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Search: id:A065941
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| A065941 |
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Triangle T[n,k] := binomial(n-floor((k+1)/2),floor(k/2)) |
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41
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| 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 3, 3, 1, 1, 1, 5, 4, 6, 3, 1, 1, 1, 6, 5, 10, 6, 4, 1, 1, 1, 7, 6, 15, 10, 10, 4, 1, 1, 1, 8, 7, 21, 15, 20, 10, 5, 1, 1, 1, 9, 8, 28, 21, 35, 20, 15, 5, 1, 1, 1, 10, 9, 36, 28, 56, 35, 35, 15, 6, 1, 1, 1, 11, 10, 45, 36, 84, 56, 70, 35, 21
(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 give the Fibonacci sequence. So do the alternating row sums.
Triangle of coefficients of polynomials defined by p(0,x)=p(1,x)=1, p(n+2,x)=x*p(n+1,x)+p(n,x) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 08 2005
T(n,k) = A108299(n,k)*A087960(k) = abs(A108299(n,k)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 01 2005
Another version of triangle in A103631 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 01 2009]
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REFERENCES
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Henry W. Gould, "A Variant of Pascal's Triangle", The Fibonacci Quarterly,3;4 Dec. 1965, pp. 257-271.
Jay Kappraff, "Beyond Measure, A Guided Tour Through Nature, Myth and Number", World Scientific, 2002; p. 490.
Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001 (Chapter 14)
Peter Steinbach, "Golden Fields: A Case for the Heptagon", Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
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LINKS
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E. Munarini and N. Z. Salvi, Binary strings without zigzags
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FORMULA
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As a square array read by antidiagonals, this is given by T1(n, k) = binomial(floor(n/2) + k, k) - Paul Barry (pbarry(AT)wit.ie), Mar 11 2003
Triangle is a reflection of that in A066170 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 16 2004
Recurrences: T(k, 0) = 1, T(k, n) = T(k-1, n) + T(k-2, n-2), or T(k, n) = T(k-1, n) + T(k-1, n-1) if n even, T(k-1, n-1) if n odd. - Ralf Stephan, May 17 2004
G.f.: sum[n, sum[k, T(k, n)x^ky^n]] = (1+xy)/(1-y-x^2y^2). sum[n>=0, T(k, n)y^n] = y^k/(1-y)^[k/2]. - Ralf Stephan, May 17 2004
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EXAMPLE
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1; 1,1; 1,1,1; 1,1,2,1; 1,1,3,2,1; ...
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CROSSREFS
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A065942 (central stalk sequence), A000045 (row sums)
Cf. A066170, A006356, A006357, A084534.
Reflected version of A046854.
Sequence in context: A152157 A039961 A108299 this_sequence A123320 A054123 A119269
Adjacent sequences: A065938 A065939 A065940 this_sequence A065942 A065943 A065944
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KEYWORD
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nonn,tabl
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AUTHOR
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Len Smiley (smiley(AT)math.uaa.alaska.edu), Nov 29 2001
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