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Search: id:A098158
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| A098158 |
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Triangle T(n,k) with diagonals T(n,n-k)=binomial(n,2k). |
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73
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| 1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 1, 6, 1, 0, 0, 0, 5, 10, 1, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0, 0, 1, 66, 495, 924
(list; table; graph; listen)
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OFFSET
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0,9
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COMMENT
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Row sums are A011782. Inverse is A065547.
Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 29 2006
Sum of entries in column k is A001519(k+1)(the odd indexed Fibonacci numbers). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2008]
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LINKS
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D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.
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FORMULA
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Triangle T(n, k)=binomial(n, 2(n-k))
Column k is generated by the polynomial sum{j=0..floor(k/2), C(k, 2j)x^(k-j)}. - Paul Barry (pbarry(AT)wit.ie), Jan 22 2005
G.f.: (1-x*y)/((1-x*y)^2 - x^2*y). - Paul D. Hanna (pauldhanna(AT)juno.com), Feb 25 2005
Sum_{k, 0<=k<=n}x^k*T(n,k)= A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 04 2006, Oct 15 2008, Oct 19 2008
T(n,k)=T(n-1,k-1)+Sum_{i, 0<=i<=k-1}T(n-2-i,k-1-i) ; T(0,0)=1 ; T(n,k)=0 if n<0, if k<0, if n<k . E.g. T(8,5)=T(7,4)+T(6,4)+T(5,3)+T(4,2)+T(3,1)+T(2,0)=7+15+5+1+0+0=28 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 04 2006
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 24 2007
Sum_{k, 0<=k<=n}T(n,k)*(-x)^(n-k)= A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x = 0,1,2,3,4,5 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 14 2008]
T(n,k)=A085478(k,n-k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2008]
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EXAMPLE
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Rows begin {1}, {0,1}, {0,1,1}, {0,0,3,1}, {0,0,1,6,1},...
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PROGRAM
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(PARI) {T(n, k)=polcoeff(polcoeff((1-x*y)/((1-x*y)^2-x^2*y)+x*O(x^n), n, x)+y*O(y^k), k, y)} (Hanna)
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CROSSREFS
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Cf. A098157.
Cf. A119900 [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2008]
Sequence in context: A062734 A117389 A122083 this_sequence A110319 A036872 A036871
Adjacent sequences: A098155 A098156 A098157 this_sequence A098159 A098160 A098161
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), Aug 29 2004
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EXTENSIONS
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Corrected first formula. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 18 2008
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