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 A051797 Partial sums of A007585. 8
 1, 12, 50, 140, 315, 616, 1092, 1800, 2805, 4180, 6006, 8372, 11375, 15120, 19720, 25296, 31977, 39900, 49210, 60060, 72611, 87032, 103500, 122200, 143325, 167076, 193662, 223300, 256215, 292640, 332816, 376992, 425425, 478380, 536130 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n-1) is the n-th antidiagonal sum of the convolution array A213835. - Clark Kimberling, Jul 04 2012 Convolution of A000027 with A001107 (excluding 0). - Bruno Berselli, Dec 07 2012 REFERENCES Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196. Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94. Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). FORMULA a(n) = binomial(n+3,3)*(2*n+1) = (n+1)*(n+2)*(n+3)*(2*n+1)/6. G.f.: (1+7*x)/(1-x)^5. a(n) = A080851(8,n). - R. J. Mathar, Jul 28 2016 E.g.f.: (6 + 66*x + 81*x^2 + 25*x^3 + 2*x^4)*exp(x)/6. - G. C. Greubel, Aug 30 2019 From Amiram Eldar, Feb 11 2022: (Start) Sum_{n>=0} 1/a(n) = (32*log(2) - 11)/10. Sum_{n>=0} (-1)^n/a(n) = (8*Pi - 56*log(2) + 23)/10. (End) MAPLE seq((2*n+1)*binomial(n+3, 3), n=0..40); # G. C. Greubel, Aug 30 2019 MATHEMATICA Table[(2*n+1)*Binomial[n+3, 3], {n, 0, 40}] (*  Vladimir Joseph Stephan Orlovsky, Apr 19 2011, modified by G. C. Greubel, Aug 30 2019 *) PROG (Magma) /* A000027 convolved with A001107 (excluding 0): */ A001107:=func; [&+[(n-i+1)*A001107(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012 (Magma) [(2*n+1)*Binomial(n+3, 3): n in [0..40]]; // G. C. Greubel, Aug 30 2019 (PARI) vector(40, n, (2*n-1)*binomial(n+2, 3)) \\ G. C. Greubel, Aug 30 2019 (Sage) [(2*n+1)*binomial(n+3, 3) for n in (0..40)] # G. C. Greubel, Aug 30 2019 (GAP) List([0..40], n-> (2*n+1)*Binomial(n+3, 3)) # G. C. Greubel, Aug 30 2019 CROSSREFS Cf. A000027, A001107, A007585, A080851. Cf. A093565 ((8, 1) Pascal, column m=4). Cf. A220212 for a list of sequences produced by the convolution of the natural numbers with the k-gonal numbers. Sequence in context: A063491 A248230 A083559 * A342482 A333276 A145886 Adjacent sequences:  A051794 A051795 A051796 * A051798 A051799 A051800 KEYWORD nonn,easy AUTHOR Barry E. Williams, Dec 11 1999 STATUS approved

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Last modified July 4 11:02 EDT 2022. Contains 355075 sequences. (Running on oeis4.)