This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A126075 Triangle T(n,k), 0 <= k <= n, read by rows, defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 2*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k+1) for k >= 1. 29
 1, 2, 1, 5, 2, 1, 12, 6, 2, 1, 30, 14, 7, 2, 1, 74, 37, 16, 8, 2, 1, 185, 90, 45, 18, 9, 2, 1, 460, 230, 108, 54, 20, 10, 2, 1, 1150, 568, 284, 128, 64, 22, 11, 2, 1, 2868, 1434, 696, 348, 150, 75, 24, 12, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Riordan array (c(x^2)/(1-2xc(x^2)),xc(x^2)) where c(x)=g.f. of Catalan numbers A000108. - Philippe Deléham, Mar 18 2007 This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1. Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; ((1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007 LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened FORMULA Sum_{k=0..n} T(n,k) = A127358(n). T(n,0)=A054341(n). Sum_{k=0..n} T(n,k)*(-k+1) = 2^n. - Philippe Deléham, Mar 25 2007 From Peter Bala, Feb 20 2018: (Start) T(n,k) = Sum_{j = 0..floor((n-k)/2)} 2^(n-k-2*j)*binomial(n, j) - Sum_{j = 0..floor((n-k-2)/2)} 2^(n-k-2-2*j)*binomial(n, j), 0 <= k <= n. - Peter Bala, Feb 20 2018 The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the rational function (1 - x^2)/(1 - 2*x) * (1 + x^2)^n about 0. For example, for n = 4, (1 - x^2)/(1 - 2*x) * (1 + x^2)^4 = (30*x^4 + 14*x*3 + 7*x^2 + 2*x + 1) + O(x^5). (End) EXAMPLE Triangle begins:      1;      2,    1;      5,    2,   1;     12,    6,   2,   1;     30,   14,   7,   2,   1;     74,   37,  16,   8,   2,  1;    185,   90,  45,  18,   9,  2,  1;    460,  230, 108,  54,  20, 10,  2,  1;   1150,  568, 284, 128,  64, 22, 11,  2, 1;   2868, 1434, 696, 348, 150, 75, 24, 12, 2, 1; MAPLE A126075 := proc (n, k) add( 2^(n-k-2*j)*binomial(n, j), j = 0..floor((n-k)/2) ) - add( 2^(n-k-2-2*j)*binomial(n, j), j = 0..floor((n-k-2)/2) ) end proc: # display sequence in triangular form for n from 0 to 10 do seq(A126075(n, k), k = 0..n) end do; # Peter Bala, Feb 20 2018 MATHEMATICA T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0, T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]]; Table[T[n, k, 2, 0], {n, 0, 49}, {k, 0, n}] // Flatten  (* G. C. Greubel, Apr 21 2017 *) CROSSREFS Cf. A000108, A054341, A127358. Sequence in context: A126125 A221876 A128514 * A134032 A137151 A048494 Adjacent sequences:  A126072 A126073 A126074 * A126076 A126077 A126078 KEYWORD nonn,tabl,easy AUTHOR Philippe Deléham, Mar 02 2007 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 18 06:34 EDT 2018. Contains 313823 sequences. (Running on oeis4.)