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A103327
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Triangle T(n, k) read by rows: binomial(2n+1, 2k+1).
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7
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1, 3, 1, 5, 10, 1, 7, 35, 21, 1, 9, 84, 126, 36, 1, 11, 165, 462, 330, 55, 1, 13, 286, 1287, 1716, 715, 78, 1, 15, 455, 3003, 6435, 5005, 1365, 105, 1, 17, 680, 6188, 19448, 24310, 12376, 2380, 136, 1, 19, 969, 11628, 50388, 92378, 75582, 27132, 3876, 171
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| A subset of Pascal's triangle A007318.
Elements have the same parity as those of Pascal's triangle.
Matrix inverse is A104033. - Paul D. Hanna (pauldhanna(AT)juno.com), Feb 28 2005
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REFERENCES
| A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224.
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FORMULA
| G.f. for column k is sum{k=0..k+1, C(2(k+1), 2j)x^j)/(1-x)^(2(k+1)) - Paul Barry (pbarry(AT)wit.ie), Feb 24 2005
G.f.: A(x, y) = (1 + x*(1-y))/( (1 + x*(1-y))^2 - 4*x ). - Paul D. Hanna (pauldhanna(AT)juno.com), Feb 28 2005
Sum_{k, 0<=k<=n} T(n, k)*A000364(n-k) = A002084(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 27 2005
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EXAMPLE
| 1
3,1
5,10,1
7,35,21,1
9,84,126,36,1
11,165,462,330,55,1
13,286,1287,1716,715,78,1
15,455,3003,6435,5005,1365,105,1
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PROG
| (PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); polcoeff(polcoeff((1+X*(1-Y))/((1+X*(1-Y))^2-4*X), n, x), k, y)} (Hanna)
(Maxima) create_list(binomial(2*n+1, 2*k+1), n, 0, 12, k, 0, n); [Emanuele Munarini, Mar 11 2011]
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CROSSREFS
| Reflected version of A091042. Cf. A086645, A103328.
Cf. A104033.
Sequence in context: A146916 A146255 A122366 * A177463 A065229 A093905
Adjacent sequences: A103324 A103325 A103326 * A103328 A103329 A103330
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KEYWORD
| nonn,tabl
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AUTHOR
| Ralf Stephan, Feb 06 2005
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