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%I M4387 #155 Feb 23 2024 08:55:17
%S 1,7,27,77,182,378,714,1254,2079,3289,5005,7371,10556,14756,20196,
%T 27132,35853,46683,59983,76153,95634,118910,146510,179010,217035,
%U 261261,312417,371287,438712,515592,602888,701624,812889,937839,1077699,1233765,1407406
%N 5-dimensional pyramidal numbers: a(n) = n*(n+1)*(n+2)*(n+3)*(2n+3)/5!.
%C Convolution of triangular numbers (A000217) and squares (A000290) (n>=1). - _Graeme McRae_, Jun 07 2006
%C p^k divides a(p^k-3), a(p^k-2), a(p^k-1) and a(p^k) for prime p > 5 and integer k > 0. p^k divides a((p^k-3)/2)) for prime p > 5 and integer k > 0. - _Alexander Adamchuk_, May 08 2007
%C If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-5) is the number of 6-subsets of X intersecting both Y and Z. - _Milan Janjic_, Sep 08 2007
%C 5-dimensional square numbers, fourth partial sums of binomial transform of [1,2,0,0,0,...]. a(n) = Sum_{i=0..n} binomial(n+4, i+4)*b(i), where b(i)=[1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
%C Antidiagonal sums of the convolution array A213550. - _Clark Kimberling_, Jun 17 2012
%C Binomial transform of (1, 6, 14, 16, 9, 2, 0, 0, 0, ...). - _Gary W. Adamson_, Jul 28 2015
%C 2*a(n) is number of ways to place 4 queens on an (n+3) X (n+3) chessboard so that they diagonally attack each other exactly 6 times. The maximal possible attack number, p=binomial(k,2)=6 for k=4 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form a corresponding complete graph. - _Antal Pinter_, Dec 27 2015
%C While adjusting for offsets, add A000389 to find the next in series A000389, A005585, A051836, A034263, A027800, A051843, A051877, A051878, A051879, A051880, A056118, A271567. (See _Bruno Berselli_'s comments in A271567.) - _Bruce J. Nicholson_, Jun 21 2018
%C Coefficients in the terminating series identity 1 - 7*n/(n + 6) + 27*n*(n - 1)/((n + 6)*(n + 7)) - 77*n*(n - 1)*(n - 2)/((n + 6)*(n + 7)*(n + 8)) + ... = 0 for n = 1,2,3,.... Cf. A002415 and A040977. - _Peter Bala_, Feb 18 2019
%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. 797.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005585/b005585.txt">Table of n, a(n) for n = 1..1000</a> (first 121 terms from Alexander Adamchuk)
%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 Paul Barry, <a href="https://arxiv.org/abs/2104.05593">On the Gap-sum and Gap-product Sequences of Integer Sequences</a>, arXiv:2104.05593 [math.CO], 2021.
%H R. K. Guy, <a href="/A005581/a005581_1.pdf">Letter to N. J. A. Sloane, Feb 1988</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H C. H. Karlson and N. J. A. Sloane, <a href="/A002790/a002790.pdf">Correspondence, 1974</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 R. P. Stanley and F. Zanello, <a href="http://arxiv.org/abs/1312.4352">The Catalan case of Armstrong's conjecture on core partitions</a>, arXiv preprint arXiv:1312.4352 [math.CO], 2013.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F G.f.: x*(1+x)/(1-x)^6.
%F a(n) = 2*C(n+4, 5) - C(n+3, 4). - _Paul Barry_, Mar 04 2003
%F a(n) = C(n+3, 5) + C(n+4, 5). - _Paul Barry_, Mar 17 2003
%F a(n) = C(n+2, 6) - C(n, 6), n >= 4. - _Zerinvary Lajos_, Jul 21 2006
%F a(n) = Sum_{k=1..n} T(k)*T(k+1)/3, where T(n) = n(n+1)/2 is a triangular number. - _Alexander Adamchuk_, May 08 2007
%F a(n-1) = (1/4)*Sum_{1 <= x_1, x_2 <= n} |x_1*x_2*det V(x_1,x_2)| = (1/4)*Sum_{1 <= i,j <= n} i*j*|i-j|, where V(x_1,x_2} is the Vandermonde matrix of order 2. First differences of A040977. - _Peter Bala_, Sep 21 2007
%F a(n) = C(n+4,4) + 2*C(n+4,5). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6), a(1)=1, a(2)=7, a(3)=27, a(4)=77, a(5)=182, a(6)=378. - _Harvey P. Dale_, Oct 04 2011
%F a(n) = (1/6)*Sum_{i=1..n+1} (i*Sum_{k=1..i} (i-1)*k). - _Wesley Ivan Hurt_, Nov 19 2014
%F E.g.f.: x*(2*x^4 + 35*x^3 + 180*x^2 + 300*x + 120)*exp(x)/120. - _Robert Israel_, Nov 19 2014
%F a(n) = A000389(n+3) + A000389(n+4). - _Bruce J. Nicholson_, Jun 21 2018
%F a(n) = -a(-3-n) for all n in Z. - _Michael Somos_, Jun 24 2018
%F From _Amiram Eldar_, Jun 28 2020: (Start)
%F Sum_{n>=1} 1/a(n) = 40*(16*log(2) - 11)/3.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 20*(8*Pi - 25)/3. (End)
%F a(n) = A004302(n+1) - A207361(n+1). - _J. M. Bergot_, May 20 2022
%F a(n) = Sum_{i=0..n+1} Sum_{j=i..n+1} i*j*(j-i)/2. - _Darío Clavijo_, Oct 11 2023
%F a(n) = (A000538(n+1) - A000330(n+1))/12. - _Yasser Arath Chavez Reyes_, Feb 21 2024
%e G.f. = x + 7*x^2 + 27*x^3 + 77*x^4 + 182*x^5 + 378*x^6 + 714*x^7 + 1254*x^8 + ... - _Michael Somos_, Jun 24 2018
%p [seq(binomial(n+2,6)-binomial(n,6), n=4..45)]; # _Zerinvary Lajos_, Jul 21 2006
%p A005585:=(1+z)/(z-1)**6; # _Simon Plouffe_ in his 1992 dissertation
%t With[{c=5!},Table[n(n+1)(n+2)(n+3)(2n+3)/c,{n,40}]] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{1,7,27,77,182,378},40] (* _Harvey P. Dale_, Oct 04 2011 *)
%t CoefficientList[Series[(1 + x) / (1 - x)^6, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 09 2013 *)
%o (Magma) I:=[1, 7, 27, 77, 182, 378]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..40]]; // _Vincenzo Librandi_, Jun 09 2013
%o (PARI) a(n)=binomial(n+3,4)*(2*n+3)/5 \\ _Charles R Greathouse IV_, Jul 28 2015
%Y a(n) = ((-1)^(n+1))*A053120(2*n+3, 5)/16, (1/16 of sixth unsigned column of Chebyshev T-triangle, zeros omitted).
%Y Partial sums of A002415.
%Y Cf. A006542, A040977, A047819, A111125 (third column).
%Y Cf. a(n) = ((-1)^(n+1))*A084960(n+1, 2)/16 (compare with the first line). - _Wolfdieter Lang_, Aug 04 2014
%Y Cf. A000389, A051836, A034263, A027800, A051843, A051877, A051878, A051879.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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