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 A212415 Number of (w,x,y,z) with all terms in {1,...,n} and w=y<=z. 3
 0, 0, 3, 17, 55, 135, 280, 518, 882, 1410, 2145, 3135, 4433, 6097, 8190, 10780, 13940, 17748, 22287, 27645, 33915, 41195, 49588, 59202, 70150, 82550, 96525, 112203, 129717, 149205, 170810, 194680, 220968, 249832, 281435, 315945 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS For a guide to related sequences, see A211795. Partial sums of A162147. - J. M. Bergot, Jun 21 2013 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). FORMULA a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). From Bruno Berselli, May 30 2012: (Start) G.f.: x^2*(3+2*x)/(1-x)^5. a(n) = (n-1)*n*(n+1)*(5*n+2)/24. (End) E.g.f.: x^2*(36 + 32*x + 5*x^2)*exp(x)/24. - G. C. Greubel, Jul 11 2019 EXAMPLE a(3) counts these (w,x,y,z): (1,2,2,2), (1,2,2,3), (1,3,3,3). MATHEMATICA t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[w < x >= y <= z, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]]; Map[t[#] &, Range[0, 40]]   (* A212415 *) Table[n*(5*n+2)*(n^2-1)/4!, {n, 0, 40}] (* G. C. Greubel, Jul 11 2019 *) PROG (PARI) vector(40, n, n--; n*(5*n+2)*(n^2-1)/4!) \\ G. C. Greubel, Jul 11 2019 (MAGMA) [n*(5*n+2)*(n^2-1)/24: n in [0..40]]; // G. C. Greubel, Jul 11 2019 (Sage) [n*(5*n+2)*(n^2-1)/24 for n in (0..40)] # G. C. Greubel, Jul 11 2019 (GAP) List([0..40], n-> n*(5*n+2)*(n^2-1)/24); # G. C. Greubel, Jul 11 2019 CROSSREFS Cf. A211795. Sequence in context: A294134 A258032 A033562 * A152457 A130857 A226719 Adjacent sequences:  A212412 A212413 A212414 * A212416 A212417 A212418 KEYWORD nonn,easy AUTHOR Clark Kimberling, May 19 2012 STATUS approved

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Last modified June 21 18:50 EDT 2021. Contains 345365 sequences. (Running on oeis4.)