf. A000332, A000566, A001622, A080852, A128064.
%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 0,3
%A _N. J. A. Sloane_
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