A162170 Matrix inverse of A162169.
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
Examples
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
Programs
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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
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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
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)
Comments