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A105615 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT(column 0 of T^((2*p-1)/2) = (2*p-1)*(column p of T), or [T^((2*p-1)/2)](m,0) = (2*p-1)*T(p+m,p+1) for all m>=1 and p>=0. 17
1, 2, 1, 10, 4, 1, 74, 26, 6, 1, 706, 226, 50, 8, 1, 8162, 2426, 522, 82, 10, 1, 110410, 30826, 6498, 1010, 122, 12, 1, 1708394, 451586, 93666, 14458, 1738, 170, 14, 1, 29752066, 7489426, 1532970, 235466, 28226, 2754, 226, 16, 1, 576037442 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Column 0 is A000698 (related to double factorials), offset 1. Column 1 is A105616 (column 0 of T^(1/2), offset 1). The matrix logarithm divided by 2 yields the integer triangle A105629.

Compare with triangular matrix A107717, which satisfies: SHIFT_LEFT(column 0 of A107717^((3*k-1)/3)) = (3*k-1)*(column k of A107717).

LINKS

Table of n, a(n) for n=0..45.

FORMULA

T(n, k) = 2*(k+1)*T(n, k+1) + Sum_{j=1..n-k-1} T(j, 0)*T(n, j+k+1) for n>k>=0, with T(n, n) = 1 for n>=0. T(n, 0) = A000698(n+1) for n>=0.

EXAMPLE

SHIFT_LEFT(column 0 of T^(-1/2)) = -1*(column 0 of T);

SHIFT_LEFT(column 0 of T^(1/2)) = 1*(column 1 of T);

SHIFT_LEFT(column 0 of T^(3/2)) = 3*(column 2 of T);

SHIFT_LEFT(column 0 of T^(5/2)) = 5*(column 3 of T).

Triangle begins:

1;

2,1;

10,4,1;

74,26,6,1;

706,226,50,8,1;

8162,2426,522,82,10,1;

110410,30826,6498,1010,122,12,1;

1708394,451586,93666,14458,1738,170,14,1;

29752066,7489426,1532970,235466,28226,2754,226,16,1; ...

Matrix square-root T^(1/2) is A105623 which begins:

1;

1,1;

4,2,1;

26,10,3,1;

226,74,19,4,1;

2426,706,167,31,5,1; ...

compare column 0 of T^(1/2) to column 1 of T;

also, column 1 of T^(1/2) equals column 0 of T.

Matrix inverse square-root T^(-1/2) is A105620 which begins:

1;

-1,1;

-2,-2,1;

-10,-4,-3,1;

-74,-20,-7,-4,1;

-706,-148,-39,-11,-5,1; ...

compare column 0 of T^(-1/2) to column 0 of T.

Matrix inverse T^-1 is A105619 which begins:

1;

-2,1;

-2,-4,1;

-10,-2,-6,1;

-74,-10,-2,-8,1;

-706,-74,-10,-2,-10,1;

-8162,-706,-74,-10,-2,-12,1; ...

PROG

(PARI) {T(n, k) = if(n<k||k<0, 0, if(n==k, 1, 2*(k+1)*T(n, k+1)+sum(j=1, n-k-1, T(j, 0)*T(n, j+k+1))))}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

(PARI) {T(n, k) = if(n<k||k<0, 0, (matrix(n+1, n+1, m, j, if(m>=j, if(m==j, 1, if(m==j+1, -2*j, polcoeff(1/sum(i=0, m-j, (2*i)!/i!/2^i*x^i)+O(x^m), m-j)))))^-1)[n+1, k+1])}

for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A000698 (column 0), A105616 (column 1), A105617 (column 2), A105618 (row sums), A105619 (T^-1), A105620 (T^(-1/2)), A105623 (T^(1/2)), A105627 (T^(3/2)), A105629 (matrix log).

Cf. A107717.

Sequence in context: A202483 A110682 A110327 * A136216 A121334 A126450

Adjacent sequences:  A105612 A105613 A105614 * A105616 A105617 A105618

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Apr 16 2005

STATUS

approved

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Last modified January 20 05:32 EST 2020. Contains 331067 sequences. (Running on oeis4.)