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%I #61 Jun 12 2019 10:56:08
%S 1,3,1,10,5,1,35,21,7,1,126,84,36,9,1,462,330,165,55,11,1,1716,1287,
%T 715,286,78,13,1,6435,5005,3003,1365,455,105,15,1,24310,19448,12376,
%U 6188,2380,680,136,17,1,92378,75582,50388
%N Right-hand side of odd-numbered rows of Pascal's triangle.
%C Riordan array (c(x)/sqrt(1-4*x),x*c(x)^2) where c(x) is g.f. of A000108. Unsigned version of A113187. Diagonal sums are A014301(n+1).
%C Triangle T(n,k),0<=k<=n, read by rows defined by :T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=3*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+2*T(n-1,k)+T(n-1,k+1) for k>=1. - _Philippe Deléham_, Mar 22 2007
%C Reversal of A122366. - _Philippe Deléham_, Mar 22 2007
%C Column k has e.g.f. exp(2x)(Bessel_I(k,2x)+Bessel_I(k+1,2x)). - _Paul Barry_, Jun 06 2007
%C This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; ((1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - _Philippe Deléham_, Sep 25 2007
%C Diagonal sums are A014301(n+1). - _Paul Barry_, Mar 08 2011
%C This triangle T(n,k) appears in the expansion of odd powers of Fibonacci numbers F=A000045 in terms of F-numbers with multiples of odd numbers as indices. See the Ozeki reference, p. 108, Lemma 2. The formula is: F_l^(2*n+1) = sum(T(n,k)*(-1)^((n-k)*(l+1))* F_{(2*k+1)*l}, k=0..n)/5^n, n >= 0, l >= 0. - _Wolfdieter Lang_, Aug 24 2012
%C Central terms give A052203. - _Reinhard Zumkeller_, Mar 14 2014
%C This triangle appears in the expansion of (4*x)^n in terms of the polynomials Todd(n, x):= T(2*n+1, sqrt(x))/sqrt(x) = sum(A084930(n,m)*x^m), n >= 0. This follows from the inversion of the lower triangular Riordan matrix built from A084930 and comparing the g.f. of the row polynomials. - _Wolfdieter Lang_, Aug 05 2014
%C From _Wolfdieter Lang_, Aug 15 2014: (Start)
%C This triangle is the inverse of the signed Riordan triangle (-1)^(n-m)*A111125(n,m).
%C This triangle T(n,k) appears in the expansion of x^n in terms of the polynomials todd(k, x):= T(2*k+1, sqrt(x)/2)/(sqrt(x)/2) = S(k, x-2) - S(k-1, x-2) with the row polynomials T and S for the triangles A053120 and A049310, respectively: x^n = sum(T(n,k)*todd(k, x), k=0..n). Compare this with the preceding comment.
%C The A- and Z-sequences for this Riordan triangle are [1, 2, 1, repeated 0] and [3, 1, repeated 0]. For A- and Z-sequences for Riordan triangles see the W. Lang link under A006232. This corresponds to the recurrences given in the Philippe Deléham, Mar 22 2007 comment above. (End)
%H Reinhard Zumkeller, <a href="/A111418/b111418.txt">Rows n = 0..125 of triangle, flattened</a>
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry5/barry223.html">On the Hurwitz Transform of Sequences</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7.
%H E. Deutsch, L. Ferrari and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.aam.2004.05.002">Production Matrices</a>, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
%H Asamoah Nkwanta and Earl R. Barnes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Nkwanta/nkwanta2.html">Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind</a>, Journal of Integer Sequences, Article 12.3.3, 2012.
%H A. Nkwanta, A. Tefera, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Nkwanta/nkwanta4.html">Curious Relations and Identities Involving the Catalan Generating Function and Numbers</a>, Journal of Integer Sequences, 16 (2013), #13.9.5.
%H K. Ozeki, <a href="http://www.fq.math.ca/Papers1/46_47-2/Ozeki.pdf">On Melham's sum</a>, The Fibonacci Quart. 46/47 (2008/2009), no. 2, 107-110.
