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Search: id:A120894
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| A120894 |
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Cascadence of 1+x+x^2; a triangle, read by rows of 2n+1 terms, that retains its original form upon convolving each row with [1,1,1] and then letting excess terms spill over from each row into the initial positions of the next row such that only 2n+1 terms remain in row n for n>=0. |
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9
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| 1, 1, 1, 1, 2, 3, 2, 1, 2, 5, 7, 6, 5, 3, 2, 5, 12, 18, 18, 14, 10, 10, 7, 5, 12, 30, 48, 50, 42, 34, 27, 22, 24, 17, 12, 30, 78, 128, 140, 126, 103, 83, 73, 63, 53, 59, 42, 30, 78, 206, 346, 394, 369, 312, 259, 219, 189, 175, 154, 131, 150, 108, 78, 206, 552, 946, 1109
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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In this case, the g.f. of column 0, H(x), satisfies: H(x) = H(x*G^2)*G/x where G satisfies: G = x*(1+G+G^2), so that G/x = g.f. of Motzkin numbers (A001006). More generally, the cascadence of polyomial F(x) of degree d, F(0)=1, is a triangle with d*n+1 terms in row n where the g.f. H(x) of column 0 satisfies: H(x) = H(x*G^d)*G/x where G = x*F(G); thus G = series_reversion(x/F(x)), or, equivalently, [x^n] G = [x^n] x*F(x)^n/n for n>=1.
Further, the g.f. of the cascadence triangle for polynomial F(x) of degree d is given by: A(x,y) = ( x*H(x) - y*H(x*y^d) )/( x*F(y) - y ), where H(x) = G*H(x*G^d)/x and G = x*F(G). - Paul D. Hanna (pauldhanna(AT)juno.com), Jul 17 2006
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FORMULA
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G.f.: A(x,y) = ( x*H(x) - y*H(x*y^2) )/( x*F(y) - y ), where H(x) = G*H(x*G^2)/x, G = x*F(G), F(x)=1+x+x^2. - Paul D. Hanna (pauldhanna(AT)juno.com), Jul 17 2006
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EXAMPLE
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Triangle begins:
1;
1,1,1;
2,3,2,1,2;
5,7,6,5,3,2,5;
12,18,18,14,10,10,7,5,12;
30,48,50,42,34,27,22,24,17,12,30;
78,128,140,126,103,83,73,63,53,59,42,30,78;
206,346,394,369,312,259,219,189,175,154,131,150,108,78,206;
552,946,1109,1075,940,790,667,583,518,460,435,389,336,392,284,206,552;
Convolution of [1,1,1] with each row produces:
[1,1,1]*[1] = [1,1,1];
[1,1,1]*[1,1,1] = [1,2,3,2,1];
[1,1,1]*[2,3,2,1,2] = [2,5,7,6,5,3,2];
[1,1,1]*[5,7,6,5,3,2,5] = [5,12,18,18,14,10,10,7,5];
[1,1,1]*[12,18,18,14,10,10,7,5,12] = [12,30,48,50,42,34,27,22,24,17,12];
These convoluted rows, when concatenated, yield the sequence:
1,1,1, 1,2,3,2,1, 2,5,7,6,5,3,2, 5,12,18,18,14,10,10,7,5, ...
which equals the concatenated rows of this original triangle:
1, 1,1,1, 2,3,2,1,2, 5,7,6,5,3,2,5, 12,18,18,14,10,10,7,5,12, ...
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PROGRAM
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(PARI) {T(n, k)=if(2*n<k|k<0, 0, if(n<=1, 1, if(k==0, T(n-1, 0)+T(n-1, 1), if(k==2*n, T(n, 0), T(n-1, k-1)+T(n-1, k)+T(n-1, k+1)))))}
(PARI) /* Generated by the G.F.: */ {T(n, k)=local(A, F=1+x+x^2, d=2, G=x, H=1+x, S=ceil(log(n+1)/log(d+1))); for(i=0, n, G=x*subst(F, x, G+x*O(x^n))); for(i=0, S, H=subst(H, x, x*G^d+x*O(x^n))*G/x); A=(x*H-y*subst(H, x, x*y^d +x*O(x^n)))/(x*subst(F, x, y)-y); polcoeff(polcoeff(A, n, x), k, y)} - Paul D. Hanna (pauldhanna(AT)juno.com), Jul 17 2006
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CROSSREFS
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Cf. A120895 (column 0), A120896 (central terms), A120897 (row sums), A001006 (Motzkin numbers); variants: A092683, A092686, A120898.
Sequence in context: A105969 A114409 A049063 this_sequence A134819 A135267 A118105
Adjacent sequences: A120891 A120892 A120893 this_sequence A120895 A120896 A120897
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jul 14 2006
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