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A029653
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Numbers in (2,1)-Pascal triangle (by row).
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51
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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
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OFFSET
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0,2
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COMMENTS
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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, 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, 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, Apr 22 2007
Triangle T(n,k), read by rows, given by (2,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - From Philippe Deléham, Nov 17 2011
A029653 is jointly generated with A208510 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+x*v(n-1)x and v(n,x)=u(n-1,x)+x*v(n-1,x)+1. See the Mathematica section. [From Clark Kimberling, Feb 28 2012]
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REFERENCES
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Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
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.
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550, 2013. - From N. J. A. Sloane, Feb 13 2013
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LINKS
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Table of n, a(n) for n=0..69.
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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). - Ralf Stephan, May 17 2004
T(n, k)=(2n-k)*binomial(n, n-k)/n, n, k>0; - Paul Barry, 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, Jul 10 2005
T(n, k) = C(n-1, k) + C(n, k) . - Philippe DELEHAM, Jul 10 2005
Equals A097806 * A007318, i.e. the pairwise operator * Pascal's Triangle as infinite lower triangular matrices. - Gary W. Adamson, Apr 22 2007
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EXAMPLE
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Triangle begins:
1
2, 1
2, 3, 1
2, 5, 4, 1
2, 7, 9, 5, 1 ...
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 16;
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A208510 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A029653 *)
(* Clark Kimberling, Feb 28 2012 *)
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CROSSREFS
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(d, 1) Pascal triangles for d=3..10: A093560-5, A093644-5.
Cf. A003945, A208510.
Sequence in context: A065158 A181842 A209564 * A067763 A087730 A126247
Adjacent sequences: A029650 A029651 A029652 * A029654 A029655 A029656
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KEYWORD
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nonn,tabl,changed
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AUTHOR
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Mohammad K. Azarian
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EXTENSIONS
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More terms from James A. Sellers
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STATUS
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approved
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