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 A210569 a(n) = (n-3)*(n-2)*(n-1)*n*(n+1)/30. 5
 0, 0, 0, 0, 4, 24, 84, 224, 504, 1008, 1848, 3168, 5148, 8008, 12012, 17472, 24752, 34272, 46512, 62016, 81396, 105336, 134596, 170016, 212520, 263120, 322920, 393120, 475020, 570024, 679644, 805504, 949344, 1113024, 1298528, 1507968, 1743588, 2007768, 2303028 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The following sequences are provided by the formula n*binomial(n,k) - binomial(n,k+1) = k*binomial(n+1,k+1): . A000217(n)   for k=1, . A007290(n+1) for k=2, . A050534(n)   for k=3, . a(n)         for k=4, . A000910(n+1) for k=5. Sum of reciprocals of a(n), for n>3: 5/16. From a(2), convolution of oblong numbers (A002378) with themselves. - Bruno Berselli, Oct 24 2016 LINKS Bruno Berselli, Table of n, a(n) for n = 0..1000 C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 3. - N. J. A. Sloane, Mar 23 2014 Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1). FORMULA G.f.: 4*x^4/(1-x)^6. a(n) = n*binomial(n,4)-binomial(n,5) = 4*binomial(n+1,5) = 4*A000389(n+1). a(n) = 2*A177747(2*n-7), with A177747(-7) = A177747(-5) = A177747(-3) = A177747(-1) = 0. (n-4)*a(n) = (n+1)*a(n-1). MAPLE f:=n->(n^5-5*n^3+4*n)/30; [seq(f(n), n=0..50)]; # N. J. A. Sloane, Mar 23 2014 MATHEMATICA LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 0, 0, 4, 24}, 39] CoefficientList[Series[4 x^4/(1 - x)^6, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *) PROG (MAGMA) [4*Binomial(n+1, 5): n in [0..38]]; (Maxima) makelist(coeff(taylor(4*x^4/(1-x)^6, x, 0, n), x, n), n, 0, 38); (PARI) a(n)=(n-3)*(n-2)*(n-1)*n*(n+1)/30 \\ Charles R Greathouse IV, Oct 07 2015 CROSSREFS Cf. A000217, A000910, A002378, A007290, A050534, A000389, A177747. First differences are in A033488. Sequence in context: A264184 A211071 A212135 * A005561 A061612 A097875 Adjacent sequences:  A210566 A210567 A210568 * A210570 A210571 A210572 KEYWORD nonn,easy AUTHOR Bruno Berselli, Mar 23 2012 STATUS approved

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Last modified December 12 12:30 EST 2019. Contains 329958 sequences. (Running on oeis4.)