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A119468
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Triangle read by rows: T(n,k) = Sum_{j=0..n-k} binomial(n,2j)*binomial(n-2j,k).
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8
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1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 8, 16, 12, 4, 1, 16, 40, 40, 20, 5, 1, 32, 96, 120, 80, 30, 6, 1, 64, 224, 336, 280, 140, 42, 7, 1, 128, 512, 896, 896, 560, 224, 56, 8, 1, 256, 1152, 2304, 2688, 2016, 1008, 336, 72, 9, 1, 512, 2560, 5760, 7680, 6720, 4032, 1680, 480, 90, 10, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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G.f.: (1 - x - xy)/(1 - 2x - 2x*y + 2x^2*y + x^2*y^2).
Number triangle T(n,k) = Sum_{j=0..n} binomial(n,j)*binomial(j,k)*(1+(-1)^(j-k))/2.
Define matrix: A(n,m,k) = If[m < n, 1, -1];
T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) - T(n-2,k-2) for n >= 2, T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 10 2013
E.g.f.: [(e^(2t)+1)/2] e^(tx) = e^(P.(x)t), so this is an Appell sequence with lowering operator D = d/dx and raising operator R = x + 2/(e^(-2D)+1), i.e., D P_n(x) = n P_{n-1}(x) and R p_n(x) = P_{n+1}(x) where P_n(x) = [(x+2)^n + x^n]/2. Also, (P.(x)+y)^n = P_n(x+y), umbrally. R = x + 1 + D - 2 D^3/3! + ... contains the e.g.f.(D) mod signs of A009006 and A155585 and signed, aerated A000182, the zag numbers, so the unsigned differential component 2/[e^(2D)+1] = 2 Sum_{n >= 0} Eta(-n) (-2D)^n/n!, where Eta(s) is the Dirichlet eta function, and 2 *(-2)^n Eta(-n) = (-1)^n (2^(n+1)-4^(n+1)) Zeta(-n) = (2^(n+1)-4^(n+1)) B(n+1)/(n+1) with Zeta(s), the Riemann zeta function, and B(n), the Bernoulli numbers. The polynomials PI_n(x) of A081733 are the umbral compositional inverses of this sequence, i.e., P_n(PI.(x)) = x^n = PI_n(P.(x)) under umbral composition. Aside from the signs and the main diagonals, multiplying this triangle by 2 gives the face-vectors of the hypercubes A038207. - Tom Copeland, Sep 27 2015
T(n,k) = 2^(n-k-1+0^(n-k))*binomial(n, k). - Peter Luschny, Nov 10 2017
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EXAMPLE
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Triangle begins
1;
1, 1;
2, 2, 1;
4, 6, 3, 1;
8, 16, 12, 4, 1;
16, 40, 40, 20, 5, 1;
32, 96, 120, 80, 30, 6, 1;
64, 224, 336, 280, 140, 42, 7, 1;
128, 512, 896, 896, 560, 224, 56, 8, 1;
256, 1152, 2304, 2688, 2016, 1008, 336, 72, 9, 1;
512, 2560, 5760, 7680, 6720, 4032, 1680, 480, 90, 10, 1;
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MAPLE
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A119468_row := proc(n) local s, t, k;
s := series(exp(z*x)/(1-tanh(x)), x, n+2);
t := factorial(n)*coeff(s, x, n); seq(coeff(t, z, k), k=(0..n)) end:
# Alternatively:
T := (n, k) -> 2^(n-k-1+0^(n-k))*binomial(n, k):
for n from 0 to 9 do seq(T(n, k), k=0..n) od; # Peter Luschny, Nov 10 2017
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MATHEMATICA
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A[k_] := Table[If[m < n, 1, -1], {m, k}, {n, k}]; a = Join[{{1}}, Table[(-1)^n*CoefficientList[CharacteristicPolynomial[A[n], x], x], {n, 1, 10}]]; Flatten[a] (* Roger L. Bagula and Gary W. Adamson, Jan 25 2009 *)
Table[Sum[Binomial[n, 2j]Binomial[n-2j, k], {j, 0, n-k}], {n, 0, 10}, {k, 0, n}]//Flatten (* Harvey P. Dale, Dec 14 2022 *)
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PROG
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(Sage)
R = PolynomialRing(QQ, 'x')
def p(n, x) :
return 1 if n==0 else add((-1)^n*binomial(n, k)*(x^(n-k)-1) for k in range(n))
x = R.gen()
return [abs(cf) for cf in list((p(n, x-1)-p(n, x+1))/2+x^n)]
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CROSSREFS
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A082137 read as triangle with rows reversed.
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KEYWORD
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AUTHOR
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STATUS
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approved
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