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 A001296 4-dimensional pyramidal numbers: a(n) = (3*n+1)*binomial(n+2, 3)/4. Also Stirling2(n+2, n). (Formerly M4385 N1845) 51
 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; text; internal format)
 OFFSET 0,3 COMMENTS Permutations avoiding 12-3 that contain the pattern 31-2 exactly once. Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005 Partial sums of A002411. - Jonathan Vos Post, Mar 16 2006 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 Janjic, Nov 23 2007 Starting with 1 = binomial transform of [1, 6, 12, 10, 3, 0, 0, 0, ...]. Equals row sums of triangle A143037. - Gary W. Adamson, Jul 18 2008 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 Half of the partial sums of A011379. [Jolley, Summation of Series, Dover (1961), page 12 eq (66).] - R. J. Mathar, Oct 03 2011 Also the number of (w,x,y,z) with all terms in {1,...,n+1} and w < x >= y > z (see A211795). - Clark Kimberling, May 19 2012 Convolution of A000027 with A000326. - Bruno Berselli, Dec 06 2012 This sequence is related to A000292 by a(n) = n*A000292(n) - Sum_{i=0..n-1} A000292(i) for n>0. - Bruno Berselli, Nov 23 2017 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, Kekulé 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. 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). 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]. P. Aluffi, Degrees of projections of rank loci, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."] S. Butler, P. Karasik, A note on nested sums, J. Int. Seq. 13 (2010), 10.4.4, page 5. 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. C. Krishnamachaki, The operator (xD)^n, J. Indian Math. Soc., 15 (1923),3-4. [Annotated scanned copy] T. Mansour, Restricted permutations by patterns of type 2-1, arXiv:math/0202219 [math.CO], 2002. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. Eric Weisstein's World of Mathematics, Stirling numbers of the 2nd kind. Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). 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=0..n} j*A000217(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). a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 3. - 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). - Jaume Oliver Lafont, Nov 23 2008 O.g.f. is D^2(x/(1-x)) = D^3(x), where D is the operator x/(1-x)*d/dx. - Peter Bala, Jul 02 2012 a(n) = A153978(n)/2. - J. M. Bergot, Aug 09 2013 a(n) = A002817(n) + A000292(n-1). - J. M. Bergot, Aug 29 2013; [corrected by Cyril Damamme, Feb 26 2018] a(n) = A000914(n+1) - 2 * A000330(n+1). - Antal Pinter, Dec 31 2015 a(n) = A080852(3,n-1). - R. J. Mathar, Jul 28 2016 a(n) = 1*(1+2+...+n) + 2*(2+3+...+n) + ... + n*n. For example, a(6) = 266 = 1(1+2+3+4+5+6) + 2*(2+3+4+5+6) + 3*(3+4+5+6) + 4*(4+5+6) + 5*(5+6) + 6*(6).- J. M. Bergot, Apr 20 2017 a(n) = A000914(-2-n) for all n in Z. - Michael Somos, Sep 04 2017 a(n) = A000292(n) + A050534(n+1). - Cyril Damamme, Feb 26 2018 From Amiram Eldar, Jul 02 2020: (Start) Sum_{n>=1} 1/a(n) = (6/5) * (47 - 3*sqrt(3)*Pi - 27*log(3)). Sum_{n>=1} (-1)^(n+1)/a(n) = (6/5) * (16*log(2) + 6*sqrt(3)*Pi - 43). (End) EXAMPLE G.f. = x + 7*x^2 + 25*x^3 + 65*x^4 + 140*x^5 + 266*x^6 + 462*x^7 + 750*x^8 + 1155*x^9 + ... MAPLE A001296:=-(1+2*z)/(z-1)**5; # Simon Plouffe in his 1992 dissertation for sequence without the leading zero MATHEMATICA Table[n*(1+n)*(2+n)*(1+3*n)/24, {n, 0, 100}] CoefficientList[Series[x (1 + 2 x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 30 2014 *) Table[StirlingS2[n+2, n], {n, 0, 40}] (* Jean-François Alcover, Jun 24 2015 *) Table[ListCorrelate[Accumulate[Range[n]], Range[n]], {n, 0, 40}]//Flatten (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 7, 25, 65}, 40] (* Harvey P. Dale, Aug 14 2017 *) PROG (PARI) t(n)=n*(n+1)/2 for(i=1, 30, print1(", "sum(j=1, i, j*t(j)))) (PARI) {a(n) = n * (n+1) * (n+2) * (3*n+1) / 24}; /* Michael Somos, Sep 04 2017 */ (Sage) [stirling_number2(n+2, n) for n in range(0, 38)] # Zerinvary Lajos, Mar 14 2009 (MAGMA) /* A000027 convolved with A000326: */ A000326:=func; [&+[(n-i+1)*A000326(i): i in [0..n]]: n in [0..40]]; // Bruno Berselli, Dec 06 2012 (MAGMA) [(3*n+1)*Binomial(n+2, 3)/4: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014 CROSSREFS a(n)=f(n, 2) where f is given in A034261. Cf. A000217, A000326, A001297, A001298, A002411, A008277, A008517, A094262. a(n)= A093560(n+3, 4), (3, 1)-Pascal column. Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers. Cf. similar sequences listed in A241765 and A254142. Cf. A000914. Sequence in context: A155290 A056685 A299262 * A000970 A247620 A240156 Adjacent sequences:  A001293 A001294 A001295 * A001297 A001298 A001299 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified October 26 23:01 EDT 2020. Contains 338033 sequences. (Running on oeis4.)