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A051740 Partial sums of A007584. 7

%I #42 May 18 2023 12:35:29

%S 1,11,45,125,280,546,966,1590,2475,3685,5291,7371,10010,13300,17340,

%T 22236,28101,35055,43225,52745,63756,76406,90850,107250,125775,146601,

%U 169911,195895,224750,256680,291896,330616,373065,419475,470085,525141

%N Partial sums of A007584.

%C Convolution of A000027 with A001106 (excluding 0). - _Bruno Berselli_, Dec 07 2012

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

%D Murray R.Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.

%H G. C. Greubel, <a href="/A051740/b051740.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_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = binomial(n+3, 3)*(7*n+4)/4.

%F a(n) = (7*n+4)*binomial(n+3, 3)/4.

%F G.f.: (1+6*x)/(1-x)^5.

%F a(n) = A080852(7,n). - _R. J. Mathar_, Jul 28 2016

%F E.g.f.: (4! + 240*x + 288*x^2 + 88*x^3 + 7*x^4)*exp(x)/4!. - _G. C. Greubel_, Aug 29 2019

%p seq((7*n+4)*binomial(n+3,3)/4, n=0..40); # _G. C. Greubel_, Aug 29 2019

%t Table[(7*n+4)*Binomial[n+3,3]/4, {n,0,40)] (* _G. C. Greubel_, Aug 29 2019 *)

%t LinearRecurrence[{5,-10,10,-5,1},{1,11,45,125,280},40] (* _Harvey P. Dale_, May 18 2023 *)

%o (Magma) /* A000027 convolved with A001106 (excluding 0): */ A001106:=func<n | n*(7*n-5)/2>; [&+[(n-i+1)*A001106(i): i in [1..n]]: n in [1..36]]; // _Bruno Berselli_, Dec 07 2012

%o (PARI) vector(40, n, (7*n-3)*binomial(n+2,3)/4) \\ _G. C. Greubel_, Aug 29 2019

%o (Sage) [(7*n+4)*binomial(n+3,3)/4 for n in (0..40)] # _G. C. Greubel_, Aug 29 2019

%o (GAP) List([0..40], n-> (7*n+4)*Binomial(n+3,3)/4); # _G. C. Greubel_, Aug 29 2019

%Y Cf. A001106, A007584.

%Y Cf. A093564 ((7, 1) Pascal, column m=4).

%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_, Dec 07 1999

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)