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.

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

Paul D. Hanna, Table of n, a(n) for n = 0..1034

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

Cf. A091174, A091175.

Sequence in context: A160001 A339549 A179750 * A378145 A101897 A208058

Adjacent sequences: A091170 A091171 A091172 * A091174 A091175 A091176

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 25 2003

Status

approved