cp's OEIS Frontend

A162170 Matrix inverse of A162169.

%I A162170 #17 Aug 16 2025 09:36:58
%S A162170 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,
%T A162170 336,0,70,0,7,1,1385,0,1708,0,350,0,28,0,1,7936,0,9792,0,2016,0,168,0,
%U A162170 9,1,50521,0,62325,0,12810,0,1050,0,45,0,1,353792,0,436480,0,89760,0,7392,0
%N A162170 Matrix inverse of A162169.
%C A162170 First column appears to be A000111. Third column is A162171. Row sums minus A000035 appears to be A062272.
%C A162170 The above remarks are correct. - _Peter Bala_, Sep 08 2021
%F A162170 From _Peter Bala_, Sep 08 2021: (Start)
%F A162170 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).
%F A162170 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)
%e A162170 Table begins:
%e A162170    1
%e A162170    1   1
%e A162170    1   0   1
%e A162170    2   0   3   1
%e A162170    5   0   6   0   1
%e A162170   16   0  20   0   5   1
%e A162170   61   0  75   0  15   0   1
%p A162170 A000111 := n -> n!*coeff(series(sec(x) + tan(x), x, n+1), x, n):
%p A162170 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
%o A162170 (PARI) T(n, k) = if (k % 2, binomial(n-1, k-1) * (-1)^floor((n+k-1)/2), if (n==k, 1, 0));
%o A162170 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
%Y A162170 Cf. A000111, A162171, A062272.
%K A162170 nonn,tabl,easy
%O A162170 1,7
%A A162170 _Mats Granvik_, Jun 27 2009