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 A111418 Right-hand side of odd-numbered rows of Pascal's triangle. 31
 1, 3, 1, 10, 5, 1, 35, 21, 7, 1, 126, 84, 36, 9, 1, 462, 330, 165, 55, 11, 1, 1716, 1287, 715, 286, 78, 13, 1, 6435, 5005, 3003, 1365, 455, 105, 15, 1, 24310, 19448, 12376, 6188, 2380, 680, 136, 17, 1, 92378, 75582, 50388 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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). 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 Reversal of A122366. - Philippe Deléham, Mar 22 2007 Column k has e.g.f. exp(2x)(Bessel_I(k,2x)+Bessel_I(k+1,2x)). - Paul Barry, Jun 06 2007 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 Diagonal sums are A014301(n+1). - Paul Barry, Mar 08 2011 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 Central terms give A052203. - Reinhard Zumkeller, Mar 14 2014 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 From Wolfdieter Lang, Aug 15 2014: (Start) This triangle is the inverse of the signed Riordan triangle (-1)^(n-m)*A111125(n,m). 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. 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) LINKS Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened Paul Barry, On the Hurwitz Transform of Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.8.7. E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122. Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Article 12.3.3, 2012. A. Nkwanta, A. Tefera, Curious Relations and Identities Involving the Catalan Generating Function and Numbers, Journal of Integer Sequences, 16 (2013), #13.9.5. K. Ozeki, On Melham's sum, The Fibonacci Quart. 46/47 (2008/2009), no. 2, 107-110. Sun, Yidong; Ma, Luping Minors of a class of Riordan arrays related to weighted partial Motzkin paths.  Eur. J. Comb. 39, 157-169 (2014), Table 2.2. FORMULA T(n, k) = C(2*n+1, n-k). Sum_{k=0..n} T(n, k) = 4^n. Sum_{k, 0<=k<=n}(-1)^k *T(n,k) = binomial(2*n,n) = A000984(n). - Philippe Deléham, Mar 22 2007 T(n,k) = sum{j=k..n, C(n,j)*2^(n-j)*C(j,floor((j-k)/2))}. - Paul Barry, Jun 06 2007 Sum_{k, k>=0} T(m,k)*T(n,k) = T(m+n,0)= A001700(m+n). - Philippe Deléham, Nov 22 2009 G.f. row polynomials: ((1+x) - (1-x)/sqrt(1-4*z))/(2*(x - (1+x)^2*z)) (see the Riordan property mentioned in a comment above). - Wolfdieter Lang, Aug 05 2014 EXAMPLE From Wolfdieter Lang, Aug 05 2014: (Start) The triangle T(n,k) begins: n\k      0      1      2      3     4     5    6    7   8  9  10 ... 0:       1 1:       3      1 2:      10      5      1 3:      35     21      7      1 4:     126     84     36      9     1 5:     462    330    165     55    11     1 6:    1716   1287    715    286    78    13    1 7:    6435   5005   3003   1365   455   105   15    1 8:   24310  19448  12376   6188  2380   680  136   17   1 9:   92378  75582  50388  27132 11628  3876  969  171  19  1 10: 352716 293930 203490 116280 54264 20349 5985 1330 210 21   1 ... Expansion examples (for the Todd polynomials see A084930 and a comment above): (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). (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) --------------------------------------------------------------------- Production matrix is 3, 1, 1, 2, 1, 0, 1, 2, 1, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1 - Paul Barry, Mar 08 2011 Application to odd powers of Fibonacci numbers F, row n=2: 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 MATHEMATICA Table[Binomial[2*n+1, n-k], {n, 0, 10}, {k, 0, n}] (* G. C. Greubel, May 22 2017 *) 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[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]]; Table[T[n, k, 3, 2], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *) PROG (Haskell) a111418 n k = a111418_tabl !! n !! k a111418_row n = a111418_tabl !! n a111418_tabl = map reverse a122366_tabl -- Reinhard Zumkeller, Mar 14 2014 CROSSREFS Cf. A000108, A113187. Columns are : A001700, A002054, A003516, A030053, A030054, A030055, A030056. Sequence in context: A078817 A316193 A091042 * A113187 A340554 A057967 Adjacent sequences:  A111415 A111416 A111417 * A111419 A111420 A111421 KEYWORD easy,nonn,tabl AUTHOR Philippe Deléham, Nov 13 2005 STATUS approved

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Last modified May 16 21:28 EDT 2021. Contains 343951 sequences. (Running on oeis4.)