%H Sun, Yidong; Ma, Luping <a href="https://doi.org/10.1016/j.ejc.2014.01.004">Minors of a class of Riordan arrays related to weighted partial Motzkin paths</a>. Eur. J. Comb. 39, 157-169 (2014), Table 2.2.
%F T(n, k) = C(2*n+1, n-k).
%F Sum_{k=0..n} T(n, k) = 4^n.
%F Sum_{k, 0<=k<=n}(-1)^k *T(n,k) = binomial(2*n,n) = A000984(n). - _Philippe Deléham_, Mar 22 2007
%F T(n,k) = sum{j=k..n, C(n,j)*2^(n-j)*C(j,floor((j-k)/2))}. - _Paul Barry_, Jun 06 2007
%F Sum_{k, k>=0} T(m,k)*T(n,k) = T(m+n,0)= A001700(m+n). - _Philippe Deléham_, Nov 22 2009
%F G.f. row polynomials: ((1+x) - (1-x)/sqrt(1-4*z))/(2*(x - (1+x)^2*z))
%F (see the Riordan property mentioned in a comment above). - _Wolfdieter Lang_, Aug 05 2014
%e From _Wolfdieter Lang_, Aug 05 2014: (Start)
%e The triangle T(n,k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 10 ...
%e 0: 1
%e 1: 3 1
%e 2: 10 5 1
%e 3: 35 21 7 1
%e 4: 126 84 36 9 1
%e 5: 462 330 165 55 11 1
%e 6: 1716 1287 715 286 78 13 1
%e 7: 6435 5005 3003 1365 455 105 15 1
%e 8: 24310 19448 12376 6188 2380 680 136 17 1
%e 9: 92378 75582 50388 27132 11628 3876 969 171 19 1
%e 10: 352716 293930 203490 116280 54264 20349 5985 1330 210 21 1
%e ...
%e Expansion examples (for the Todd polynomials see A084930 and a comment above):
%e (4*x)^2 = 10*Todd(n, 0) + 5*Todd(n, 1) + 1*Todd(n, 2) = 10*1 + 5*(-3 + 4*x) + 1*(5 - 20*x + 16*x^2).
%e (4*x)^3 = 35*1 + 21*(-3 + 4*x) + 7*(5 - 20*x + 16*x^2) + (-7 + 56*x - 112*x^2 +64*x^3)*1. (End)
%e ---------------------------------------------------------------------
%e Production matrix is
%e 3, 1,
%e 1, 2, 1,
%e 0, 1, 2, 1,
%e 0, 0, 1, 2, 1,
%e 0, 0, 0, 1, 2, 1,
%e 0, 0, 0, 0, 1, 2, 1,
%e 0, 0, 0, 0, 0, 1, 2, 1,
%e 0, 0, 0, 0, 0, 0, 1, 2, 1,
%e 0, 0, 0, 0, 0, 0, 0, 1, 2, 1
%e - _Paul Barry_, Mar 08 2011
%e Application to odd powers of Fibonacci numbers F, row n=2:
%e F_l^5 = (10*(-1)^(2*(l+1))*F_l + 5*(-1)^(1*(l+1))*F_{3*l} + 1*F_{5*l})/5^2, l >= 0. - _Wolfdieter Lang_, Aug 24 2012
%t Table[Binomial[2*n+1, n-k], {n,0,10}, {k,0,n}] (* _G. C. Greubel_, May 22 2017 *)
%t T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0,
%t T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
%t Table[T[n, k, 3, 2], {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, May 22 2017 *)
%o (Haskell)
%o a111418 n k = a111418_tabl !! n !! k
%o a111418_row n = a111418_tabl !! n
%o a111418_tabl = map reverse a122366_tabl
%o -- _Reinhard Zumkeller_, Mar 14 2014
%Y Cf. A000108, A113187.
%Y Columns are : A001700, A002054, A003516, A030053, A030054, A030055, A030056.
%K easy,nonn,tabl
%O 0,2
%A _Philippe Deléham_, Nov 13 2005
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