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A162170
Matrix inverse of A162169.
4
1, 1, 1, 1, 0, 1, 2, 0, 3, 1, 5, 0, 6, 0, 1, 16, 0, 20, 0, 5, 1, 61, 0, 75, 0, 15, 0, 1, 272, 0, 336, 0, 70, 0, 7, 1, 1385, 0, 1708, 0, 350, 0, 28, 0, 1, 7936, 0, 9792, 0, 2016, 0, 168, 0, 9, 1, 50521, 0, 62325, 0, 12810, 0, 1050, 0, 45, 0, 1, 353792, 0, 436480, 0, 89760, 0, 7392, 0
OFFSET
1,7
COMMENTS
First column appears to be A000111. Third column is A162171. Row sums minus A000035 appears to be A062272.
The above remarks are correct. - Peter Bala, Sep 08 2021
FORMULA
From Peter Bala, Sep 08 2021: Start:
Assuming an offset of 0: T(2*n+1,2*n+1) = 1 for n >= 0 else otherwise T(n,k) = (1 + (-1)^k)/2*binomial(n,k)*A000111(n-k).
E.g.f.: (sec(x) + tan(x))*cosh(t*x) + sinh(t*x) = 1 + (1 + t)*x + (1 + t^2)*x^2/2! + (2 + 3*t^2 + t^3)*x^3/3! + .... (End)
EXAMPLE
Table begins:
.1
.1...1
.1...0...1
.2...0...3...1
.5...0...6...0...1
16...0..20...0...5...1
61...0..75...0..15...0...1
MAPLE
A000111 := n -> n!*coeff(series(sec(x) + tan(x), x, n+1), x, n):
seq(seq(0^(n-k)*((1 - (-1)^k)*(1/2))*((1 - (-1)^n)*(1/2)) + ((1 + (-1)^k)*(1/2))*binomial(n, k)*A000111(n-k), k = 0..n), n = 0..11); # - Peter Bala, Sep 08 2021
PROG
(PARI) T(n, k) = if (k % 2, binomial(n-1, k-1) * (-1)^floor((n+k-1)/2), if (n==k, 1 , 0));
tabl(nn) = {m = matrix(nn, nn, n, k, if (n>=k, T(n, k), 0)); m = m^(-1); for (n=1, nn, for (k=1, n, print1(m[n, k], ", "); ); print(); ); } \\ Michel Marcus, Jun 17 2015
CROSSREFS
KEYWORD
nonn,tabl,easy
AUTHOR
Mats Granvik, Jun 27 2009
STATUS
approved