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Search: id:A111418
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| A111418 |
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Right-hand side of odd-numbered rows of Pascal's triangle. |
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25
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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
(list; table; graph; listen)
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OFFSET
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0,2
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COMMENT
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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 (kolotoko(AT)wanadoo.fr), Mar 22 2007
Reversal of A122366 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007
Column k has e.g.f. exp(2x)(Bessel_I(k,2x)+Bessel_I(k+1,2x)); - Paul Barry (pbarry(AT)wit.ie), 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 (kolotoko(AT)wanadoo.fr), Sep 25 2007
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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 (kolotoko(AT)wanadoo.fr), Mar 22 2007
T(n,k)=sum{j=k..n, C(n,j)*2^(n-j)*C(j,floor((j-k)/2))}; - Paul Barry (pbarry(AT)wit.ie), Jun 06 2007
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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;
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CROSSREFS
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Cf. A000108, A113187.
Columns are : A001700, A002054, A003516, A030053, A030054, A030055, A030056.
Adjacent sequences: A111415 A111416 A111417 this_sequence A111419 A111420 A111421
Sequence in context: A107870 A078817 A091042 this_sequence A113187 A057967 A132964
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 13 2005
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