A001296 - OEIS

%I M4385 N1845 #183 Sep 08 2022 08:44:29

%S 0,1,7,25,65,140,266,462,750,1155,1705,2431,3367,4550,6020,7820,9996,

%T 12597,15675,19285,23485,28336,33902,40250,47450,55575,64701,74907,

%U 86275,98890,112840,128216,145112,163625,183855,205905,229881,255892,284050,314470

%N 4-dimensional pyramidal numbers: a(n) = (3*n+1)*binomial(n+2, 3)/4. Also Stirling2(n+2, n).

%C Permutations avoiding 12-3 that contain the pattern 31-2 exactly once.

%C Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Nov 18 2005

%C Partial sums of A002411. - _Jonathan Vos Post_, Mar 16 2006

%C 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

%C 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

%C 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

%C Half of the partial sums of A011379. [Jolley, Summation of Series, Dover (1961), page 12 eq (66).] - _R. J. Mathar_, Oct 03 2011

%C 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

%C Convolution of A000027 with A000326. - _Bruno Berselli_, Dec 06 2012

%C 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

%C a(n-2) is the maximum number of intersections made from the perpendicular bisectors of all pair combinations of n points. - _Ian Tam_, Dec 22 2020

%D 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.

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

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.

%D 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).

%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.

%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 T. D. Noe, <a href="/A001296/b001296.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H P. Aluffi, <a href="http://arxiv.org/abs/1408.1702">Degrees of projections of rank loci</a>, 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]."]

%H S. Butler, P. Karasik, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Butler/butler7.html">A note on nested sums</a>, J. Int. Seq. 13 (2010), 10.4.4, page 5.

%H M. Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Griffiths2/griffiths17.html">Remodified Bessel Functions via Coincidences and Near Coincidences</a>, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.

%H L. Hogben, <a href="https://archive.org/details/chanceandchoiceb029729mbp">Choice and Chance by Cardpack and Chessboard</a>, Vol. 1, Chanticleer Press, NY, 1950, p. 36.

%H C. Krishnamachaki, <a href="/A001296/a001296.pdf">The operator (xD)^n</a>, J. Indian Math. Soc., 15 (1923),3-4. [Annotated scanned copy]

%H T. Mansour, <a href="https://arxiv.org/abs/math/0202219">Restricted permutations by patterns of type 2-1</a>, arXiv:math/0202219 [math.CO], 2002.

%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">Stirling numbers of the 2nd kind</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) = n*(1+n)*(2+n)*(1+3*n)/24. - _T. D. Noe_, Jan 21 2008

%F G.f.: x*(1+2*x)/(1-x)^5. - _Paul Barry_, Jul 23 2003

%F a(n) = Sum_{j=0..n} j*A000217(j). - _Jon Perry_, Jul 28 2003

%F 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.

%F E.g.f. x*exp(x)*(24 + 60*x + 28*x^2 + 3*x^3)/24 (above e.g.f. differentiated).

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 3. - _Kieren MacMillan_, Sep 29 2008

%F 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

%F 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

%F a(n) = A153978(n)/2. - _J. M. Bergot_, Aug 09 2013

%F a(n) = A002817(n) + A000292(n-1). - _J. M. Bergot_, Aug 29 2013; [corrected by _Cyril Damamme_, Feb 26 2018]

%F a(n) = A000914(n+1) - 2 * A000330(n+1). - _Antal Pinter_, Dec 31 2015

%F a(n) = A080852(3,n-1). - _R. J. Mathar_, Jul 28 2016

%F 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

%F a(n) = A000914(-2-n) for all n in Z. - _Michael Somos_, Sep 04 2017

%F a(n) = A000292(n) + A050534(n+1). - _Cyril Damamme_, Feb 26 2018

%F From _Amiram Eldar_, Jul 02 2020: (Start)

%F Sum_{n>=1} 1/a(n) = (6/5) * (47 - 3*sqrt(3)*Pi - 27*log(3)).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (6/5) * (16*log(2) + 6*sqrt(3)*Pi - 43). (End)

%e 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 + ...

%p A001296:=-(1+2*z)/(z-1)**5; # _Simon Plouffe_ in his 1992 dissertation for sequence without the leading zero

%t Table[n*(1+n)*(2+n)*(1+3*n)/24, {n, 0, 100}]

%t CoefficientList[Series[x (1 + 2 x)/(1 - x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 30 2014 *)

%t Table[StirlingS2[n+2, n], {n, 0, 40}] (* _Jean-François Alcover_, Jun 24 2015 *)

%t 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 *)

%o (PARI) t(n)=n*(n+1)/2

%o for(i=1,30,print1(","sum(j=1,i,j*t(j))))

%o (PARI) {a(n) = n * (n+1) * (n+2) * (3*n+1) / 24}; /* _Michael Somos_, Sep 04 2017 */

%o (Sage) [stirling_number2(n+2,n) for n in range(0,38)] # _Zerinvary Lajos_, Mar 14 2009

%o (Magma) /* A000027 convolved with A000326: */ A000326:=func<n | n*(3*n-1)/2>; [&+[(n-i+1)*A000326(i): i in [0..n]]: n in [0..40]]; // _Bruno Berselli_, Dec 06 2012

%o (Magma) [(3*n+1)*Binomial(n+2,3)/4: n in [0..40]]; // _Vincenzo Librandi_, Jul 30 2014

%Y a(n)=f(n, 2) where f is given in A034261.

%Y Cf. A000217, A000326, A001297, A001298, A002411, A008277, A008517, A094262.

%Y a(n)= A093560(n+3, 4), (3, 1)-Pascal column.

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

%Y Cf. similar sequences listed in A241765 and A254142.

%Y Cf. A000914.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_