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%I #38 Sep 08 2022 08:44:59
%S 1,12,57,182,462,1008,1974,3564,6039,9724,15015,22386,32396,45696,
%T 63036,85272,113373,148428,191653,244398,308154,384560,475410,582660,
%U 708435,855036,1024947,1220842,1445592,1702272,1994168,2324784,2697849,3117324,3587409
%N Partial sums of A051740.
%C Convolution of triangular numbers (A000217) and enneagonal numbers (A001106). - _Bruno Berselli_, Jul 21 2015
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%D Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-16.
%H G. C. Greubel, <a href="/A051877/b051877.txt">Table of n, a(n) for n = 0..1000</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 a(n) = C(n+4, 4)*(7*n+5)/5.
%F G.f.: (1+6*x)/(1-x)^6.
%F From _Wesley Ivan Hurt_, May 02 2015: (Start)
%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).
%F a(n) = (n+1)*(n+2)*(n+3)*(n+4)*(7*n+5)/120. (End)
%F E.g.f.: (5! +1320*x +2040*x^2 +920*x^3 +145*x^4 +7*x^5)*exp(x)/5!
%p A051877:=n->binomial(n+4,4)*(7*n+5)/5: seq(A051877(n), n=0..40); # _Wesley Ivan Hurt_, May 02 2015
%t Table[(n+1)(n+2)(n+3)(n+4)(7n+5)/120, {n, 0, 40}] (* _Vincenzo Librandi_, May 03 2015 *)
%t LinearRecurrence[{6,-15,20,-15,6,-1},{1,12,57,182,462,1008},40] (* _Harvey P. Dale_, May 05 2022 *)
%o (Magma) [(n+1)*(n+2)*(n+3)*(n+4)*(7*n+5)/120 : n in [0..40]]; // _Wesley Ivan Hurt_, May 02 2015
%o (PARI) vector(40, n, (7*n-2)*binomial(n+3,4)/5) \\ _G. C. Greubel_, Aug 29 2019
%o (Sage) [(7*n+5)*binomial(n+4,4)/5 for n in (0..40)] # _G. C. Greubel_, Aug 29 2019
%o (GAP) List([0..40], n-> (7*n+5)*Binomial(n+4,4)/5); # _G. C. Greubel_, Aug 29 2019
%Y Cf. A051740; A000217, A001106.
%Y Cf. A093564 ((7, 1) Pascal, column m=5).
%K nonn,easy
%O 0,2
%A _Barry E. Williams_, Dec 14 1999
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