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%I #37 Apr 19 2020 08:35:49
%S 1,1,1,2,2,1,2,4,3,1,3,6,7,4,1,3,9,13,11,5,1,4,12,22,24,16,6,1,4,16,
%T 34,46,40,22,7,1,5,20,50,80,86,62,29,8,1,5,25,70,130,166,148,91,37,9,
%U 1,6,30,95,200,296,314,239,128,46,10,1
%N A128174 * A007318.
%C Row Sums = A000975: (1, 2, 5, 10, 21, 42, 85, 170, ...).
%C A007318 * A128174 = A128175.
%C From _Peter Bala_, Aug 14 2014: (Start)
%C Riordan array ( 1/((1 - x^2)*(1 - x)), x/(1 - x) ).
%C Let B_n be the set of length n nonzero binary words ending in an even number (possibly 0) of 0's. Then T(n,k) is the number of words in B_n having k 1's. An example is given below. (End)
%H G. C. Greubel, <a href="/A128176/b128176.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%H Georg Cantor, <a href="http://resolver.sub.uni-goettingen.de/purl?PPN237853094">Gesammelte Abhandlungen mathematischen und philosophischen Inhalts</a>, Part IV, 4. Mitteilungen zur Lehre vom Transfiniten, VIII Nr. 13, Springer, Berlin, 1932. See p. 434.
%F A128174 * A007318 (Pascal's triangle), as infinite lower triangular matrices.
%F From _Peter Bala_, Aug 14 2014: (Start)
%F Working with a row and column offset of 0 we have T(n,k) = Sum_{i = 0..floor(n/2)} binomial(n - 2*i,k).
%F O.g.f.: 1/( (1 - z^2)*(1 - z*(1 + x)) ) = Sum_{n >= 0} R(n,x)*z^n = 1 + (1 + x)*z + (2 + 2*x + x^2)*z^2 + ....
%F The row polynomials satisfy R(n+2,x) - R(n,x) = (1 + x)^(n+1). (End)
%F From _Hartmut F. W. Hoft_, Mar 15 2017: (Start)
%F Using offset 0, the triangle has the Pascal Triangle recursion pattern:
%F T(n, 0) = 1 + floor(n/2) and T(n, n) = 1, for n >= 0;
%F T(n, k) = T(n-1, k-1) + T(n-1, k) for n > 0 and 0 < k < n. (End)
%e First few rows of the triangle are:
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 2, 4, 3, 1;
%e 3, 6, 7, 4, 1;
%e 3, 9, 13, 11, 5, 1;
%e 4, 12, 22, 24, 16, 6, 1;
%e 4, 16, 34, 46, 40, 22, 7, 1;
%e ...
%e From _Peter Bala_, Aug 14 2014: (Start)
%e Row 4: [2,4,3,1].
%e k Binary words in B_4 with k 1's Number
%e - - - - - - - - - - - - - - - - - - - - - - - - - -
%e 1 0001, 0100 2
%e 2 0011, 0101, 1001, 1100 4
%e 3 0111, 1011, 1101 3
%e 4 1111 1
%e - - - - - - - - - - - - - - - - - - - - - - - - - -
%e The infinitesimal generator matrix begins
%e 0
%e 1 0
%e 1 2 0
%e -1 1 3 0
%e 1 -1 1 4 0
%e -1 1 -1 1 5 0
%e ...
%e Cf. A132440. (End)
%t (* Dot product of two lower triangular matrices *)
%t dotRow[r_, s_, n_] := Map[Sum[r[n, k] s[k, #], {k, #, n}]&, Range[0, n]]
%t dotTriangle[r_, s_, n_] := Map[dotRow[r, s, #]&, Range[0, n]]
%t (* The pure function in the first argument computes A128174 *)
%t a128176[r_] := dotTriangle[If[EvenQ[#1 + #2], 1, 0]&, Binomial, r]
%t TableForm[a128176[7]] (* triangle *)
%t Flatten[a128176[9]] (* data *) (* _Hartmut F. W. Hoft_, Mar 15 2017 *)
%t T[n_, n_] := 1; T[n_, 0] := 1 + Floor[n/2]; T[n_, k_] := T[n, k] = T[n - 1, k - 1] + T[n - 1, k]; Table[T[n, k], {n,0,20}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Sep 30 2017 *)
%o (PARI) for(n=0, 10, for(k=0,n, print1(sum(i=0,floor(n/2), binomial(n - 2*i,k)), ", "))) \\ _G. C. Greubel_, Sep 30 2017
%Y Cf. A000975, A128175, A007318.
%Y Cf. A035317 (mirror). [_Johannes W. Meijer_, Jul 20 2011]
%K nonn,tabl
%O 1,4
%A _Gary W. Adamson_, Feb 17 2007
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