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Search: id:A029653
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| A029653 |
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Numbers in (2,1)-Pascal triangle (by row). |
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+0
44
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| 1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 7, 9, 5, 1, 2, 9, 16, 14, 6, 1, 2, 11, 25, 30, 20, 7, 1, 2, 13, 36, 55, 50, 27, 8, 1, 2, 15, 49, 91, 105, 77, 35, 9, 1, 2, 17, 64, 140, 196, 182, 112, 44, 10, 1, 2, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 2, 21, 100, 285, 540, 714, 672, 450, 210
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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Reverse of A029635. Row sums are A003945. Diagonal sums are Fib(n+2)=sum{k=0..floor(n/2), (2n-3k)C(n-k,n-2k)/(n-k)}. - Paul Barry (pbarry(AT)wit.ie), Jan 30 2005
Riordan array ((1+x)/(1-x), x/(1-x)). The signed triangle (-1)^(n-k)T(n,k) or ((1-x)/(1+x), x/(1+x)) is the inverse of A055248. Row sums are A003945. Diagonal sums are F(n+2). - Paul Barry (pbarry(AT)wit.ie), Feb 03 2005
Row sums = A003945: (1, 3, 6, 12, 24, 48, 96...) = (1, 3, 7, 15, 31, 63, 127...) - (0, 0, 1, 3, 7, 15, 31,...); where (1, 3, 7, 15,...) = A000225. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 22 2007
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REFERENCES
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B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 39.
H. Hosoya, Pascal's triangle, non-adjacent numbers and D-dimensional atomic orbitals, J. Math. Chemistry, vol. 23, 1998, 169-178.
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FORMULA
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T(n, k) = C(n-2, k-1)+C(n-2, k)+C(n-1, k-1)+C(n-1, k).
G.f.: (1+x+y+xy)/(1-y-xy). - R. Stephan, May 17 2004
T(n, k)=(2n-k)*binomial(n, n-k)/n, n, k>0; - Paul Barry (pbarry(AT)wit.ie), Jan 30 2005
Sum_{0<=k<=n} T(n, k)*x^k are A003945-A003954 for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 10 2005
T(n, k) = C(n-1, k) + C(n, k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 10 2005
Equals A097806 * A007318, i.e. the pairwise operator * Pascal's Triangle as infinite lower triangular matrices. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 22 2007
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EXAMPLE
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1; 2 1; 2 3 1; 2 5 4 1; 2 7 9 5 1; ...
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CROSSREFS
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(d, 1) Pascal triangles for d=3..10: A093560-5, A093644-5.
Cf. A003945.
Sequence in context: A134628 A064882 A065158 this_sequence A067763 A087730 A126247
Adjacent sequences: A029650 A029651 A029652 this_sequence A029654 A029655 A029656
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KEYWORD
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nonn,tabl
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AUTHOR
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Mohammad K. Azarian (ma3(AT)evansville.edu)
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu)
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