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 A241765 a(n) = n*(n + 1)*(n + 2)*(3*n + 17)/24. 11
 0, 5, 23, 65, 145, 280, 490, 798, 1230, 1815, 2585, 3575, 4823, 6370, 8260, 10540, 13260, 16473, 20235, 24605, 29645, 35420, 41998, 49450, 57850, 67275, 77805, 89523, 102515, 116870, 132680, 150040, 169048, 189805, 212415, 236985, 263625, 292448 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Equivalently, Sum_{i=0..n} (i+4)*A000217(i). Sequences of the type Sum_{i=0..n} (i+k)*A000217(i): k = 0,  A001296: 0,  1,  7, 25,  65, 140, 266, 462, ... k = 1,  A000914: 0,  2, 11, 35,  85, 175, 322, 546, ... k = 2,  A050534: 0,  3, 15, 45, 105, 210, 378, 630, ... (deleting two 0) k = 3,  A215862: 0,  4, 19, 55, 125, 245, 434, 714, ... k = 4,     a(n): 0,  5, 23, 65, 145, 280, 490, 798, ... k = 5,  A239568: 0,  6, 27, 75, 165, 315, 546, 882, ... Antidiagonal sums (without 0) give A034263: 1, 9, 39, 119, 294, ... Diagonal: 1, 11, 45, 125, 280, 546, ... is A051740. Also: k = -1 gives A050534 deleting a 0; k = -2 gives 0 followed by A059302. After 0, partial sums of A212343 and third column of A118788. This sequence is even related to A005286 by a(n) = n*A005286(n) - Sum_{i=0..n-1} A005286(i). LINKS Bruno Berselli, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1). FORMULA G.f.: x*(5 - 2*x)/(1 - x)^5. a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). a(n) = A227342(A055998(n+1)). a(n) = Sum_{j=0..n+2} (-1)^(n-j)*binomial(-j,-n-2)*S1(j,n), S1 Stirling cycle numbers A132393. - Peter Luschny, Apr 10 2016 EXAMPLE a(7) = 4*0 + 5*1 + 6*3 + 7*6 + 8*10 + 9*15 + 10*21 + 11*28 = 798. MAPLE A241765:=n->n*(n + 1)*(n + 2)*(3*n + 17)/24; seq(A241765(n), n=0..40); # Wesley Ivan Hurt, May 09 2014 MATHEMATICA Table[n (n + 1) (n + 2) (3 n + 17)/24, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 23, 65, 145}, 40] CoefficientList[Series[x (5 - 2 x)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 09 2014 *) PROG (Sage) [n*(n+1)*(n+2)*(3*n+17)/24 for n in (0..40)] (Maxima) makelist(coeff(taylor(x*(5-2*x)/(1-x)^5, x, 0, n), x, n), n, 0, 40); (MAGMA) /* By first comment: */ k:=4; A000217:=func; [&+[(i+k)*A000217(i): i in [0..n]]: n in [0..40]]; (PARI) a(n)=n*(n+1)*(n+2)*(3*n+17)/24 \\ Charles R Greathouse IV, Oct 07 2015 (PARI) x='x+O('x^99); concat(0, Vec(x*(5-2*x)/(1-x)^5)) \\ Altug Alkan, Apr 10 2016 CROSSREFS Cf. A000217, A005286, A118788, A212343, A227342. Cf. similar sequences A000914, A001296, A050534, A059302, A215862, A239568 (see table in Comments lines). Sequence in context: A159241 A179094 A176874 * A106956 A084671 A284648 Adjacent sequences:  A241762 A241763 A241764 * A241766 A241767 A241768 KEYWORD nonn,easy AUTHOR Bruno Berselli, Apr 28 2014 STATUS approved

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Last modified September 16 02:16 EDT 2019. Contains 327088 sequences. (Running on oeis4.)