A119258
Triangle read by rows: T(n,0) = T(n,n) = 1 and for 0
1, 1, 1, 1, 3, 1, 1, 5, 7, 1, 1, 7, 17, 15, 1, 1, 9, 31, 49, 31, 1, 1, 11, 49, 111, 129, 63, 1, 1, 13, 71, 209, 351, 321, 127, 1, 1, 15, 97, 351, 769, 1023, 769, 255, 1, 1, 17, 127, 545, 1471, 2561, 2815, 1793, 511, 1, 1, 19, 161, 799, 2561, 5503, 7937, 7423, 4097, 1023, 1
Offset: 0
Examples
Triangle begins as: 1; 1, 1; 1, 3, 1; 1, 5, 7, 1; 1, 7, 17, 15, 1; 1, 9, 31, 49, 31, 1;
Links
- Reinhard Zumkeller, Rows n=0..120 of triangle, flattened
- A. Björner and V. Welker, The homology of "k-equal" manifolds and related partition lattices, Adv. Math., 110 (1995), 277-313.
- Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, Pinnacle sets of signed permutations, arXiv:2301.02628 [math.CO], 2023.
- R. M. Green, Homology representations arising from the half cube, Adv. Math., 222 (2009), 216-239.
- R. M. Green, Homology representations arising from the half cube, II, J. Combin. Theory Ser. A, 117 (2010), 1037-1048.
- R. M. Green, Homology representations arising from the half cube, II, arXiv:0812.1208 [math.RT], 2008.
- R. M. Green and Jacob T. Harper, Morse matchings on polytopes, arXiv preprint arXiv:1107.4993 [math.GT], 2011.
- OEIS Wiki, Sequence of the Day for November 3.
- K. Petras, On the Smolyak cubature error for analytic functions, Adv. Comput. Math., 12 (2000), 71-93.
- M. Shattuck and T. Waldhauser, Proofs of some binomial identities using the method of last squares, Fib. Q., 48 (2010), 290-297.
- M. Shattuck and T. Waldhauser, Proofs of some binomial identities using the method of last squares, arXiv:1107.1063 [math.CO], 2010.
- Index entries for triangles and arrays related to Pascal's triangle
Crossrefs
Programs
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Haskell
a119258 n k = a119258_tabl !! n !! k a119258_row n = a119258_tabl !! n a119258_list = concat a119258_tabl a119258_tabl = iterate (\row -> zipWith (+) ([0] ++ init row ++ [0]) $ zipWith (+) ([0] ++ row) (row ++ [0])) [1] -- Reinhard Zumkeller, Nov 15 2011
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Magma
function T(n,k) if k eq 0 or k eq n then return 1; else return 2*T(n-1,k-1) + T(n-1,k); end if; return T; end function; [T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 18 2019
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Maple
# Case m = 2 of the more general: A119258 := (n,k,m) -> (1-m)^(-n+k)-m^(k+1)*pochhammer(n-k, k+1) *hypergeom([1,n+1],[k+2],m)/(k+1)!; seq(seq(round(evalf(A119258(n,k,2))),k=0..n), n=0..10); # Peter Luschny, Jul 25 2014
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Mathematica
T[n_, k_] := Binomial[n, k] Hypergeometric2F1[-k, n-k, n-k+1, -1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 10 2017 *) -
PARI
T(n,k) = if(k==0 || k==n, 1, 2*T(n-1, k-1) + T(n-1,k) ); \\ G. C. Greubel, Nov 18 2019
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Sage
@CachedFunction def T(n, k): if (k==0 or k==n): return 1 else: return 2*T(n-1, k-1) + T(n-1, k) [[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 18 2019
Formula
T(2*n,n-1) = T(2*n-1,n) for n > 0;
T(n,0) = T(n,n) = 1;
T(n,1) = A005408(n-1) for n > 0;
T(n,2) = A056220(n-1) for n > 1;
T(n,n-4) = A027608(n-4) for n > 3;
T(n,n-3) = A055580(n-3) for n > 2;
T(n,n-2) = A000337(n-1) for n > 1;
T(n,n-1) = A000225(n) for n > 0.
T(n,k) = [k<=n]*(-1)^k*Sum_{i=0..k} (-1)^i*C(k-n,k-i)*C(n,i). - Paul Barry, Sep 28 2007
From Richard M. Green, Jul 26 2011: (Start)
T(n,k) = [k<=n] Sum_{i=n-k..n} (-1)^(n-k-i)*2^(n-i)*C(n,i).
T(n,k) = [k<=n] Sum_{i=n-k..n} C(n,i)*C(i-1,n-k-1).
G.f. for T(n,n-k): x^k/(((1-2x)^k)*(1-x)). (End)
T(n,k) = R(n,k,2) where R(n, k, m) = (1-m)^(-n+k)-m^(k+1)*Pochhammer(n-k,k+1)* hyper2F1([1,n+1], [k+2], m)/(k+1)!. - Peter Luschny, Jul 25 2014
From Peter Bala, Mar 05 2018: (Start)
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 + 2*x)^n/(1 + x) about 0. For example, for n = 4 we have (1 + 2*x)^4/(1 + x) = 1 + 7*x + 17*x^2 + 15*x^3 + x^4 + O(x^5).
Row reverse of A112857. (End)
Comments