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A055268 a(n) = (11*n + 4)*C(n+3, 3)/4.

%I A055268 #34 Jan 05 2025 19:51:36
%S A055268 1,15,65,185,420,826,1470,2430,3795,5665,8151,11375,15470,20580,26860,
%T A055268 34476,43605,54435,67165,82005,99176,118910,141450,167050,195975,
%U A055268 228501,264915,305515,350610,400520,455576,516120,582505,655095,734265
%N A055268 a(n) = (11*n + 4)*C(n+3, 3)/4.
%C A055268 a(n) is the number of compositions of n when there are 9 types of each natural number. - _Milan Janjic_, Aug 13 2010
%C A055268 Convolution of A000027 with A051865 (excluding 0). - _Bruno Berselli_, Dec 07 2012
%D A055268 A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
%H A055268 G. C. Greubel, <a href="/A055268/b055268.txt">Table of n, a(n) for n = 0..1000</a>
%H A055268 I. Adler, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), pp. 181-193.
%H A055268 E. I. Emerson, <a href="https://web.archive.org/web/2024*/https://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 A055268 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H A055268 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A055268 G.f.: (1 + 10*x)/(1-x)^5. - _R. J. Mathar_, Oct 26 2011
%F A055268 From _G. C. Greubel_, Jan 17 2020:(Start)
%F A055268 a(n) = 11*binomial(n+4,4) - 10*binomial(n+3,3).
%F A055268 E.g.f.: (24 + 336*x + 432*x^2 + 136*x^3 + 11*x^4)*exp(x)/24. (End)
%p A055268 seq( (11*n+4)*binomial(n+3,3)/4, n=0..30); # _G. C. Greubel_, Jan 17 2020
%t A055268 Table[11*Binomial[n+4,4] -10*Binomial[n+3,3], {n,0,30}] (* _G. C. Greubel_, Jan 17 2020 *)
%o A055268 (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 A055268 (Python)
%o A055268 A055268_list, m = [], [11, 1, 1, 1, 1]
%o A055268 for _ in range(10**2):
%o A055268     A055268_list.append(m[-1])
%o A055268     for i in range(4):
%o A055268         m[i+1] += m[i] # _Chai Wah Wu_, Jan 24 2016
%o A055268 (PARI) a(n) = (11*n+4)*binomial(n+3, 3)/4; \\ _Michel Marcus_, Sep 07 2017
%o A055268 (Sage) [(11*n+4)*binomial(n+3,3)/4 for n in (0..30)] # _G. C. Greubel_, Jan 17 2020
%o A055268 (GAP) List([0..30], n-> (11*n+4)*Binomial(n+3,3)/4 ); # _G. C. Greubel_, Jan 17 2020
%Y A055268 Partial sums of A050441.
%Y A055268 Cf. A000292, A051865.
%Y A055268 Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers.
%K A055268 nonn,easy
%O A055268 0,2
%A A055268 _Barry E. Williams_, May 10 2000