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 A005993 Expansion of (1+x^2)/((1-x)^2*(1-x^2)^2). (Formerly M1576) 36
 1, 2, 6, 10, 19, 28, 44, 60, 85, 110, 146, 182, 231, 280, 344, 408, 489, 570, 670, 770, 891, 1012, 1156, 1300, 1469, 1638, 1834, 2030, 2255, 2480, 2736, 2992, 3281, 3570, 3894, 4218, 4579, 4940, 5340, 5740, 6181, 6622, 7106, 7590, 8119, 8648, 9224, 9800 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Alkane (or paraffin) numbers l(6,n). Dimension of the space of homogeneous degree n polynomials in (x1, y1, x2, y2) invariant under permutation of variables x1<->y1, x2<->y2. Also multidigraphs with loops on 2 nodes with n arcs (see A138107). - Vladeta Jovovic, Dec 27 1999 Euler transform of finite sequence [2,3,0,-1]. - Michael Somos, Mar 17 2004 a(n-2) is the number of plane partitions with trace 2. - Michael Somos, Mar 17 2004 With offset 4, a(n) is the number of bracelets with n beads, 3 of which are red, 1 of which is blue. For odd n, a(n) = C(n-1,3)/2. For even n, a(n) = C(n-1,3)/2 +(n-2)/4. For n >= 6, with K = (n-1)(n-2)/((n-5)(n-4)), for odd n, a(n) = K*a(n-2). For even n, a(n) = K*a(n-2) -(n-2)/(n-5). - Washington Bomfim, Aug 05 2008 Equals (1,2,3,4,...) convolved with (1,0,3,0,5,...). - Gary W. Adamson, Feb 16 2009 Equals row sums of triangle A177878. Equals (1/2)*((1, 4, 10, 20, 35, 56, ...) + (1, 0, 2 0, 3, 0, 4, ...)). From Ctibor O. Zizka, Nov 21 2014: (Start) With offset 4, a(n) is the number of different patterns of the 2-color 4-partition of n. P(n)_(k;t) gives the number of different patterns of the t-color, k-partition of n. P(n)_(k;t) = 1 + Sum(i=2..n) Sum(j=2..i) Sum(r=1..m) c_(i,j)*v_r*F_r(X_1,...,X_i). P(n;i;j) = Sum(r=1..m) c_(i,j)*v_r*F_r(X_1,...,X_i). m partition number of i. c_(i,j) number of different coloring patterns on the r-th form (X_1,...,X_i) of i-partition with j-colors. v_r number of i-partitions of n of the r-th form (X_1,...,X_i). F_r(X_1,...,X_i) number of different patterns of the r-th form i-partition of n. Some simple results: P(1)_(k;t)=1, P(2)_(k;t)=2, P(3)_(k;t)=4, P(4)_(k;t)=11, etc. P(n;1;1) = P(n;n;n) = 1 for all n; P(n;2;2) = floor(n/2) (A004526); P(n;3;2) = (n*n - 2*n + n mod 2)/4 (A002620). This sequence is a(n) = P(n;4;2). 2-coloring of 4-partition is (A,B,A,B) or (B,A,B,A). Each 4-partition of n has one of the form (X_1,X_1,X_1,X_1),(X_1,X_1,X_1,X_2), (X_1,X_1,X_2,X_2),(X_1,X_1,X_2,X_3),(X_1,X_2,X_3,X_4). The number of forms is m=5 which is the partition number of k=4. Partition form (X_1,X_1,X_1,X_1) gives 1 pattern ((X_1A,X_1B,X_1A,X_1B), (X_1,X_1,X_1,X_2) gives 2 patterns, (X_1,X_1,X_2,X_2) gives 4 patterns, (X_1,X_1,X_2,X_3) gives 6 patterns and (X_1,X_2,X_3,X_4) gives 12 patterns. Thus a(n) = P(n;4;2) = 1*1*v_1 + 1*2*v_2 + 1*4*v_3 + 1*6*v_4 + 1*12*v_5 where v_r is the number of different 4-partitions of the r-th form (X_1,X_2,X_3,X_4) for a given n. Example: The 4-partitions of 8 are (2,2,2,2), (1,1,1,5), (1,1,3,3), (1,1,2,4), and (1,2,2,3): (2,2,2,2) 1 pattern (1,1,1,5), (1,1,5,1) 2 patterns (1,1,3,3), (1,3,3,1), (3,1,1,3), (1,3,1,3) 4 patterns (1,1,2,4), (1,1,4,2), (1,2,1,4), (1,2,4,1), (1,4,1,2), (2,1,1,4) 6 patterns (2,2,1,3), (2,2,3,1), (2,1,2,3), (2,1,3,2), (2,3,2,1), (1,2,2,3) 6 patterns Thus a(8) = P(8,4,2) = 1 + 2 + 4 + 6 + 6 = 19. (End) a(n) = length of run n+2 of consecutive 1's in A254338. - Reinhard Zumkeller, Feb 27 2015 Take a chessboard of (n+2) X (n+2) unit squares in which the a1 square is black. a(n) is the number of composite squares having black unit squares on their vertices. - Ivan N. Ianakiev, Jul 19 2018 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). L. Smith, Polynomial Invariants of Finite Groups, A K Peters, 1995, p. 96. LINKS T. D. Noe, Table of n, a(n) for n = 0..1000 M. Benoumhani, M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, Lemma 6 3rd line. Washington Bomfim, The 19 bracelets with 8 beads - one blue, three reds and four blacks. [From Washington Bomfim, Aug 05 2008] T. M. Brown, On the unimodality of convolutions of sequences of binomial coefficients, arXiv:1810.08235 [math.CO] (2018). Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017. Dragomir Z. Djokovic, Poincaré series of some pure and mixed trace algebras of two generic matrices, arXiv:math/0609262 [math.AC], 2006. See Table 8. Naihuan Jing, Kailash Misra, Carla Savage, On multi-color partitions and the generalized Rogers-Ramanujan identities, arXiv:math/9907183 [math.CO], 1999. S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy) N. J. A. Sloane, Classic Sequences L. Smith, Polynomial invariants of finite groups. A survey of recent developments. Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 3, 211-250. See page 218. MR1433171 (98i:13009). Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1). FORMULA l(c, r) = 1/2 C(c+r-3, r) + 1/2 d(c, r), where d(c, r) is C((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, C((c + r - 4)/2, r/2) if c is even and r is even, C((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd. G.f.: (1+x^2)/((1-x)^2*(1-x^2)^2) = (1+x^2)/((1+x)^2*(x-1)^4) = (1/(1-x)^4 +1/(1-x^2)^2)/2. a(2n) = (n+1)(2n^2+4n+3)/3, a(2n+1) = (n+1)(n+2)(2n+3)/3. a(-4-n) = -a(n). From Yosu Yurramendi, Sep 12 2008: (Start) a(n+1) = a(n) + A008794(n+3) with a(1)=1. a(n) = A027656(n) + 2*A006918(n). a(n+2) = a(n) + A000982(n+2) with a(1)=1, a(2)=2. (End) a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6). - Jaume Oliver Lafont, Dec 05 2008 a(n) = (n^3 + 6*n^2 + 11*n + 6)/12 + ((n+2)/4)[n even] (the bracket means that the second term is added if and only if n is even). - Benoit Jubin, Mar 31 2012 a(n) = (1/12)*n*(n+1)*(n+2) + (1/4)*(n+1)*(1/2)*(1-(-1)^n), with offset 1. - Yosu Yurramendi, Jun 20 2013 a(n) = Sum_{i=0..n+1} ceiling(i/2) * round(i/2) = Sum_{i=0..n+2} floor(i/2)^2. - Bruno Berselli, Aug 30 2013 a(n) = (n + 2)*(3*(-1)^n + 2*n^2 + 8*n + 9)/24. - Ilya Gutkovskiy, May 04 2016 Recurrence formula:  a(n) = ((n+2)*a(n-2)+2*a(n-1)-n)/(n-2), a(1)=1, a(2)=2. - Gerry Martens, Jun 10 2018 E.g.f.: exp(-x)*(6 - 3*x + exp(2*x)*(18 + 39*x + 18*x^2 + 2*x^3))/24. - Stefano Spezia, Feb 23 2020 EXAMPLE a(2) = 6, since ( x1*y1, x2*y2, x1*x1+y1*y1, x2*x2+y2*y2, x1*x2+y1*y2, x1*y2+x2*y1 ) are a basis for homogeneous quadratic invariant polynomials. MAPLE g := proc(n) local i; add(floor(i/2)^2, i=1..n+1) end: # Joseph S. Riel (joer(AT)k-online.com), Mar 22 2002 a:= n-> (Matrix([[1, 0\$3, -1, -2]]).Matrix(6, (i, j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -4, 1, 2, -1][i] else 0 fi)^n)[1, 1]; seq (a(n), n=0..44); # Alois P. Heinz, Jul 31 2008 MATHEMATICA CoefficientList[Series[(1+x^2)/((1-x)^2*(1-x^2)^2), {x, 0, 44}], x]  (* Jean-François Alcover, Apr 08 2011 *) LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 2, 6, 10, 19, 28}, 50] (* Harvey P. Dale, Feb 20 2012 *) PROG (Haskell)   Following Gary W. Adamson. import Data.List (inits, intersperse) a005993 n = a005994_list !! n a005993_list = map (sum . zipWith (*) (intersperse 0 [1, 3 ..]) . reverse) \$                    tail \$ inits [1..] -- Reinhard Zumkeller, Feb 27 2015 (MAGMA) I:=[1, 2, 6, 10, 19, 28]; [n le 6 select I[n] else 2*Self(n-1)+Self(n-2)-4*Self(n-3)+Self(n-4)+2*Self(n-5)-Self(n-6): n in [1..60]]; // Vincenzo Librandi, Jul 19 2015 (PARI) a(n)=polcoeff((1+x^2)/(1-x)^2/(1-x^2)^2+x*O(x^n), n) (PARI) a(n) = (binomial(n+3, n) + (1-n%2)*binomial((n+2)/2, n>>1))/2 \\ Washington Bomfim, Aug 05 2008 (PARI) a = vector(50); a[1]=1; a[2]=2; for(n=3, 50, a[n] = ((n+2)*a[n-2]+2*a[n-1]-n)/(n-2)); a \\ Gerry Martens, Jun 03 2018 (Sage) def A005993():     a, b, to_be = 0, 0, True     while True:         yield (a*(a*(2*a+9)+13)+b*(b+1)*(2*b+1)+6)//6         if to_be: b += 1         else: a += 1         to_be = not to_be a = A005993() [next(a) for _ in range(48)] # Peter Luschny, May 04 2016 CROSSREFS Cf. A177878. Partial sums of A008794 (without 0). - Bruno Berselli, Aug 30 2013 Cf. A254338, A002260, A005408, A282011. Sequence in context: A054273 A127567 A169643 * A028247 A209535 A065054 Adjacent sequences:  A005990 A005991 A005992 * A005994 A005995 A005996 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu) STATUS approved

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Last modified January 21 12:45 EST 2021. Contains 340350 sequences. (Running on oeis4.)