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A153764 Triangle T(n,k), 0<=k<=n, read by rows, given by [1,0,-1,0,0,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. 1
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 3, 1, 1, 0, 1, 3, 3, 4, 1, 1, 0, 1, 3, 6, 4, 5, 1, 1, 0, 1, 4, 6, 10, 5, 6, 1, 1, 0, 1, 4, 10, 10, 15, 6, 7, 1, 1, 0, 1, 5, 10, 20, 15, 21, 7, 8, 1, 1, 0, 1, 5, 15, 20, 35, 21, 28, 8, 9, 1, 1, 0, 1, 6, 15, 35, 35, 56, 28, 36, 9, 10, 1, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,12

COMMENTS

A130595*A153342 as infinite lower triangular matrices . Reflected version of A103631 . Another version of A046854 . Row sums are Fibonacci numbers (A000045) .

A055830*A130595 as infinite lower triangular matrices.

LINKS

G. C. Greubel, Table of n, a(n) for the first 45 rows

FORMULA

T(n,k) = binomial(floor((n+k-1)/2),k).

Sum_{k, 0<=k<=n} T(n,k)*x^k = A122335(n-1), A039834(n-2), A000012(n), A000045(n+1), A001333(n), A003688(n), A015448(n), A015449(n), A015451(n), A015453(n), A015454(n), A015455(n), A015456(n), A015457(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively. - Philippe Deléham, Dec 17 2011

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A152163(n), A000007(n), A000045(n+1), A026597(n), A122994(n+1), A158608(n), A122995(n+1), A158797(n), A122996(n+1), A158798(n), A158609(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Dec 17 2011

G.f.: (1+(1-y)*x)/(1-y*x-x^2). - Philippe Deléham, Dec 17 2011

T(n,k) = T(n-1,k-1) + T(n-2,k), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 09 2013

EXAMPLE

Triangle begins :

1 ;

1, 0 ;

1, 1, 0 ;

1, 1, 1, 0 ;

1, 2, 1, 1, 0 ;

1, 2, 3, 1, 1, 0 ;

1, 3, 3, 4, 1, 1, 0 ; ...

MATHEMATICA

Table[Binomial[Floor[(n + k - 1)/2], k], {n, 0, 45}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 27 2016 *)

PROG

(MAGMA) / As triangle / [[Binomial(Floor((n+k-1)/2), k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 28 2016

CROSSREFS

Cf. A046854, A103631.

Sequence in context: A004602 A247418 A229899 * A294509 A059571 A027052

Adjacent sequences:  A153761 A153762 A153763 * A153765 A153766 A153767

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Jan 01 2009

STATUS

approved

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Last modified October 20 02:07 EDT 2019. Contains 328244 sequences. (Running on oeis4.)