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 A108561 Triangle read by rows: T(n,0)=1, T(n,n)=(-1)^n, T(n+1,k)=T(n,k-1)+T(n,k) for 0 < k < n. 23
 1, 1, -1, 1, 0, 1, 1, 1, 1, -1, 1, 2, 2, 0, 1, 1, 3, 4, 2, 1, -1, 1, 4, 7, 6, 3, 0, 1, 1, 5, 11, 13, 9, 3, 1, -1, 1, 6, 16, 24, 22, 12, 4, 0, 1, 1, 7, 22, 40, 46, 34, 16, 4, 1, -1, 1, 8, 29, 62, 86, 80, 50, 20, 5, 0, 1, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, -1, 1, 10, 46, 128, 239, 314, 296, 200, 95, 30, 6, 0, 1, 1, 11, 56, 174, 367 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,12 COMMENTS Sum_{k=0..n} T(n,k) = A078008(n); Sum_{k=0..n} abs(T(n,k) = A052953(n-1) for n > 0; T(n,1) = n - 2 for n > 1; T(n,2) = A000124(n-3) for n > 2; T(n,3) = A003600(n-4) for n > 4; T(n,n-6) = A001753(n-6) for n > 6; T(n,n-5) = A001752(n-5) for n > 5; T(n,n-4) = A002623(n-4) for n > 4; T(n,n-3) = A002620(n-1) for n > 3; T(n,n-2) = A008619(n-2) for n > 2; T(n,n-1) = n mod 2 for n > 0; T(2*n,n) = A072547(n+1). Sum_{k=0..n} T(n,k)*x^k = A232015(n), A078008(n), A000012(n), A040000(n), A001045(n+2), A140725(n+1) for x = 2, 1, 0, -1, -2, -3 respectively. - Philippe Deléham, Nov 17 2013, Nov 19 2013 (1,a^n) Pascal triangle with a = -1. - Philippe Deléham, Dec 27 2013 T(n,k) = A112465(n,n-k). - Reinhard Zumkeller, Jan 03 2014 LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened C. Merino, S. D. Noble, M. Ramirez-Ibanez, R. Villarroel-Flores, On the structure of the h-vector of a paving matroid, Eur. J. Comb. 33, No. 8, 1787-1799 (2012). FORMULA G.f.: (1-y*x)/(1-x-(y+y^2)*x). - Philippe Deléham, Nov 17 2013 T(n,k) = T(n-1,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0)=T(1,0)=1, T(1,1)=-1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 17 2013 From Peter Bala, Feb 18 2018: (Start) T(n,k) = Sum_{i = 0..k} binomial(n,i)*(-2)^(k-i), 0 <= k <= n. The n-th row polynomial is the n-th degree Taylor polynomial of the rational function (1 + x)^n/(1 + 2*x) about 0. For example, for n = 4, (1 + x)^4/(1 + 2*x) = 1 + 2*x + 2*x^2 + x^4 + O(x^5). (End) EXAMPLE From Philippe Deléham, Nov 17 2013: (Start) Triangle begins:   1;   1, -1;   1,  0,  1;   1,  1,  1, -1;   1,  2,  2,  0,  1;   1,  3,  4,  2,  1, -1;   1,  4,  7,  6,  3,  0,  1; (End) MAPLE A108561 := (n, k) -> add(binomial(n, i)*(-2)^(k-i), i = 0..k): seq(seq(A108561(n, k), k = 0..n), n = 0..12); # Peter Bala, Feb 18 2018 MATHEMATICA Clear[t]; t[n_, 0] = 1; t[n_, n_] := t[n, n] = (-1)^Mod[n, 2]; t[n_, k_] := t[n, k] = t[n-1, k] + t[n-1, k-1]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013 *) PROG (Haskell) a108561 n k = a108561_tabl !! n !! k a108561_row n = a108561_tabl !! n a108561_tabl = map reverse a112465_tabl -- Reinhard Zumkeller, Jan 03 2014 (Sage) def A108561_row(n):     @cached_function     def prec(n, k):         if k==n: return 1         if k==0: return 0         return -prec(n-1, k-1)-sum(prec(n, k+i-1) for i in (2..n-k+1))     return [(-1)^k*prec(n, k) for k in (1..n-1)]+[(-1)^(n+1)] for n in (1..12): print(A108561_row(n)) # Peter Luschny, Mar 16 2016 (GAP) Flat(List([0..13], n->List([0..n], k->Sum([0..k], i->Binomial(n, i)*(-2)^(k-i))))); # Muniru A Asiru, Feb 19 2018 CROSSREFS Cf. A007318 (a=1), A008949(a=2), A164844(a=10). Similar to the triangles A035317, A059259, A080242, A112555. Cf. A228196 Cf. A072547 (central terms). Sequence in context: A172371 A279006 A112555 * A174626 A264909 A104579 Adjacent sequences:  A108558 A108559 A108560 * A108562 A108563 A108564 KEYWORD sign,tabl AUTHOR Reinhard Zumkeller, Jun 10 2005 EXTENSIONS Definition corrected by Philippe Deléham, Dec 26 2013 STATUS approved

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Last modified August 3 11:57 EDT 2020. Contains 336198 sequences. (Running on oeis4.)