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 A005803 Second-order Eulerian numbers: a(n) = 2^n - 2*n. (Formerly M1838) 39
 1, 0, 0, 2, 8, 22, 52, 114, 240, 494, 1004, 2026, 4072, 8166, 16356, 32738, 65504, 131038, 262108, 524250, 1048536, 2097110, 4194260, 8388562, 16777168, 33554382, 67108812, 134217674, 268435400, 536870854, 1073741764, 2147483586 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Starting with n=2, a(n) is the second-order Eulerian number <> (see A008517). Also, number of 3 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1), (01;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i11. - Johannes W. Meijer, Oct 16 2009 a(0)=1, a(1)=0, a(2)=0, a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3). - Harvey P. Dale, May 21 2011 a(n) = A000225(n+1) - A081494(n+1), n > 1. In other words, a(n) equals the sum of the elements in a Pascal triangle of depth n+1 minus the sum of the elements on its perimeter. - Ivan N. Ianakiev, Jun 01 2014 a(n) = A165900(n-1) + Sum_{i=0..n-1} a(i), for n > 0. - Ivan N. Ianakiev, Nov 24 2014 a(n) = A000225(n) - A005408(n-1). - Miquel Cerda, Nov 25 2016 E.g.f.: exp(x)*(exp(x) - 2*x). - Ilya Gutkovskiy, Nov 25 2016 EXAMPLE G.f. = 1 + 2*x^3 + 8*x^4 + 22*x^5 + 52*x^6 + 114*x^7 + 240*x^8 + 494*x^9 + ... MAPLE A005803:=-2*z/(2*z-1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence except for three leading terms MATHEMATICA Table[2^n-2n, {n, 0, 50}] (* or *) LinearRecurrence[{4, -5, 2}, {1, 0, 0}, 51] (* Harvey P. Dale, May 21 2011 *) PROG (PARI) {a(n) = if( n<0, 0, 2^n - 2*n)}; /* Michael Somos, Oct 13 2002 */ (Haskell) a005803 n = 2 ^ n - 2 * n a005803_list = 1 : f 1 [0, 2 ..] where f x (z:zs@(z':_)) = y : f y zs where y = (x + z) * 2 - z' -- Reinhard Zumkeller, Jan 19 2014 (Magma) [2^n-2*n: n in [0..30]]; // Wesley Ivan Hurt, Jun 04 2014 CROSSREFS Equivalent to second column of A008517. a(n) = A070313 + 1 = A052515 + 2. Bisection of A077866. Equals for n =>3 the third right hand column of A163936. Cf. A000918 (first differences). Cf. A000079, A000325, A005843, A130102. Sequence in context: A006696 A094939 A006732 * A145654 A221880 A295141 Adjacent sequences: A005800 A005801 A005802 * A005804 A005805 A005806 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified December 8 08:22 EST 2022. Contains 358693 sequences. (Running on oeis4.)