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Search: id:A086645
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| A086645 |
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Triangle read by rows: T(n; k) = Binomial(2*n; 2*k). |
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10
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| 1, 1, 1, 1, 6, 1, 1, 15, 15, 1, 1, 28, 70, 28, 1, 1, 45, 210, 210, 45, 1, 1, 66, 495, 924, 495, 66, 1, 1, 91, 1001, 3003, 3003, 1001, 91, 1, 1, 120, 1820, 8008, 12870, 8008, 1820, 120, 1, 1, 153, 3060, 18564, 43758, 43758, 18564, 3060, 153, 1, 1, 190, 4845, 38760
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Elements have the same parity as those of Pascal's triangle.
Coefficients of polynomials 1/2[(1+x^(1/2))^(2n) + (1-x^(1/2))^(2n)].
Number of compositions of 2n having k parts greater than 1; example : T(3, 2) = 15 because we have 4+2, 2+4, 3+2+1, 3+1+2, 2+3+1, 2+1+3, 1+3+2, 1+2+3, 2+2+1+1, 2+1+2+1, 2+1+1+2, 1+2+2+1, 1+2+1+2, 1+1+2+2, 3+3 . - Philippe DELEHAM, May 18 2005
Number of binary words of length 2n - 1 having k runs of consecutive 1's; example : T(3,2) = 15 because we have 00101, 01001, 01010, 01011, 01101, 10001, 10010, 10011, 10100, 10110, 10111, 11001, 11010, 11011, 11101 . - Philippe DELEHAM, May 18 2005
Let M_n be the n X n matrix M_n(i, j) = T(i, j-1); then for n>0 det(M_n) = A000364(n), Euler numbers; example : det([1, 1, 0, 0; 1, 6, 1, 0; 1, 15, 15, 1; 1, 28, 70, 28 ]) = 1385 = A000364(4) . -Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 04 2005
Equals ConvOffsStoT transform of the hexagonal numbers, A000384: (1, 6, 15, 28, 45,...); e.g. ConvOffs transform of (1, 6, 15, 28) = (1, 28, 70, 28, 1). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 22 2008
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REFERENCES
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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
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T(n; k) = (2*n)!/((2*(n-k))!*(2*k)!) row sums = A081294. COLUMNS :A000012, A000384
Sum_{k>=0} T(n, k)*A000364(k) = A000795(n) = (2^n)*A005647(n).
Sum_{k>=0} T(n, k)*2^k = A001541(n). Sum_{k>=0} T(n, k)*3^k = 2^n*A001075(n). Sum_{k>=0} T(n, k)*4^k = A083884(n). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 29 2004
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MAPLE
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T:=(n, k)->binomial(2*n, 2*k): seq(seq(T(n, k), k=0..n), n=0..12);
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PROGRAM
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(PARI) T(n, k)=binomial(2*n, 2*k)
(PARI) T(n, k)=sum(i=0, min(k, n-k), 4^i*C(n, 2*i)*C(n-2*i, k-i)) /* Michael Somos May 26 2005 */
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CROSSREFS
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Cf. A000012 A000384 A081294.
Cf. A000384.
Sequence in context: A131063 A081579 A082105 this_sequence A109001 A119726 A103999
Adjacent sequences: A086642 A086643 A086644 this_sequence A086646 A086647 A086648
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 26 2003
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EXTENSIONS
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More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 24 2004
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