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A111418
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Right-hand side of odd-numbered rows of Pascal's triangle.
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29
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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
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OFFSET
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0,2
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COMMENTS
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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 DELEHAM, Mar 22 2007
Reversal of A122366 . - Philippe DELEHAM, 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 DELEHAM, Sep 25 2007
Diagonal sums are A014301(n+1). [Paul Barry, Mar 8 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
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REFERENCES
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E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in 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. - From N. J. A. Sloane, Sep 16 2012
K. Ozeki, On Melham's sum, The Fibonacci Quart. 46/47 (2008/2009), no. 2, 107-110.
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LINKS
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Table of n, a(n) for n=0..47.
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FORMULA
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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 DELEHAM, 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). [From Philippe DELEHAM, Nov 22 2009]
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EXAMPLE
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Triangle begins:
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, 27132, 11628, 3876, 969, 171, 19, 1;
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 8 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
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CROSSREFS
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Cf. A000108, A113187.
Columns are : A001700, A002054, A003516, A030053, A030054, A030055, A030056.
Sequence in context: A107870 A078817 A091042 * A113187 A057967 A132964
Adjacent sequences: A111415 A111416 A111417 * A111419 A111420 A111421
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Philippe DELEHAM, Nov 13 2005
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STATUS
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approved
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