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A097710
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Lower triangular matrix T, read by rows, such that row (n) is formed from the sums of adjacent terms in row (n-1) of the matrix square T^2, with T(0,0)=1.
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11
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1, 1, 1, 2, 3, 1, 7, 13, 7, 1, 41, 88, 61, 15, 1, 397, 951, 781, 257, 31, 1, 6377, 16691, 15566, 6231, 1041, 63, 1, 171886, 484490, 500057, 231721, 48303, 4161, 127, 1, 7892642, 23701698, 26604323, 13843968, 3406505, 374127, 16577, 255, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Column 0 forms the number of tournament sequences (A008934). Column 1 forms A097711. Row sums form A093657. This triangle has the same row sums and first column terms as in rows 2^n, for n>=0, of triangle A093654.
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FORMULA
| T(n, k) = T^2(n-1, k-1) + T^2(n-1, k) for n>=1 and k>1; T(n, 1) = T^2(n-1, 1).
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EXAMPLE
| Rows of T begin:
[1],
[1,1],
[2,3,1],
[7,13,7,1],
[41,88,61,15,1],
[397,951,781,257,31,1],
[6377,16691,15566,6231,1041,63,1],
[171886,484490,500057,231721,48303,4161,127,1],
[7892642,23701698,26604323,13843968,3406505,374127,16577,255,1],...
Rows of T^2 begin:
[1],
[2,1],
[7,6,1],
[41,47,14,1],
[397,554,227,30,1],
[6377,10314,5252,979,62,1],
[171886,312604,187453,44268,4035,126,1],
[7892642,15809056,10795267,3048701,357804,16323,254,1],
[627340987,1362986823,1035658531,337316292,47984869,2853660,65539,510,1],...
The sums of adjacent terms in row (n) of T^2 forms row (n+1) of T:
T(5,0) = T^2(4,0) = 397;
T(5,1) = T^2(4,0) + T^2(4,1) = 397 + 554 = 951;
T(5,2) = T^2(4,1) + T^2(4,2) = 554 + 227 = 781.
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PROG
| (PARI) {T(n, k)=if(n<0|k>n, 0, if(n==k, 1, if(k==0, sum(j=0, n-1, T(n-1, j)*T(j, 0)), sum(j=0, n-1, T(n-1, j)*T(j, k-1))+sum(j=0, n-1, T(n-1, j)*T(j, k)); )))}
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CROSSREFS
| Cf. A008934, A097711, A093657, A093654.
Sequence in context: A103364 A104027 A192363 * A171024 A109198 A081320
Adjacent sequences: A097707 A097708 A097709 * A097711 A097712 A097713
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Aug 22 2004
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