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 A227430 Expansion of x^2*(1-x)^3/((1-2*x)*(1-x+x^2)*(1-3*x+3x^2)). 1
 0, 0, 1, 3, 6, 10, 15, 21, 29, 45, 90, 220, 561, 1365, 3095, 6555, 13110, 25126, 46971, 87381, 164921, 320001, 640002, 1309528, 2707629, 5592405, 11450531, 23166783, 46333566, 91869970, 181348455, 357913941, 708653429, 1410132405, 2820264810, 5662052980 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Consider the binomial transform of 0, 0, 0, 0, 0, 1 (period 6) with its differences: 0, 0, 0, 0, 0, 1, 6, 21, 56, 126,... d(n): after 0, it is A192080. 0, 0, 0, 0, 1, 5, 15, 35, 70, 126,... e(n) 0, 0, 0, 1, 4, 10, 20, 35, 56, 85,... f(n) 0, 0, 1, 3, 6, 10, 15, 21, 29, 45,... a(n) 0, 1, 2, 3, 4, 5, 6, 8, 16, 45,... b(n) 1, 1, 1, 1, 1, 1, 2, 8, 29, 85,... c(n) 0, 0, 0, 0, 0, 1, 6, 21, 56, 126,... d(n). a(n) + d(n) = A024495(n), b(n) + e(n) = A131708(n), c(n) + f(n) = A024493(n). a(n) - d(n) = 0, 0, 1, 3, 6, 9, 9, 0,... A057083(n-2) b(n) - e(n) = 0, 1, 2, 3, 3, 0, -9, -27,... A057682(n) c(n) - f(n) = 1, 1, 1, 0, -3, -9, -18, -27,... A057681(n) d(n) - a(n) = 0, 0, -1, -3, -6, -9, -9, 0,... -A057083(n-2) e(n) - b(n) = 0, -1, -2, -3, -3, 0, 9, 27,... -A057682(n) f(n) - c(n) = -1, -1, -1, 0, 3, 9, 18, 27,... -A057681(n). The first column is A131531(n). The first two trisections are multiples of 3. Is the third (1, 10, 29,...) mod 9 A029898(n)? LINKS Seiichi Manyama, Table of n, a(n) for n = 0..3000 Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6). FORMULA a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) for n>5, a(0)=a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(5)=10. a(n) = A024495(n) - A192080(n-5) for n>4. G.f.: -(x^5 - 3*x^4 + 3*x^3 - x^2)/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)). - Ralf Stephan, Jul 13 2013 a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k+2). - Seiichi Manyama, Mar 23 2019 EXAMPLE a(6)=6*10-15*6+20*3-15*1+6*0=15, a(7)=90-150+120-45+6=21. MATHEMATICA Join[{0}, LinearRecurrence[{6, -15, 20, -15, 6}, {0, 1, 3, 6, 10}, 40]] (* Harvey P. Dale, Dec 17 2014 *) PROG (PARI) {a(n) = sum(k=0, n\6, binomial(n, 6*k+2))} \\ Seiichi Manyama, Mar 23 2019 CROSSREFS Sequence in context: A175724 A335184 A101551 * A051166 A188278 A188279 Adjacent sequences: A227427 A227428 A227429 * A227431 A227432 A227433 KEYWORD nonn,easy AUTHOR Paul Curtz, Jul 11 2013 EXTENSIONS Definition uses the g.f. of Ralf Stephan. More terms from Harvey P. Dale, Dec 17 2014 STATUS approved

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Last modified June 19 23:45 EDT 2024. Contains 373510 sequences. (Running on oeis4.)