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A091173
Triangle, read by rows, where the n-th row lists the coefficients of the polynomial of degree n, with root -1, that generates the n-th diagonal of this sequence.
3
1, 1, 1, 1, 2, 1, 2, 4, 3, 1, 4, 10, 9, 4, 1, 10, 28, 30, 16, 5, 1, 30, 90, 108, 68, 25, 6, 1, 106, 328, 426, 304, 130, 36, 7, 1, 420, 1338, 1842, 1444, 700, 222, 49, 8, 1, 1818, 6024, 8706, 7320, 3930, 1404, 350, 64, 9, 1, 8530, 29626, 44736, 39700, 23110, 9150, 2548
OFFSET
0,5
COMMENTS
The leftmost column (A091174) is determined by the condition that the root of each row polynomial is -1. The next column is T(n,1)=A091175(n+1) (n>=0).
LINKS
FORMULA
T(n+k, k) = Sum_{j=0..n} T(n, j) * k^j, with T(0,0)=1, T(0,n)=1 and T(n,0) = -Sum_{j=1..n} T(n, j) * (-1)^j.
EXAMPLE
For n=3, k=2, T(n+k,k) = T(5,2) = 30 = (2) + (4)2 + (3)2^2 + (1)2^3.
For n=4, k=3, T(n+k,k) = T(7,3) = 304 = (4) + (10)3 + (9)3^2 + (4)3^3 + (1)3^4.
Rows begin with n=0:
1;
1, 1;
1, 2, 1;
2, 4, 3, 1;
4, 10, 9, 4, 1;
10, 28, 30, 16, 5, 1;
30, 90, 108, 68, 25, 6, 1;
106, 328, 426, 304, 130, 36, 7, 1;
420, 1338, 1842, 1444, 700, 222, 49, 8, 1;
1818, 6024, 8706, 7320, 3930, 1404, 350, 64, 9, 1;
8530, 29626, 44736, 39700, 23110, 9150, 2548, 520, 81, 10, 1;
43430, 158012, 248466, 230424, 142890, 61680, 18970, 4288, 738, 100, 11, 1;
240208, 909010, 1483398, 1429236, 931500, 431646, 144858, 35976, 6804, 1010, 121, 12, 1; ...
MATHEMATICA
T[0, _] = 1; T[n_, 0] := T[n, 0] = -Sum[T[n, j]*(-1)^j, {j, 1, n}]; T[n_, k_] := T[n, k] = Sum[T[n-k, j]*k^j, {j, 0, n-k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 21 2015 *)
PROG
(PARI) {T(n, k)=if(n==k, 1, if(n>k&k>0, sum(j=0, n-k, T(n-k, j)*k^j), if(k==0, -sum(j=1, n, T(n, j)*(-1)^j))))}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))
CROSSREFS
Sequence in context: A160001 A339549 A179750 * A101897 A208058 A078142
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Dec 25 2003
STATUS
approved