A162169 -id:A162169 - cp's OEIS frontend

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

Mats Granvik, Jun 27 2009

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

			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

Cf. A000111, A162171, A062272.

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

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

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)

A178616 Triangle by columns, odd columns of Pascal's triangle A007318, otherwise (1, 0, 0, 0, ...).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 0, 1, 0, 4, 0, 4, 1, 0, 5, 0, 10, 0, 1, 0, 6, 0, 20, 0, 6, 1, 0, 7, 0, 35, 0, 21, 0, 1, 0, 8, 0, 56, 0, 56, 0, 8, 0, 1, 0, 9, 0, 84, 0, 126, 0, 36, 0, 1, 0, 10, 0, 120, 0, 252, 0, 120, 0, 10, 1

Offset: 0

Gary W. Adamson, May 30 2010

Row sums = a variant of A052950, starting (1, 1, 3, 4, 9, 16, 33, ...); whereas A052950 starts (2, 1, 3, 4, 9, ...).

Column 1 of the inverse of A178616 is a signed variant of A065619 prefaced with a 0; where A065619 = (1, 2, 3, 8, 25, 96, 427, ...).

			First few rows of the triangle:
  1,
  0,  1;
  0,  2, 1;
  0,  3, 0,   1
  0,  4, 0,   4, 1;
  0,  5, 0,  10, 0,   1;
  0,  6, 0,  20, 0,   6, 1;
  0,  7, 0,  35, 0,  21, 0,   1;
  0,  8, 0,  56, 0,  56, 0,   8, 1;
  0,  9, 0,  84, 0, 126, 0,  36, 0,  1;
  0, 10, 0, 120, 0, 252, 0, 120, 0, 10, 1;
  0, 11, 0, 165, 0, 462, 0, 330, 0, 55, 0, 1;
  ...

Cf. A162109, A065619, A052950, A095704.

Triangle, odd columns of Pascal's triangle; (1, 0, 0, 0, ...) as even columns k.
Alternatively, (since A178616 + A162169 - Identity matrix) = Pascal's triangle,
we can begin with Pascal's triangle, subtract A162169, then add the Identity
matrix to obtain A178616.

cp's OEIS Frontend

A162170 Matrix inverse of A162169.

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