|
%I #32 Sep 08 2022 08:45:01
%S 1,15,65,185,420,826,1470,2430,3795,5665,8151,11375,15470,20580,26860,
%T 34476,43605,54435,67165,82005,99176,118910,141450,167050,195975,
%U 228501,264915,305515,350610,400520,455576,516120,582505,655095,734265
%N a(n) = (11*n + 4)*C(n+3, 3)/4.
%C a(n) is the number of compositions of n when there are 9 types of each natural number. - _Milan Janjic_, Aug 13 2010
%C Convolution of A000027 with A051865 (excluding 0). - _Bruno Berselli_, Dec 07 2012
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
%H G. C. Greubel, <a href="/A055268/b055268.txt">Table of n, a(n) for n = 0..1000</a>
%H I. Adler, <a href="http://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), pp. 181-193.
%H E. I. Emerson, <a href="http://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242.
%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 G.f.: (1 + 10*x)/(1-x)^5. - _R. J. Mathar_, Oct 26 2011
%F From _G. C. Greubel_, Jan 17 2020:(Start)
%F a(n) = 11*binomial(n+4,4) - 10*binomial(n+3,3).
%F E.g.f.: (24 + 336*x + 432*x^2 + 136*x^3 + 11*x^4)*exp(x)/24. (End)
%p seq( (11*n+4)*binomial(n+3,3)/4, n=0..30); # _G. C. Greubel_, Jan 17 2020
%t Table[11*Binomial[n+4,4] -10*Binomial[n+3,3], {n,0,30}] (* _G. C. Greubel_, Jan 17 2020 *)
%o (Magma) /* A000027 convolved with A051865 (excluding 0): */ A051865:=func<n | n*(11*n-9)/2>; [&+[(n-i+1)*A051865(i): i in [1..n]]: n in [1..35]]; // _Bruno Berselli_, Dec 07 2012
%o (Python)
%o A055268_list, m = [], [11, 1, 1, 1, 1]
%o for _ in range(10**2):
%o A055268_list.append(m[-1])
%o for i in range(4):
%o m[i+1] += m[i] # _Chai Wah Wu_, Jan 24 2016
%o (PARI) a(n) = (11*n+4)*binomial(n+3, 3)/4; \\ _Michel Marcus_, Sep 07 2017
%o (Sage) [(11*n+4)*binomial(n+3,3)/4 for n in (0..30)] # _G. C. Greubel_, Jan 17 2020
%o (GAP) List([0..30], n-> (11*n+4)*Binomial(n+3,3)/4 ); # _G. C. Greubel_, Jan 17 2020
%Y Partial sums of A050441.
%Y Cf. A000292, A051865.
%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 0,2
%A _Barry E. Williams_, May 10 2000
|
